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(* Title: ZF/ex/Bin.ML
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1994 University of Cambridge
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For Bin.thy. Arithmetic on binary integers.
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*)
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open Bin;
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(*Perform induction on l, then prove the major premise using prems. *)
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fun bin_ind_tac a prems i =
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EVERY [res_inst_tac [("x",a)] bin.induct i,
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rename_last_tac a ["1"] (i+3),
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ares_tac prems i];
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(** bin_rec -- by Vset recursion **)
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goal Bin.thy "bin_rec(Plus,a,b,h) = a";
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by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
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by (rewrite_goals_tac bin.con_defs);
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by (simp_tac rank_ss 1);
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val bin_rec_Plus = result();
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goal Bin.thy "bin_rec(Minus,a,b,h) = b";
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by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
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by (rewrite_goals_tac bin.con_defs);
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by (simp_tac rank_ss 1);
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val bin_rec_Minus = result();
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goal Bin.thy "bin_rec(w$$x,a,b,h) = h(w, x, bin_rec(w,a,b,h))";
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by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
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by (rewrite_goals_tac bin.con_defs);
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by (simp_tac rank_ss 1);
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val bin_rec_Bcons = result();
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(*Type checking*)
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val prems = goal Bin.thy
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"[| w: bin; \
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\ a: C(Plus); b: C(Minus); \
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\ !!w x r. [| w: bin; x: bool; r: C(w) |] ==> h(w,x,r): C(w$$x) \
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\ |] ==> bin_rec(w,a,b,h) : C(w)";
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by (bin_ind_tac "w" prems 1);
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by (ALLGOALS
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(asm_simp_tac (ZF_ss addsimps (prems@[bin_rec_Plus,bin_rec_Minus,
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bin_rec_Bcons]))));
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val bin_rec_type = result();
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(** Versions for use with definitions **)
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val [rew] = goal Bin.thy
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"[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Plus) = a";
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by (rewtac rew);
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by (rtac bin_rec_Plus 1);
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val def_bin_rec_Plus = result();
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val [rew] = goal Bin.thy
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"[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Minus) = b";
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by (rewtac rew);
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by (rtac bin_rec_Minus 1);
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val def_bin_rec_Minus = result();
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val [rew] = goal Bin.thy
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"[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(w$$x) = h(w,x,j(w))";
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by (rewtac rew);
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by (rtac bin_rec_Bcons 1);
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val def_bin_rec_Bcons = result();
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fun bin_recs def = map standard
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([def] RL [def_bin_rec_Plus, def_bin_rec_Minus, def_bin_rec_Bcons]);
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(** Type checking **)
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val bin_typechecks0 = bin_rec_type :: bin.intrs;
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goalw Bin.thy [integ_of_bin_def]
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"!!w. w: bin ==> integ_of_bin(w) : integ";
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by (typechk_tac (bin_typechecks0@integ_typechecks@
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nat_typechecks@[bool_into_nat]));
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val integ_of_bin_type = result();
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goalw Bin.thy [bin_succ_def]
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"!!w. w: bin ==> bin_succ(w) : bin";
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by (typechk_tac (bin_typechecks0@bool_typechecks));
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val bin_succ_type = result();
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goalw Bin.thy [bin_pred_def]
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"!!w. w: bin ==> bin_pred(w) : bin";
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by (typechk_tac (bin_typechecks0@bool_typechecks));
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val bin_pred_type = result();
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goalw Bin.thy [bin_minus_def]
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"!!w. w: bin ==> bin_minus(w) : bin";
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by (typechk_tac ([bin_pred_type]@bin_typechecks0@bool_typechecks));
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val bin_minus_type = result();
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goalw Bin.thy [bin_add_def]
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"!!v w. [| v: bin; w: bin |] ==> bin_add(v,w) : bin";
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by (typechk_tac ([bin_succ_type,bin_pred_type]@bin_typechecks0@
