(* Title: ZF/ex/Bin.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
For Bin.thy. Arithmetic on binary integers.
*)
open Bin;
Addsimps bin.case_eqns;
(*Perform induction on l, then prove the major premise using prems. *)
fun bin_ind_tac a prems i =
EVERY [res_inst_tac [("x",a)] bin.induct i,
rename_last_tac a ["1"] (i+3),
ares_tac prems i];
(** bin_rec -- by Vset recursion **)
Goal "bin_rec(Plus,a,b,h) = a";
by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
by (rewrite_goals_tac bin.con_defs);
by (simp_tac rank_ss 1);
qed "bin_rec_Plus";
Goal "bin_rec(Minus,a,b,h) = b";
by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
by (rewrite_goals_tac bin.con_defs);
by (simp_tac rank_ss 1);
qed "bin_rec_Minus";
Goal "bin_rec(Bcons(w,x),a,b,h) = h(w, x, bin_rec(w,a,b,h))";
by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
by (rewrite_goals_tac bin.con_defs);
by (simp_tac rank_ss 1);
qed "bin_rec_Bcons";
(*Type checking*)
val prems = goal Bin.thy
"[| w: bin; \
\ a: C(Plus); b: C(Minus); \
\ !!w x r. [| w: bin; x: bool; r: C(w) |] ==> h(w,x,r): C(Bcons(w,x)) \
\ |] ==> bin_rec(w,a,b,h) : C(w)";
by (bin_ind_tac "w" prems 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps (prems@[bin_rec_Plus, bin_rec_Minus,
bin_rec_Bcons]))));
qed "bin_rec_type";
(** Versions for use with definitions **)
val [rew] = goal Bin.thy
"[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Plus) = a";
by (rewtac rew);
by (rtac bin_rec_Plus 1);
qed "def_bin_rec_Plus";
val [rew] = goal Bin.thy
"[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Minus) = b";
by (rewtac rew);
by (rtac bin_rec_Minus 1);
qed "def_bin_rec_Minus";
val [rew] = goal Bin.thy
"[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Bcons(w,x)) = h(w,x,j(w))";
by (rewtac rew);
by (rtac bin_rec_Bcons 1);
qed "def_bin_rec_Bcons";
fun bin_recs def = map standard
([def] RL [def_bin_rec_Plus, def_bin_rec_Minus, def_bin_rec_Bcons]);
Goalw [norm_Bcons_def] "norm_Bcons(Plus,0) = Plus";
by (Asm_simp_tac 1);
qed "norm_Bcons_Plus_0";
Goalw [norm_Bcons_def] "norm_Bcons(Plus,1) = Bcons(Plus,1)";
by (Asm_simp_tac 1);
qed "norm_Bcons_Plus_1";
Goalw [norm_Bcons_def] "norm_Bcons(Minus,0) = Bcons(Minus,0)";
by (Asm_simp_tac 1);
qed "norm_Bcons_Minus_0";
Goalw [norm_Bcons_def] "norm_Bcons(Minus,1) = Minus";
by (Asm_simp_tac 1);
qed "norm_Bcons_Minus_1";
Goalw [norm_Bcons_def]
"norm_Bcons(Bcons(w,x),b) = Bcons(Bcons(w,x),b)";
by (asm_simp_tac (simpset() addsimps bin.case_eqns) 1);
qed "norm_Bcons_Bcons";
val norm_Bcons_simps = [norm_Bcons_Plus_0, norm_Bcons_Plus_1,
norm_Bcons_Minus_0, norm_Bcons_Minus_1,
norm_Bcons_Bcons];
(** Type checking **)
val bin_typechecks0 = bin_rec_type :: bin.intrs;
Goalw [integ_of_bin_def]
"!!w. w: bin ==> integ_of_bin(w) : integ";
