author | paulson |
Thu, 24 Sep 1998 17:07:24 +0200 | |
changeset 5551 | ed5e19bc7e32 |
parent 5540 | 0f16c3b66ab4 |
child 5562 | 02261e6880d1 |
permissions | -rw-r--r-- |
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(* Title: HOL/Integ/Bin.ML |
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Authors: Lawrence C Paulson, Cambridge University Computer Laboratory |
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David Spelt, University of Twente |
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Copyright 1994 University of Cambridge |
5 |
Copyright 1996 University of Twente |
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7 |
Arithmetic on binary integers. |
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8 |
*) |
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9 |
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(** extra rules for bin_succ, bin_pred, bin_add, bin_mult **) |
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qed_goal "NCons_Pls_0" Bin.thy |
13 |
"NCons Pls False = Pls" |
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(fn _ => [(Simp_tac 1)]); |
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5512 | 16 |
qed_goal "NCons_Pls_1" Bin.thy |
17 |
"NCons Pls True = Pls BIT True" |
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(fn _ => [(Simp_tac 1)]); |
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5512 | 20 |
qed_goal "NCons_Min_0" Bin.thy |
21 |
"NCons Min False = Min BIT False" |
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(fn _ => [(Simp_tac 1)]); |
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5512 | 24 |
qed_goal "NCons_Min_1" Bin.thy |
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"NCons Min True = Min" |
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(fn _ => [(Simp_tac 1)]); |
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5512 | 28 |
qed_goal "bin_succ_1" Bin.thy |
29 |
"bin_succ(w BIT True) = (bin_succ w) BIT False" |
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(fn _ => [(Simp_tac 1)]); |
1632 | 31 |
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5512 | 32 |
qed_goal "bin_succ_0" Bin.thy |
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"bin_succ(w BIT False) = NCons w True" |
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(fn _ => [(Simp_tac 1)]); |
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5512 | 36 |
qed_goal "bin_pred_1" Bin.thy |
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"bin_pred(w BIT True) = NCons w False" |
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(fn _ => [(Simp_tac 1)]); |
1632 | 39 |
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5512 | 40 |
qed_goal "bin_pred_0" Bin.thy |
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"bin_pred(w BIT False) = (bin_pred w) BIT True" |
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(fn _ => [(Simp_tac 1)]); |
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qed_goal "bin_minus_1" Bin.thy |
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"bin_minus(w BIT True) = bin_pred (NCons (bin_minus w) False)" |
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(fn _ => [(Simp_tac 1)]); |
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5512 | 48 |
qed_goal "bin_minus_0" Bin.thy |
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"bin_minus(w BIT False) = (bin_minus w) BIT False" |
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(fn _ => [(Simp_tac 1)]); |
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5491 | 52 |
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1632 | 53 |
(*** bin_add: binary addition ***) |
54 |
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5512 | 55 |
qed_goal "bin_add_BIT_11" Bin.thy |
56 |
"bin_add (v BIT True) (w BIT True) = \ |
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57 |
\ NCons (bin_add v (bin_succ w)) False" |
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(fn _ => [(Simp_tac 1)]); |
1632 | 59 |
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5512 | 60 |
qed_goal "bin_add_BIT_10" Bin.thy |
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"bin_add (v BIT True) (w BIT False) = NCons (bin_add v w) True" |
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(fn _ => [(Simp_tac 1)]); |
1632 | 63 |
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5512 | 64 |
qed_goal "bin_add_BIT_0" Bin.thy |
65 |
"bin_add (v BIT False) (w BIT y) = NCons (bin_add v w) y" |
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5491 | 66 |
(fn _ => [Auto_tac]); |
1632 | 67 |
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5551 | 68 |
Goal "bin_add w Pls = w"; |
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by (induct_tac "w" 1); |
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by Auto_tac; |
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qed "bin_add_Pls_right"; |
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1632 | 72 |
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5512 | 73 |
qed_goal "bin_add_BIT_Min" Bin.thy |
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"bin_add (v BIT x) Min = bin_pred (v BIT x)" |
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(fn _ => [(Simp_tac 1)]); |
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5512 | 77 |
qed_goal "bin_add_BIT_BIT" Bin.thy |
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"bin_add (v BIT x) (w BIT y) = \ |
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\ NCons(bin_add v (if x & y then (bin_succ w) else w)) (x~= y)" |