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bool_typechecks@ZF_typechecks));
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val bin_add_type = result();
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goalw Bin.thy [bin_mult_def]
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"!!v w. [| v: bin; w: bin |] ==> bin_mult(v,w) : bin";
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by (typechk_tac ([bin_minus_type,bin_add_type]@bin_typechecks0@
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bool_typechecks));
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val bin_mult_type = result();
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val bin_typechecks = bin_typechecks0 @
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[integ_of_bin_type, bin_succ_type, bin_pred_type,
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bin_minus_type, bin_add_type, bin_mult_type];
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val bin_ss = integ_ss
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addsimps([bool_1I, bool_0I,
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bin_rec_Plus, bin_rec_Minus, bin_rec_Bcons] @
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bin_recs integ_of_bin_def @ bool_simps @ bin_typechecks);
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val typechecks = bin_typechecks @ integ_typechecks @ nat_typechecks @
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[bool_subset_nat RS subsetD];
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(**** The carry/borrow functions, bin_succ and bin_pred ****)
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(** Lemmas **)
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goal Integ.thy
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"!!z v. [| z $+ v = z' $+ v'; \
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\ z: integ; z': integ; v: integ; v': integ; w: integ |] \
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\ ==> z $+ (v $+ w) = z' $+ (v' $+ w)";
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by (asm_simp_tac (integ_ss addsimps ([zadd_assoc RS sym])) 1);
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val zadd_assoc_cong = result();
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goal Integ.thy
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"!!z v w. [| z: integ; v: integ; w: integ |] \
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\ ==> z $+ (v $+ w) = v $+ (z $+ w)";
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by (REPEAT (ares_tac [zadd_commute RS zadd_assoc_cong] 1));
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val zadd_assoc_swap = result();
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val zadd_cong =
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read_instantiate_sg (sign_of Integ.thy) [("t","op $+")] subst_context2;
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val zadd_kill = (refl RS zadd_cong);
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val zadd_assoc_swap_kill = zadd_kill RSN (4, zadd_assoc_swap RS trans);
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(*Pushes 'constants' of the form $#m to the right -- LOOPS if two!*)
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val zadd_assoc_znat = standard (znat_type RS zadd_assoc_swap);
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goal Integ.thy
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"!!z w. [| z: integ; w: integ |] \
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\ ==> w $+ (z $+ (w $+ z)) = w $+ (w $+ (z $+ z))";
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by (REPEAT (ares_tac [zadd_kill, zadd_assoc_swap] 1));
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val zadd_swap_pairs = result();
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val carry_ss = bin_ss addsimps
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(bin_recs bin_succ_def @ bin_recs bin_pred_def);
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goal Bin.thy
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"!!w. w: bin ==> integ_of_bin(bin_succ(w)) = $#1 $+ integ_of_bin(w)";
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by (etac bin.induct 1);
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by (simp_tac (carry_ss addsimps [zadd_0_right]) 1);
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by (simp_tac (carry_ss addsimps [zadd_zminus_inverse]) 1);
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by (etac boolE 1);
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by (ALLGOALS (asm_simp_tac (carry_ss addsimps [zadd_assoc])));
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by (REPEAT (ares_tac (zadd_swap_pairs::typechecks) 1));
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val integ_of_bin_succ = result();
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goal Bin.thy
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"!!w. w: bin ==> integ_of_bin(bin_pred(w)) = $~ ($#1) $+ integ_of_bin(w)";
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by (etac bin.induct 1);
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by (simp_tac (carry_ss addsimps [zadd_0_right]) 1);
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by (simp_tac (carry_ss addsimps [zadd_zminus_inverse]) 1);
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by (etac boolE 1);
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by (ALLGOALS
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(asm_simp_tac
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(carry_ss addsimps [zadd_assoc RS sym,
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zadd_zminus_inverse, zadd_zminus_inverse2])));
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by (REPEAT (ares_tac ([zadd_commute, zadd_cong, refl]@typechecks) 1));
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val integ_of_bin_pred = result();
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(*These two results replace the definitions of bin_succ and bin_pred*)
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(*** bin_minus: (unary!) negation of binary integers ***)
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val bin_minus_ss =
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bin_ss addsimps (bin_recs bin_minus_def @
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[integ_of_bin_succ, integ_of_bin_pred]);
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goal Bin.thy
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"!!w. w: bin ==> integ_of_bin(bin_minus(w)) = $~ integ_of_bin(w)";
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by (etac bin.induct 1);
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by (simp_tac (bin_minus_ss addsimps [zminus_0]) 1);