by (typechk_tac (bin_typechecks0@integ_typechecks@
nat_typechecks@[bool_into_nat]));
qed "integ_of_bin_type";
Goalw [norm_Bcons_def]
"!!w. [| w: bin; b: bool |] ==> norm_Bcons(w,b) : bin";
by (etac bin.elim 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps bin.case_eqns)));
by (typechk_tac (bin_typechecks0@bool_typechecks));
qed "norm_Bcons_type";
Goalw [bin_succ_def]
"!!w. w: bin ==> bin_succ(w) : bin";
by (typechk_tac ([norm_Bcons_type]@bin_typechecks0@bool_typechecks));
qed "bin_succ_type";
Goalw [bin_pred_def]
"!!w. w: bin ==> bin_pred(w) : bin";
by (typechk_tac ([norm_Bcons_type]@bin_typechecks0@bool_typechecks));
qed "bin_pred_type";
Goalw [bin_minus_def]
"!!w. w: bin ==> bin_minus(w) : bin";
by (typechk_tac ([bin_pred_type]@bin_typechecks0@bool_typechecks));
qed "bin_minus_type";
Goalw [bin_add_def]
"!!v w. [| v: bin; w: bin |] ==> bin_add(v,w) : bin";
by (typechk_tac ([norm_Bcons_type, bin_succ_type, bin_pred_type]@
bin_typechecks0@ bool_typechecks@ZF_typechecks));
qed "bin_add_type";
Goalw [bin_mult_def]
"!!v w. [| v: bin; w: bin |] ==> bin_mult(v,w) : bin";
by (typechk_tac ([norm_Bcons_type, bin_minus_type, bin_add_type]@
bin_typechecks0@ bool_typechecks));
qed "bin_mult_type";
val bin_typechecks = bin_typechecks0 @
[integ_of_bin_type, norm_Bcons_type, bin_succ_type, bin_pred_type,
bin_minus_type, bin_add_type, bin_mult_type];
Addsimps ([bool_1I, bool_0I,
bin_rec_Plus, bin_rec_Minus, bin_rec_Bcons] @
bin_recs integ_of_bin_def @ bin_typechecks);
val typechecks = bin_typechecks @ integ_typechecks @ nat_typechecks @
[bool_subset_nat RS subsetD];
(**** The carry/borrow functions, bin_succ and bin_pred ****)
(** Lemmas **)
goal Integ.thy
"!!z v. [| z $+ v = z' $+ v'; \
\ z: integ; z': integ; v: integ; v': integ; w: integ |] \
\ ==> z $+ (v $+ w) = z' $+ (v' $+ w)";
by (asm_simp_tac (simpset() addsimps ([zadd_assoc RS sym])) 1);
qed "zadd_assoc_cong";
goal Integ.thy
"!!z v w. [| z: integ; v: integ; w: integ |] \
\ ==> z $+ (v $+ w) = v $+ (z $+ w)";
by (REPEAT (ares_tac [zadd_commute RS zadd_assoc_cong] 1));
qed "zadd_assoc_swap";
(*Pushes 'constants' of the form $#m to the right -- LOOPS if two!*)
bind_thm ("zadd_assoc_znat", (znat_type RS zadd_assoc_swap));
Addsimps (bin_recs bin_succ_def @ bin_recs bin_pred_def);
(*norm_Bcons preserves the integer value of its argument*)
Goal
"!!w. [| w: bin; b: bool |] ==> \
\ integ_of_bin(norm_Bcons(w,b)) = integ_of_bin(Bcons(w,b))";
by (etac bin.elim 1);
by (asm_simp_tac (simpset() addsimps norm_Bcons_simps) 3);
by (ALLGOALS (etac boolE));
by (ALLGOALS (asm_simp_tac (simpset() addsimps (norm_Bcons_simps))));
qed "integ_of_bin_norm_Bcons";
Goal
"!!w. w: bin ==> integ_of_bin(bin_succ(w)) = $#1 $+ integ_of_bin(w)";
by (etac bin.induct 1);