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(fn _ => [(Simp_tac 1)]); |
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82 |
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(*** bin_add: binary multiplication ***) |
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84 |
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qed_goal "bin_mult_1" Bin.thy |
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"bin_mult (v BIT True) w = bin_add (NCons (bin_mult v w) False) w" |
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(fn _ => [(Simp_tac 1)]); |
1632 | 88 |
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5512 | 89 |
qed_goal "bin_mult_0" Bin.thy |
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"bin_mult (v BIT False) w = NCons (bin_mult v w) False" |
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(fn _ => [(Simp_tac 1)]); |
1632 | 92 |
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93 |
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(**** The carry/borrow functions, bin_succ and bin_pred ****) |
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(**** integ_of ****) |
1632 | 98 |
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5512 | 99 |
qed_goal "integ_of_NCons" Bin.thy |
100 |
"integ_of(NCons w b) = integ_of(w BIT b)" |
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5184 | 101 |
(fn _ =>[(induct_tac "w" 1), |
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(ALLGOALS Asm_simp_tac) ]); |
1632 | 103 |
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5512 | 104 |
Addsimps [integ_of_NCons]; |
1632 | 105 |
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qed_goal "integ_of_succ" Bin.thy |
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"integ_of(bin_succ w) = $#1 + integ_of w" |
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(fn _ =>[(rtac bin.induct 1), |
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(ALLGOALS(asm_simp_tac (simpset() addsimps zadd_ac))) ]); |
5491 | 110 |
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qed_goal "integ_of_pred" Bin.thy |
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"integ_of(bin_pred w) = - ($#1) + integ_of w" |
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(fn _ =>[(rtac bin.induct 1), |
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(ALLGOALS(asm_simp_tac (simpset() addsimps zadd_ac))) ]); |
1632 | 115 |
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5491 | 116 |
Goal "integ_of(bin_minus w) = - (integ_of w)"; |
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by (rtac bin.induct 1); |
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by (Simp_tac 1); |
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by (Simp_tac 1); |
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by (asm_simp_tac (simpset() |
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delsimps [bin_pred_Pls, bin_pred_Min, bin_pred_BIT] |
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addsimps [integ_of_succ,integ_of_pred, |
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zadd_assoc]) 1); |
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qed "integ_of_minus"; |
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val bin_add_simps = [bin_add_BIT_BIT, |
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integ_of_succ, integ_of_pred]; |
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Goal "! w. integ_of(bin_add v w) = integ_of v + integ_of w"; |
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by (induct_tac "v" 1); |
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by (simp_tac (simpset() addsimps bin_add_simps) 1); |
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by (simp_tac (simpset() addsimps bin_add_simps) 1); |
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by (rtac allI 1); |
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by (induct_tac "w" 1); |
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by (ALLGOALS (asm_simp_tac (simpset() addsimps bin_add_simps @ zadd_ac))); |
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qed_spec_mp "integ_of_add"; |
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val bin_mult_simps = [zmult_zminus, integ_of_minus, integ_of_add]; |
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Goal "integ_of(bin_mult v w) = integ_of v * integ_of w"; |
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by (induct_tac "v" 1); |
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by (simp_tac (simpset() addsimps bin_mult_simps) 1); |
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by (simp_tac (simpset() addsimps bin_mult_simps) 1); |
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by (asm_simp_tac |
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(simpset() addsimps bin_mult_simps @ [zadd_zmult_distrib] @ zadd_ac) 1); |
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qed "integ_of_mult"; |
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(** Simplification rules with integer constants **) |
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Goal "#0 + z = z"; |
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by (Simp_tac 1); |
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qed "zadd_0"; |
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Goal "z + #0 = z"; |
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by (Simp_tac 1); |
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qed "zadd_0_right"; |
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Goal "z + (- z) = #0"; |