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by (simp_tac (bin_minus_ss addsimps [zadd_0_right]) 1);
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by (etac boolE 1);
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by (ALLGOALS
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(asm_simp_tac (bin_minus_ss addsimps [zminus_zadd_distrib, zadd_assoc])));
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val integ_of_bin_minus = result();
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(*** bin_add: binary addition ***)
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goalw Bin.thy [bin_add_def] "!!w. w: bin ==> bin_add(Plus,w) = w";
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by (asm_simp_tac bin_ss 1);
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val bin_add_Plus = result();
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goalw Bin.thy [bin_add_def] "!!w. w: bin ==> bin_add(Minus,w) = bin_pred(w)";
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by (asm_simp_tac bin_ss 1);
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val bin_add_Minus = result();
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goalw Bin.thy [bin_add_def] "bin_add(v$$x,Plus) = v$$x";
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by (simp_tac bin_ss 1);
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val bin_add_Bcons_Plus = result();
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goalw Bin.thy [bin_add_def] "bin_add(v$$x,Minus) = bin_pred(v$$x)";
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by (simp_tac bin_ss 1);
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val bin_add_Bcons_Minus = result();
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goalw Bin.thy [bin_add_def]
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"!!w y. [| w: bin; y: bool |] ==> \
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\ bin_add(v$$x, w$$y) = \
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\ bin_add(v, cond(x and y, bin_succ(w), w)) $$ (x xor y)";
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by (asm_simp_tac bin_ss 1);
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val bin_add_Bcons_Bcons = result();
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val bin_add_rews = [bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus,
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bin_add_Bcons_Minus, bin_add_Bcons_Bcons,
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integ_of_bin_succ, integ_of_bin_pred];
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val bin_add_ss = bin_ss addsimps ([bool_subset_nat RS subsetD] @ bin_add_rews);
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goal Bin.thy
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"!!v. v: bin ==> \
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\ ALL w: bin. integ_of_bin(bin_add(v,w)) = \
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\ integ_of_bin(v) $+ integ_of_bin(w)";
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by (etac bin.induct 1);
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by (simp_tac bin_add_ss 1);
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by (simp_tac bin_add_ss 1);
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by (rtac ballI 1);
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by (bin_ind_tac "wa" [] 1);
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by (asm_simp_tac (bin_add_ss addsimps [zadd_0_right]) 1);
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by (asm_simp_tac bin_add_ss 1);
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by (REPEAT (ares_tac (zadd_commute::typechecks) 1));
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by (etac boolE 1);
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by (asm_simp_tac (bin_add_ss addsimps [zadd_assoc, zadd_swap_pairs]) 2);
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by (REPEAT (ares_tac ([refl, zadd_kill, zadd_assoc_swap_kill]@typechecks) 2));
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by (etac boolE 1);
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by (ALLGOALS (asm_simp_tac (bin_add_ss addsimps [zadd_assoc,zadd_swap_pairs])));
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by (REPEAT (ares_tac ([refl, zadd_kill, zadd_assoc_swap_kill RS sym]@
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typechecks) 1));
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val integ_of_bin_add_lemma = result();
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val integ_of_bin_add = integ_of_bin_add_lemma RS bspec;
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(*** bin_add: binary multiplication ***)
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val bin_mult_ss =
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bin_ss addsimps (bin_recs bin_mult_def @
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[integ_of_bin_minus, integ_of_bin_add]);
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val major::prems = goal Bin.thy
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"[| v: bin; w: bin |] ==> \
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\ integ_of_bin(bin_mult(v,w)) = \
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\ integ_of_bin(v) $* integ_of_bin(w)";
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by (cut_facts_tac prems 1);
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by (bin_ind_tac "v" [major] 1);
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by (asm_simp_tac (bin_mult_ss addsimps [zmult_0]) 1);
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by (asm_simp_tac (bin_mult_ss addsimps [zmult_1,zmult_zminus]) 1);
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by (etac boolE 1);
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by (asm_simp_tac (bin_mult_ss addsimps [zadd_zmult_distrib]) 2);
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by (asm_simp_tac
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(bin_mult_ss addsimps [zadd_zmult_distrib, zmult_1, zadd_assoc]) 1);
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by (REPEAT (ares_tac ([zadd_commute, zadd_assoc_swap_kill RS sym]@
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typechecks) 1));
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val integ_of_bin_mult = result();
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(**** Computations ****)
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(** extra rules for bin_succ, bin_pred **)
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val [bin_succ_Plus, bin_succ_Minus, _] = bin_recs bin_succ_def;
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val [bin_pred_Plus, bin_pred_Minus, _] = bin_recs bin_pred_def;