by (Simp_tac 1);
by (Simp_tac 1);
by (etac boolE 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps integ_of_bin_norm_Bcons::zadd_ac)));
qed "integ_of_bin_succ";
Goal
"!!w. w: bin ==> integ_of_bin(bin_pred(w)) = $~ ($#1) $+ integ_of_bin(w)";
by (etac bin.induct 1);
by (Simp_tac 1);
by (Simp_tac 1);
by (etac boolE 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps integ_of_bin_norm_Bcons::zadd_ac)));
qed "integ_of_bin_pred";
(*These two results replace the definitions of bin_succ and bin_pred*)
(*** bin_minus: (unary!) negation of binary integers ***)
Addsimps (bin_recs bin_minus_def @
[integ_of_bin_succ, integ_of_bin_pred]);
Goal
"!!w. w: bin ==> integ_of_bin(bin_minus(w)) = $~ integ_of_bin(w)";
by (etac bin.induct 1);
by (Simp_tac 1);
by (Simp_tac 1);
by (etac boolE 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps (zadd_ac@[zminus_zadd_distrib]))));
qed "integ_of_bin_minus";
(*** bin_add: binary addition ***)
Goalw [bin_add_def] "!!w. w: bin ==> bin_add(Plus,w) = w";
by (Asm_simp_tac 1);
qed "bin_add_Plus";
Goalw [bin_add_def] "!!w. w: bin ==> bin_add(Minus,w) = bin_pred(w)";
by (Asm_simp_tac 1);
qed "bin_add_Minus";
Goalw [bin_add_def] "bin_add(Bcons(v,x),Plus) = Bcons(v,x)";
by (Simp_tac 1);
qed "bin_add_Bcons_Plus";
Goalw [bin_add_def] "bin_add(Bcons(v,x),Minus) = bin_pred(Bcons(v,x))";
by (Simp_tac 1);
qed "bin_add_Bcons_Minus";
Goalw [bin_add_def]
"!!w y. [| w: bin; y: bool |] ==> \
\ bin_add(Bcons(v,x), Bcons(w,y)) = \
\ norm_Bcons(bin_add(v, cond(x and y, bin_succ(w), w)), x xor y)";
by (Asm_simp_tac 1);
qed "bin_add_Bcons_Bcons";
Addsimps [bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus,
bin_add_Bcons_Minus, bin_add_Bcons_Bcons,
integ_of_bin_succ, integ_of_bin_pred,
integ_of_bin_norm_Bcons];
Addsimps [bool_subset_nat RS subsetD];
Goal
"!!v. v: bin ==> \
\ ALL w: bin. integ_of_bin(bin_add(v,w)) = \
\ integ_of_bin(v) $+ integ_of_bin(w)";
by (etac bin.induct 1);
by (Simp_tac 1);
by (Simp_tac 1);
by (rtac ballI 1);
by (bin_ind_tac "wa" [] 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps zadd_ac setloop (etac boolE))));
val integ_of_bin_add_lemma = result();
bind_thm("integ_of_bin_add", integ_of_bin_add_lemma RS bspec);
(*** bin_add: binary multiplication ***)
Addsimps (bin_recs bin_mult_def @
[integ_of_bin_minus, integ_of_bin_add,
integ_of_bin_norm_Bcons]);
val major::prems = goal Bin.thy
"[| v: bin; w: bin |] ==> \
\ integ_of_bin(bin_mult(v,w)) = \
\ integ_of_bin(v) $* integ_of_bin(w)";
by (cut_facts_tac prems 1);
by (bin_ind_tac "v" [major] 1);
by (Asm_simp_tac 1);
by (Asm_simp_tac 1);
by (etac boolE 1);
by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib]) 2);
by (asm_simp_tac
(simpset() addsimps ([zadd_zmult_distrib, zmult_1] @ zadd_ac)) 1);