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by (Simp_tac 1); |
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qed "zadd_zminus_inverse"; |
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Goal "(- z) + z = #0"; |
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by (Simp_tac 1); |
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qed "zadd_zminus_inverse2"; |
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(*These rewrite to $# 0. Henceforth we should rewrite to #0 *) |
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Delsimps [zadd_zminus_inverse_nat, zadd_zminus_inverse_nat2]; |
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Addsimps [zadd_0, zadd_0_right, zadd_zminus_inverse, zadd_zminus_inverse2]; |
173 |
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Goal "- (#0) = #0"; |
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by (Simp_tac 1); |
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qed "zminus_0"; |
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177 |
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Addsimps [zminus_0]; |
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179 |
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Goal "#0 * z = #0"; |
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by (Simp_tac 1); |
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qed "zmult_0"; |
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183 |
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Goal "#1 * z = z"; |
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by (Simp_tac 1); |
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qed "zmult_1"; |
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187 |
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Goal "#2 * z = z+z"; |
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by (simp_tac (simpset() addsimps [zadd_zmult_distrib]) 1); |
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qed "zmult_2"; |
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191 |
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Goal "z * #0 = #0"; |
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by (Simp_tac 1); |
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qed "zmult_0_right"; |
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195 |
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196 |
Goal "z * #1 = z"; |
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197 |
by (Simp_tac 1); |
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qed "zmult_1_right"; |
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Goal "z * #2 = z+z"; |
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by (simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1); |
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qed "zmult_2_right"; |
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Addsimps [zmult_0, zmult_0_right, |
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zmult_1, zmult_1_right, |
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zmult_2, zmult_2_right]; |
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Goal "(w < z + #1) = (w<z | w=z)"; |
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by (simp_tac (simpset() addsimps [zless_add_nat1_eq]) 1); |
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qed "zless_add1_eq"; |
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Goal "(w + #1 <= z) = (w<z)"; |
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by (simp_tac (simpset() addsimps [add_nat1_zle_eq]) 1); |
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qed "add1_zle_eq"; |
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Addsimps [add1_zle_eq]; |
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216 |
||
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Goal "neg x = (x < #0)"; |
218 |
by (simp_tac (simpset() addsimps [neg_eq_less_nat0]) 1); |
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qed "neg_eq_less_0"; |
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Goal "(~neg x) = ($# 0 <= x)"; |
222 |
by (simp_tac (simpset() addsimps [not_neg_eq_ge_nat0]) 1); |
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qed "not_neg_eq_ge_0"; |
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(** Simplification rules for comparison of binary numbers (Norbert Voelker) **) |
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228 |
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229 |
(** Equals (=) **) |
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1632 | 230 |
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5491 | 231 |
Goalw [iszero_def] |
232 |
"(integ_of x = integ_of y) \ |
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\ = iszero(integ_of (bin_add x (bin_minus y)))"; |
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by (simp_tac (simpset() addsimps |
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235 |
(zcompare_rls @ [integ_of_add, integ_of_minus])) 1); |
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236 |
qed "eq_integ_of_eq"; |
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237 |
||
5512 | 238 |
Goalw [iszero_def] "iszero (integ_of Pls)"; |
5491 | 239 |
by (Simp_tac 1); |
5512 | 240 |
qed "iszero_integ_of_Pls"; |
5491 | 241 |
|
5512 | 242 |
Goalw [iszero_def] "~ iszero(integ_of Min)"; |
5491 | 243 |
by (Simp_tac 1); |
5512 | 244 |
qed "nonzero_integ_of_Min"; |
5491 | 245 |
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246 |
Goalw [iszero_def] |