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goal Bin.thy "bin_succ(w$$1) = bin_succ(w) $$ 0";
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by (simp_tac carry_ss 1);
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val bin_succ_Bcons1 = result();
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goal Bin.thy "bin_succ(w$$0) = w$$1";
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by (simp_tac carry_ss 1);
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val bin_succ_Bcons0 = result();
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goal Bin.thy "bin_pred(w$$1) = w$$0";
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by (simp_tac carry_ss 1);
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val bin_pred_Bcons1 = result();
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goal Bin.thy "bin_pred(w$$0) = bin_pred(w) $$ 1";
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by (simp_tac carry_ss 1);
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val bin_pred_Bcons0 = result();
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(** extra rules for bin_minus **)
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val [bin_minus_Plus, bin_minus_Minus, _] = bin_recs bin_minus_def;
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goal Bin.thy "bin_minus(w$$1) = bin_pred(bin_minus(w) $$ 0)";
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by (simp_tac bin_minus_ss 1);
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val bin_minus_Bcons1 = result();
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goal Bin.thy "bin_minus(w$$0) = bin_minus(w) $$ 0";
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by (simp_tac bin_minus_ss 1);
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val bin_minus_Bcons0 = result();
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(** extra rules for bin_add **)
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goal Bin.thy
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"!!w. w: bin ==> bin_add(v$$1, w$$1) = bin_add(v, bin_succ(w)) $$ 0";
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by (asm_simp_tac bin_add_ss 1);
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val bin_add_Bcons_Bcons11 = result();
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goal Bin.thy
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"!!w. w: bin ==> bin_add(v$$1, w$$0) = bin_add(v,w) $$ 1";
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by (asm_simp_tac bin_add_ss 1);
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val bin_add_Bcons_Bcons10 = result();
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goal Bin.thy
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"!!w y.[| w: bin; y: bool |] ==> bin_add(v$$0, w$$y) = bin_add(v,w) $$ y";
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by (asm_simp_tac bin_add_ss 1);
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val bin_add_Bcons_Bcons0 = result();
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(** extra rules for bin_mult **)
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val [bin_mult_Plus, bin_mult_Minus, _] = bin_recs bin_mult_def;
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334 |
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goal Bin.thy "bin_mult(v$$1, w) = bin_add(bin_mult(v,w)$$0, w)";
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by (simp_tac bin_mult_ss 1);
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val bin_mult_Bcons1 = result();
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338 |
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goal Bin.thy "bin_mult(v$$0, w) = bin_mult(v,w)$$0";
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by (simp_tac bin_mult_ss 1);
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val bin_mult_Bcons0 = result();
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342 |
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343 |
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344 |
(*** The computation simpset ***)
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val bin_comp_ss = integ_ss
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addsimps [bin_succ_Plus, bin_succ_Minus,
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bin_succ_Bcons1, bin_succ_Bcons0,
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bin_pred_Plus, bin_pred_Minus,
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bin_pred_Bcons1, bin_pred_Bcons0,
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bin_minus_Plus, bin_minus_Minus,
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bin_minus_Bcons1, bin_minus_Bcons0,
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bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus,
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bin_add_Bcons_Minus, bin_add_Bcons_Bcons0,
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bin_add_Bcons_Bcons10, bin_add_Bcons_Bcons11,
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bin_mult_Plus, bin_mult_Minus,
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bin_mult_Bcons1, bin_mult_Bcons0]
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setsolver (type_auto_tac ([bool_1I, bool_0I] @ bin_typechecks0));
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359 |
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360 |
(*** Examples of performing binary arithmetic by simplification ***)
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361 |
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proof_timing := true;
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363 |
(*All runtimes below are on a SPARCserver 10*)
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364 |
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365 |
(* 13+19 = 32 *)
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366 |
goal Bin.thy
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367 |
"bin_add(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = Plus$$1$$0$$0$$0$$0$$0";
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368 |
by (simp_tac bin_comp_ss 1); (*0.6 secs*)
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369 |
result();
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370 |
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371 |
bin_add(binary_of_int 13, binary_of_int 19);
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372 |
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373 |
(* 1234+5678 = 6912 *)
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374 |
goal Bin.thy
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375 |
"bin_add(Plus$$1$$0$$0$$1$$1$$0$$1$$0$$0$$1$$0, \
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376 |
\ Plus$$1$$0$$1$$1$$0$$0$$0$$1$$0$$1$$1$$1$$0) = \
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377 |
