src/HOL/Matrix/LP.thy
 author wenzelm Fri Jan 14 15:44:47 2011 +0100 (2011-01-14) changeset 41550 efa734d9b221 parent 41413 64cd30d6b0b8 permissions -rw-r--r--
eliminated global prems;
tuned proofs;
```     1 (*  Title:      HOL/Matrix/LP.thy
```
```     2     Author:     Steven Obua
```
```     3 *)
```
```     4
```
```     5 theory LP
```
```     6 imports Main "~~/src/HOL/Library/Lattice_Algebras"
```
```     7 begin
```
```     8
```
```     9 lemma le_add_right_mono:
```
```    10   assumes
```
```    11   "a <= b + (c::'a::ordered_ab_group_add)"
```
```    12   "c <= d"
```
```    13   shows "a <= b + d"
```
```    14   apply (rule_tac order_trans[where y = "b+c"])
```
```    15   apply (simp_all add: assms)
```
```    16   done
```
```    17
```
```    18 lemma linprog_dual_estimate:
```
```    19   assumes
```
```    20   "A * x \<le> (b::'a::lattice_ring)"
```
```    21   "0 \<le> y"
```
```    22   "abs (A - A') \<le> \<delta>A"
```
```    23   "b \<le> b'"
```
```    24   "abs (c - c') \<le> \<delta>c"
```
```    25   "abs x \<le> r"
```
```    26   shows
```
```    27   "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"
```
```    28 proof -
```
```    29   from assms have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
```
```    30   from assms have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
```
```    31   have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: algebra_simps)
```
```    32   from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
```
```    33   have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
```
```    34     by (simp only: 4 estimate_by_abs)
```
```    35   have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"
```
```    36     by (simp add: abs_le_mult)
```
```    37   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"
```
```    38     by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
```
```    39   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"
```
```    40     by (simp add: abs_triangle_ineq mult_right_mono)
```
```    41   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"
```
```    42     by (simp add: abs_le_mult mult_right_mono)
```
```    43   have 10: "c'-c = -(c-c')" by (simp add: algebra_simps)
```
```    44   have 11: "abs (c'-c) = abs (c-c')"
```
```    45     by (subst 10, subst abs_minus_cancel, simp)
```
```    46   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"
```
```    47     by (simp add: 11 assms mult_right_mono)
```
```    48   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"
```
```    49     by (simp add: assms mult_right_mono mult_left_mono)
```
```    50   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"
```
```    51     apply (rule mult_left_mono)
```
```    52     apply (simp add: assms)
```
```    53     apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
```
```    54     apply (rule mult_left_mono[of "0" "\<delta>A", simplified])
```
```    55     apply (simp_all)
```
```    56     apply (rule order_trans[where y="abs (A-A')"], simp_all add: assms)
```
```    57     apply (rule order_trans[where y="abs (c-c')"], simp_all add: assms)
```
```    58     done
```
```    59   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"
```
```    60     by (simp)
```
```    61   show ?thesis
```
```    62     apply (rule le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
```
```    63     apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified assms]])
```
```    64     done
```
```    65 qed
```
```    66
```
```    67 lemma le_ge_imp_abs_diff_1:
```
```    68   assumes
```
```    69   "A1 <= (A::'a::lattice_ring)"
```
```    70   "A <= A2"
```
```    71   shows "abs (A-A1) <= A2-A1"
```
```    72 proof -
```
```    73   have "0 <= A - A1"
```
```    74   proof -
```
```    75     have 1: "A - A1 = A + (- A1)" by simp
```
```    76     show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified assms])
```
```    77   qed
```
```    78   then have "abs (A-A1) = A-A1" by (rule abs_of_nonneg)
```
```    79   with assms show "abs (A-A1) <= (A2-A1)" by simp
```
```    80 qed
```
```    81
```
```    82 lemma mult_le_prts:
```
```    83   assumes
```
```    84   "a1 <= (a::'a::lattice_ring)"
```
```    85   "a <= a2"
```
```    86   "b1 <= b"
```
```    87   "b <= b2"
```
```    88   shows
```
```    89   "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
```
```    90 proof -
```
```    91   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
```
```    92     apply (subst prts[symmetric])+
```
```    93     apply simp
```
```    94     done
```
```    95   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
```
```    96     by (simp add: algebra_simps)
```
```    97   moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
```
```    98     by (simp_all add: assms mult_mono)
```
```    99   moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
```
```   100   proof -
```
```   101     have "pprt a * nprt b <= pprt a * nprt b2"
```
```   102       by (simp add: mult_left_mono assms)
```
```   103     moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
```
```   104       by (simp add: mult_right_mono_neg assms)
```
```   105     ultimately show ?thesis
```
```   106       by simp
```
```   107   qed
```
```   108   moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
```
```   109   proof -
```
```   110     have "nprt a * pprt b <= nprt a2 * pprt b"
```
```   111       by (simp add: mult_right_mono assms)
```
```   112     moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
```
```   113       by (simp add: mult_left_mono_neg assms)
```
```   114     ultimately show ?thesis
```
```   115       by simp
```
```   116   qed
```
```   117   moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
```
```   118   proof -
```
```   119     have "nprt a * nprt b <= nprt a * nprt b1"
```
```   120       by (simp add: mult_left_mono_neg assms)
```
```   121     moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
```
```   122       by (simp add: mult_right_mono_neg assms)
```
```   123     ultimately show ?thesis
```
```   124       by simp
```
```   125   qed
```
```   126   ultimately show ?thesis
```
```   127     by - (rule add_mono | simp)+
```
```   128 qed
```
```   129
```
```   130 lemma mult_le_dual_prts:
```
```   131   assumes
```
```   132   "A * x \<le> (b::'a::lattice_ring)"
```
```   133   "0 \<le> y"
```
```   134   "A1 \<le> A"
```
```   135   "A \<le> A2"
```
```   136   "c1 \<le> c"
```
```   137   "c \<le> c2"
```
```   138   "r1 \<le> x"
```
```   139   "x \<le> r2"
```
```   140   shows
```
```   141   "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)"
```
```   142   (is "_ <= _ + ?C")
```
```   143 proof -
```
```   144   from assms have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
```
```   145   moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: algebra_simps)
```
```   146   ultimately have "c * x + (y * A - c) * x <= y * b" by simp
```
```   147   then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
```
```   148   then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: algebra_simps)
```
```   149   have s2: "c - y * A <= c2 - y * A1"
```
```   150     by (simp add: diff_minus assms add_mono mult_left_mono)
```
```   151   have s1: "c1 - y * A2 <= c - y * A"
```
```   152     by (simp add: diff_minus assms add_mono mult_left_mono)
```
```   153   have prts: "(c - y * A) * x <= ?C"
```
```   154     apply (simp add: Let_def)
```
```   155     apply (rule mult_le_prts)
```
```   156     apply (simp_all add: assms s1 s2)
```
```   157     done
```
```   158   then have "y * b + (c - y * A) * x <= y * b + ?C"
```
```   159     by simp
```
```   160   with cx show ?thesis
```
```   161     by(simp only:)
```
```   162 qed
```
```   163
```
`   164 end`