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5512 | 247 |
"iszero (integ_of (w BIT x)) = (~x & iszero (integ_of w))"; |
5491 | 248 |
by (Simp_tac 1); |
249 |
by (int_case_tac "integ_of w" 1); |
|
250 |
by (ALLGOALS (asm_simp_tac |
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5540 | 251 |
(simpset() addsimps zcompare_rls @ |
252 |
[zminus_zadd_distrib RS sym, |
|
253 |
add_nat]))); |
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5512 | 254 |
qed "iszero_integ_of_BIT"; |
5491 | 255 |
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256 |
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257 |
(** Less-than (<) **) |
|
258 |
||
259 |
Goalw [zless_def,zdiff_def] |
|
260 |
"integ_of x < integ_of y \ |
|
5540 | 261 |
\ = neg (integ_of (bin_add x (bin_minus y)))"; |
5491 | 262 |
by (simp_tac (simpset() addsimps bin_mult_simps) 1); |
5540 | 263 |
qed "less_integ_of_eq_neg"; |
5491 | 264 |
|
5540 | 265 |
Goal "~ neg (integ_of Pls)"; |
5491 | 266 |
by (Simp_tac 1); |
5512 | 267 |
qed "not_neg_integ_of_Pls"; |
5491 | 268 |
|
5540 | 269 |
Goal "neg (integ_of Min)"; |
5491 | 270 |
by (Simp_tac 1); |
5512 | 271 |
qed "neg_integ_of_Min"; |
5491 | 272 |
|
5540 | 273 |
Goal "neg (integ_of (w BIT x)) = neg (integ_of w)"; |
5491 | 274 |
by (Asm_simp_tac 1); |
275 |
by (int_case_tac "integ_of w" 1); |
|
276 |
by (ALLGOALS (asm_simp_tac |
|
5540 | 277 |
(simpset() addsimps [add_nat, neg_eq_less_nat0, |
278 |
symmetric zdiff_def] @ zcompare_rls))); |
|
5512 | 279 |
qed "neg_integ_of_BIT"; |
5491 | 280 |
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281 |
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282 |
(** Less-than-or-equals (<=) **) |
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283 |
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284 |
Goal "(integ_of x <= integ_of y) = (~ integ_of y < integ_of x)"; |
|
285 |
by (simp_tac (simpset() addsimps [zle_def]) 1); |
|
286 |
qed "le_integ_of_eq_not_less"; |
|
287 |
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5540 | 288 |
(*Delete the original rewrites, with their clumsy conditional expressions*) |
5551 | 289 |
Delsimps [bin_succ_BIT, bin_pred_BIT, bin_minus_BIT, |
290 |
NCons_Pls, NCons_Min, bin_add_BIT, bin_mult_BIT]; |
|
5491 | 291 |
|
292 |
(*Hide the binary representation of integer constants*) |
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5540 | 293 |
Delsimps [integ_of_Pls, integ_of_Min, integ_of_BIT]; |
5491 | 294 |
|
295 |
(*Add simplification of arithmetic operations on integer constants*) |
|
296 |
Addsimps [integ_of_add RS sym, |
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297 |
integ_of_minus RS sym, |
|
298 |
integ_of_mult RS sym, |
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5512 | 299 |
bin_succ_1, bin_succ_0, |
300 |
bin_pred_1, bin_pred_0, |
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301 |
bin_minus_1, bin_minus_0, |
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5551 | 302 |
bin_add_Pls_right, bin_add_BIT_Min, |
5512 | 303 |
bin_add_BIT_0, bin_add_BIT_10, bin_add_BIT_11, |
304 |
bin_mult_1, bin_mult_0, |
|
305 |
NCons_Pls_0, NCons_Pls_1, |
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306 |
NCons_Min_0, NCons_Min_1, |
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307 |
NCons_BIT]; |
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5491 | 309 |
(*... and simplification of relational operations*) |
5512 | 310 |
Addsimps [eq_integ_of_eq, iszero_integ_of_Pls, nonzero_integ_of_Min, |
311 |
iszero_integ_of_BIT, |
|
5540 | 312 |
less_integ_of_eq_neg, |
5512 | 313 |
not_neg_integ_of_Pls, neg_integ_of_Min, neg_integ_of_BIT, |
5491 | 314 |
le_integ_of_eq_not_less]; |
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315 |
|
5491 | 316 |
Goalw [zdiff_def] |
317 |
"integ_of v - integ_of w = integ_of(bin_add v (bin_minus w))"; |
|
318 |
by (Simp_tac 1); |
|
319 |
qed "diff_integ_of_eq"; |
|
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320 |
|
5491 | 321 |
(*... and finally subtraction*) |
322 |
Addsimps [diff_integ_of_eq]; |
|
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|
323 |
|
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|
324 |
|
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|
325 |
(** Simplification of inequalities involving numerical constants **) |
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|
326 |
|
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327 |
Goal "(w <= z + #1) = (w<=z | w = z + #1)"; |
5540 | 328 |
by (simp_tac (simpset() addsimps [integ_le_less, zless_add1_eq]) 1); |
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|
329 |
qed "zle_add1_eq"; |
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|
330 |
|
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331 |
Goal "(w <= z - #1) = (w<z)"; |
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|
332 |
by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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|
333 |
qed "zle_diff1_eq"; |
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|
334 |
Addsimps [zle_diff1_eq]; |
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|
335 |
|
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336 |
(*2nd premise can be proved automatically if v is a literal*) |
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337 |
Goal "[| w <= z; #0 <= v |] ==> w <= z + v"; |
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|
338 |
by (dtac zadd_zle_mono 1); |
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|
339 |
by (assume_tac 1); |
