(* Title: HOL/ex/BinEx.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Examples of performing binary arithmetic by simplification
Also a proof that binary arithmetic on normalized operands
yields normalized results.
*)
set proof_timing;
(** Addition **)
Goal "#13 + #19 = #32";
by (Simp_tac 1);
result();
Goal "#1234 + #5678 = #6912";
by (Simp_tac 1);
result();
Goal "#1359 + #-2468 = #-1109";
by (Simp_tac 1);
result();
Goal "#93746 + #-46375 = #47371";
by (Simp_tac 1);
result();
(** Negation **)
Goal "- #65745 = #-65745";
by (Simp_tac 1);
result();
Goal "- #-54321 = #54321";
by (Simp_tac 1);
result();
(** Multiplication **)
Goal "#13 * #19 = #247";
by (Simp_tac 1);
result();
Goal "#-84 * #51 = #-4284";
by (Simp_tac 1);
result();
Goal "#255 * #255 = #65025";
by (Simp_tac 1);
result();
Goal "#1359 * #-2468 = #-3354012";
by (Simp_tac 1);
result();
Goal "#89 * #10 ~= #889";
by (Simp_tac 1);
result();
Goal "#13 < #18 - #4";
by (Simp_tac 1);
result();
Goal "#-345 < #-242 + #-100";
by (Simp_tac 1);
result();
Goal "#13557456 < #18678654";
by (Simp_tac 1);
result();
Goal "#999999 <= (#1000001 + #1)-#2";
by (Simp_tac 1);
result();
Goal "#1234567 <= #1234567";
by (Simp_tac 1);
result();
(** Testing the cancellation of complementary terms **)
Goal "y + (x + -x) = #0 + y";
by (Simp_tac 1);
result();
Goal "y + (-x + (- y + x)) = #0";
by (Simp_tac 1);
result();
Goal "-x + (y + (- y + x)) = #0";
by (Simp_tac 1);
result();
Goal "x + (x + (- x + (- x + (- y + - z)))) = #0 - y - z";
by (Simp_tac 1);
result();
Goal "x + x - x - x - y - z = #0 - y - z";
by (Simp_tac 1);
result();
Goal "x + y + z - (x + z) = y - #0";
by (Simp_tac 1);
result();
Goal "x+(y+(y+(y+ (-x + -x)))) = #0 + y - x + y + y";
by (Simp_tac 1);
result();
Goal "x+(y+(y+(y+ (-y + -x)))) = y + #0 + y";
by (Simp_tac 1);
result();
Goal "x + y - x + z - x - y - z + x < #1";
by (Simp_tac 1);
result();
Addsimps normal.intrs;
Goal "(w BIT b): normal ==> (w BIT b BIT c): normal";
by (case_tac "c" 1);
by Auto_tac;
qed "normal_BIT_I";
Addsimps [normal_BIT_I];
Goal "w BIT b: normal ==> w: normal";
by (etac normal.elim 1);
by Auto_tac;
qed "normal_BIT_D";
Goal "w : normal --> NCons w b : normal";
by (induct_tac "w" 1);
by (auto_tac (claset(), simpset() addsimps [NCons_Pls, NCons_Min]));
qed_spec_mp "NCons_normal";
Addsimps [NCons_normal];
Goal "NCons w True ~= Pls";
by (induct_tac "w" 1);
by Auto_tac;
qed "NCons_True";
Goal "NCons w False ~= Min";
by (induct_tac "w" 1);
by Auto_tac;
qed "NCons_False";
Goal "w: normal ==> bin_succ w : normal";
by (etac normal.induct 1);
by (exhaust_tac "w" 4);
by (auto_tac (claset(), simpset() addsimps [NCons_True, bin_succ_BIT]));
qed "bin_succ_normal";
Goal "w: normal ==> bin_pred w : normal";
by (etac normal.induct 1);
by (exhaust_tac "w" 3);
by (auto_tac (claset(), simpset() addsimps [NCons_False, bin_pred_BIT]));
qed "bin_pred_normal";
Addsimps [bin_succ_normal, bin_pred_normal];
Goal "w : normal --> (ALL z. z: normal --> bin_add w z : normal)";
by (induct_tac "w" 1);
by (Simp_tac 1);
by (Simp_tac 1);
by (rtac impI 1);
by (rtac allI 1);
by (induct_tac "z" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [bin_add_BIT])));
by (safe_tac (claset() addSDs [normal_BIT_D]));
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "bin_add_normal";
Goal "w: normal ==> (w = Pls) = (integ_of w = #0)";
by (etac normal.induct 1);
by Auto_tac;
qed "normal_Pls_eq_0";
Goal "w : normal ==> bin_minus w : normal";
by (etac normal.induct 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [bin_minus_BIT])));
by (resolve_tac normal.intrs 1);
by (assume_tac 1);
by (asm_full_simp_tac (simpset() addsimps [normal_Pls_eq_0]) 1);
by (asm_full_simp_tac
(simpset_of Int.thy addsimps [integ_of_minus, iszero_def,
zminus_exchange]) 1);
qed "bin_minus_normal";
Goal "w : normal ==> z: normal --> bin_mult w z : normal";
by (etac normal.induct 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [bin_minus_normal, bin_mult_BIT])));
by (safe_tac (claset() addSDs [normal_BIT_D]));
by (asm_simp_tac (simpset() addsimps [bin_add_normal]) 1);
qed_spec_mp "bin_mult_normal";