\ Plus$$1$$1$$0$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0";
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by (simp_tac bin_comp_ss 1); (*2.6 secs*)
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379 |
result();
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380 |
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381 |
bin_add(binary_of_int 1234, binary_of_int 5678);
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382 |
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383 |
(* 1359-2468 = ~1109 *)
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384 |
goal Bin.thy
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385 |
"bin_add(Plus$$1$$0$$1$$0$$1$$0$$0$$1$$1$$1$$1, \
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386 |
\ Minus$$0$$1$$1$$0$$0$$1$$0$$1$$1$$1$$0$$0) = \
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387 |
\ Minus$$1$$0$$1$$1$$1$$0$$1$$0$$1$$0$$1$$1";
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388 |
by (simp_tac bin_comp_ss 1); (*2.3 secs*)
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389 |
result();
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390 |
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391 |
bin_add(binary_of_int 1359, binary_of_int ~2468);
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392 |
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393 |
(* 93746-46375 = 47371 *)
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394 |
goal Bin.thy
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395 |
"bin_add(Plus$$1$$0$$1$$1$$0$$1$$1$$1$$0$$0$$0$$1$$1$$0$$0$$1$$0, \
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396 |
\ Minus$$0$$1$$0$$0$$1$$0$$1$$0$$1$$1$$0$$1$$1$$0$$0$$1) = \
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397 |
\ Plus$$0$$1$$0$$1$$1$$1$$0$$0$$1$$0$$0$$0$$0$$1$$0$$1$$1";
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398 |
by (simp_tac bin_comp_ss 1); (*3.9 secs*)
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399 |
result();
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400 |
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401 |
bin_add(binary_of_int 93746, binary_of_int ~46375);
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402 |
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|
403 |
(* negation of 65745 *)
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|
404 |
goal Bin.thy
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|
405 |
"bin_minus(Plus$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1$$1$$0$$1$$0$$0$$0$$1) = \
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|
406 |
\ Minus$$0$$1$$1$$1$$1$$1$$1$$1$$1$$0$$0$$1$$0$$1$$1$$1$$1";
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|
407 |
by (simp_tac bin_comp_ss 1); (*0.6 secs*)
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|
408 |
result();
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|
409 |
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|
410 |
bin_minus(binary_of_int 65745);
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|
411 |
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|
412 |
(* negation of ~54321 *)
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|
413 |
goal Bin.thy
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|
414 |
"bin_minus(Minus$$0$$0$$1$$0$$1$$0$$1$$1$$1$$1$$0$$0$$1$$1$$1$$1) = \
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|
415 |
\ Plus$$0$$1$$1$$0$$1$$0$$1$$0$$0$$0$$0$$1$$1$$0$$0$$0$$1";
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|
416 |
by (simp_tac bin_comp_ss 1); (*0.7 secs*)
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|
417 |
result();
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418 |
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|
419 |
bin_minus(binary_of_int ~54321);
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|
420 |
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|
421 |
(* 13*19 = 247 *)
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|
422 |
goal Bin.thy "bin_mult(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = \
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|
423 |
\ Plus$$1$$1$$1$$1$$0$$1$$1$$1";
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|
424 |
by (simp_tac bin_comp_ss 1); (*1.5 secs*)
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|
425 |
result();
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|
426 |
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|
427 |
bin_mult(binary_of_int 13, binary_of_int 19);
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|
428 |
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|
429 |
(* ~84 * 51 = ~4284 *)
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|
430 |
goal Bin.thy
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|
431 |
"bin_mult(Minus$$0$$1$$0$$1$$1$$0$$0, Plus$$1$$1$$0$$0$$1$$1) = \
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|
432 |
\ Minus$$0$$1$$1$$1$$1$$0$$1$$0$$0$$0$$1$$0$$0";
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|
433 |
by (simp_tac bin_comp_ss 1); (*2.6 secs*)
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|
434 |
result();
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|
435 |
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|
436 |
bin_mult(binary_of_int ~84, binary_of_int 51);
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|
437 |
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|
438 |
(* 255*255 = 65025; the worst case for 8-bit operands *)
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|
439 |
goal Bin.thy
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|
440 |
"bin_mult(Plus$$1$$1$$1$$1$$1$$1$$1$$1, \
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|
441 |
\ Plus$$1$$1$$1$$1$$1$$1$$1$$1) = \
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|
442 |
\ Plus$$1$$1$$1$$1$$1$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1";
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|
443 |
by (simp_tac bin_comp_ss 1); (*9.8 secs*)
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|
444 |
result();
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|
445 |
|
|
446 |
bin_mult(binary_of_int 255, binary_of_int 255);
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|
447 |
|
|
448 |
(* 1359 * ~2468 = ~3354012 *)
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|
449 |
goal Bin.thy
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|
450 |
"bin_mult(Plus$$1$$0$$1$$0$$1$$0$$0$$1$$1$$1$$1, \
|
|
451 |
\ Minus$$0$$1$$1$$0$$0$$1$$0$$1$$1$$1$$0$$0) = \
|
|
452 |
\ Minus$$0$$0$$1$$1$$0$$0$$1$$1$$0$$1$$0$$0$$1$$0$$0$$1$$1$$0$$0$$1$$0$$0";
|
|
453 |
by (simp_tac bin_comp_ss 1); (*13.7 secs*)
|
|
454 |
result();
|
|
455 |
|
|
456 |
bin_mult(binary_of_int 1359, binary_of_int ~2468);
|