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|
340 |
by (Full_simp_tac 1); |
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|
341 |
qed "zle_imp_zle_zadd"; |
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|
342 |
|
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|
343 |
Goal "w <= z ==> w <= z + #1"; |
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|
344 |
by (asm_simp_tac (simpset() addsimps [zle_imp_zle_zadd]) 1); |
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345 |
qed "zle_imp_zle_zadd1"; |
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|
346 |
|
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347 |
(*2nd premise can be proved automatically if v is a literal*) |
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|
348 |
Goal "[| w < z; #0 <= v |] ==> w < z + v"; |
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|
349 |
by (dtac zadd_zless_mono 1); |
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|
350 |
by (assume_tac 1); |
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|
351 |
by (Full_simp_tac 1); |
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|
352 |
qed "zless_imp_zless_zadd"; |
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|
353 |
|
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|
354 |
Goal "w < z ==> w < z + #1"; |
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|
355 |
by (asm_simp_tac (simpset() addsimps [zless_imp_zless_zadd]) 1); |
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|
356 |
qed "zless_imp_zless_zadd1"; |
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|
357 |
|
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|
358 |
Goal "(w < z + #1) = (w<=z)"; |
5540 | 359 |
by (simp_tac (simpset() addsimps [zless_add1_eq, integ_le_less]) 1); |
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|
360 |
qed "zle_add1_eq_le"; |
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|
361 |
Addsimps [zle_add1_eq_le]; |
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|
362 |
|
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|
363 |
Goal "(z = z + w) = (w = #0)"; |
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|
364 |
by (rtac trans 1); |
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|
365 |
by (rtac zadd_left_cancel 2); |
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|
366 |
by (simp_tac (simpset() addsimps [eq_sym_conv]) 1); |
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|
367 |
qed "zadd_left_cancel0"; |
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|
368 |
Addsimps [zadd_left_cancel0]; |
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|
369 |
|
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|
370 |
(*LOOPS as a simprule!*) |
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|
371 |
Goal "[| w + v < z; #0 <= v |] ==> w < z"; |
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|
372 |
by (dtac zadd_zless_mono 1); |
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|
373 |
by (assume_tac 1); |
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|
374 |
by (full_simp_tac (simpset() addsimps zadd_ac) 1); |
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|
375 |
qed "zless_zadd_imp_zless"; |
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|
376 |
|
5540 | 377 |
(*LOOPS as a simprule! Analogous to Suc_lessD*) |
5510
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|
378 |
Goal "w + #1 < z ==> w < z"; |
5540 | 379 |
by (dtac zless_zadd_imp_zless 1); |
380 |
by (assume_tac 2); |
|
5510
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|
381 |
by (Simp_tac 1); |
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|
382 |
qed "zless_zadd1_imp_zless"; |
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|
383 |
|
5551 | 384 |
Goal "w + #-1 = w - #1"; |
385 |
by (simp_tac (simpset() addsimps zadd_ac@zcompare_0_rls) 1); |
|
386 |
qed "zplus_minus1_conv"; |
|
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|
387 |
|
5551 | 388 |
(*Eliminates neg from the subgoal, introduced e.g. by zcompare_0_rls*) |
389 |
val no_neg_ss = |
|
390 |
simpset() |
|
391 |
delsimps [less_integ_of_eq_neg] (*loops: it introduces neg*) |
|
392 |
addsimps [zadd_assoc RS sym, zplus_minus1_conv, |
|
393 |
neg_eq_less_0, iszero_def] @ zcompare_rls; |
|
394 |
||
395 |
||
396 |
(*** nat_of ***) |
|
397 |
||
398 |
Goal "#0 <= z ==> $# (nat_of z) = z"; |
|
399 |
by (asm_full_simp_tac |
|
400 |
(simpset() addsimps [neg_eq_less_0, zle_def, not_neg_nat_of]) 1); |
|
401 |
qed "nat_of_0_le"; |
|
402 |
||
403 |
Goal "z < #0 ==> nat_of z = 0"; |
|
404 |
by (asm_full_simp_tac |
|
405 |
(simpset() addsimps [neg_eq_less_0, zle_def, neg_nat_of]) 1); |
|
406 |
qed "nat_of_less_0"; |
|
407 |
||
408 |
Addsimps [nat_of_0_le, nat_of_less_0]; |
|
409 |
||
410 |
Goal "#0 <= w ==> (nat_of w = m) = (w = $# m)"; |
|
411 |
by Auto_tac; |
|
412 |
qed "nat_of_eq_iff"; |
|
413 |
||
414 |
Goal "#0 <= w ==> (nat_of w < m) = (w < $# m)"; |
|
415 |
by (rtac iffI 1); |
|
416 |
by (asm_full_simp_tac |
|
417 |
(simpset() delsimps [zless_eq_less] addsimps [zless_eq_less RS sym]) 2); |
|
418 |
by (etac (nat_of_0_le RS subst) 1); |
|
419 |
by (Simp_tac 1); |
|
420 |
qed "nat_of_less_iff"; |
|
421 |
||
422 |
Goal "#0 <= w ==> (nat_of w < nat_of z) = (w<z)"; |
|
423 |
by (case_tac "neg z" 1); |
|
424 |
by (auto_tac (claset(), simpset() addsimps [nat_of_less_iff])); |
|
425 |
by (auto_tac (claset() addIs [zless_trans], |
|
426 |
simpset() addsimps [neg_eq_less_0, integ_of_Pls, zle_def])); |
|
427 |
qed "nat_of_less_eq_zless"; |