qed "integ_of_bin_mult";
(**** Computations ****)
(** extra rules for bin_succ, bin_pred **)
val [bin_succ_Plus, bin_succ_Minus, _] = bin_recs bin_succ_def;
val [bin_pred_Plus, bin_pred_Minus, _] = bin_recs bin_pred_def;
Goal "bin_succ(Bcons(w,1)) = Bcons(bin_succ(w), 0)";
by (Simp_tac 1);
qed "bin_succ_Bcons1";
Goal "bin_succ(Bcons(w,0)) = norm_Bcons(w,1)";
by (Simp_tac 1);
qed "bin_succ_Bcons0";
Goal "bin_pred(Bcons(w,1)) = norm_Bcons(w,0)";
by (Simp_tac 1);
qed "bin_pred_Bcons1";
Goal "bin_pred(Bcons(w,0)) = Bcons(bin_pred(w), 1)";
by (Simp_tac 1);
qed "bin_pred_Bcons0";
(** extra rules for bin_minus **)
val [bin_minus_Plus, bin_minus_Minus, _] = bin_recs bin_minus_def;
Goal "bin_minus(Bcons(w,1)) = bin_pred(Bcons(bin_minus(w), 0))";
by (Simp_tac 1);
qed "bin_minus_Bcons1";
Goal "bin_minus(Bcons(w,0)) = Bcons(bin_minus(w), 0)";
by (Simp_tac 1);
qed "bin_minus_Bcons0";
(** extra rules for bin_add **)
Goal
"!!w. w: bin ==> bin_add(Bcons(v,1), Bcons(w,1)) = \
\ norm_Bcons(bin_add(v, bin_succ(w)), 0)";
by (Asm_simp_tac 1);
qed "bin_add_Bcons_Bcons11";
Goal
"!!w. w: bin ==> bin_add(Bcons(v,1), Bcons(w,0)) = \
\ norm_Bcons(bin_add(v,w), 1)";
by (Asm_simp_tac 1);
qed "bin_add_Bcons_Bcons10";
Goal
"!!w y. [| w: bin; y: bool |] ==> \
\ bin_add(Bcons(v,0), Bcons(w,y)) = norm_Bcons(bin_add(v,w), y)";
by (Asm_simp_tac 1);
qed "bin_add_Bcons_Bcons0";
(** extra rules for bin_mult **)
val [bin_mult_Plus, bin_mult_Minus, _] = bin_recs bin_mult_def;
Goal
"bin_mult(Bcons(v,1), w) = bin_add(norm_Bcons(bin_mult(v,w),0), w)";
by (Simp_tac 1);
qed "bin_mult_Bcons1";
Goal "bin_mult(Bcons(v,0), w) = norm_Bcons(bin_mult(v,w),0)";
by (Simp_tac 1);
qed "bin_mult_Bcons0";
(*** The computation simpset ***)
val bin_comp_ss = simpset_of Integ.thy
addsimps [integ_of_bin_add RS sym, (*invoke bin_add*)
integ_of_bin_minus RS sym, (*invoke bin_minus*)
integ_of_bin_mult RS sym, (*invoke bin_mult*)
bin_succ_Plus, bin_succ_Minus,
bin_succ_Bcons1, bin_succ_Bcons0,
bin_pred_Plus, bin_pred_Minus,
bin_pred_Bcons1, bin_pred_Bcons0,
bin_minus_Plus, bin_minus_Minus,
bin_minus_Bcons1, bin_minus_Bcons0,
bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus,
bin_add_Bcons_Minus, bin_add_Bcons_Bcons0,
bin_add_Bcons_Bcons10, bin_add_Bcons_Bcons11,
bin_mult_Plus, bin_mult_Minus,
bin_mult_Bcons1, bin_mult_Bcons0] @
norm_Bcons_simps
setSolver (type_auto_tac ([bool_1I, bool_0I] @ bin_typechecks0));
(*** Examples of performing binary arithmetic by simplification ***)
set proof_timing;
(*All runtimes below are on a SPARCserver 10*)
Goal "#13 $+ #19 = #32";
by (simp_tac bin_comp_ss 1); (*0.4 secs*)
result();
bin_add(binary_of_int 13, binary_of_int 19);
Goal "#1234 $+ #5678 = #6912";
by (simp_tac bin_comp_ss 1); (*1.3 secs*)
result();
bin_add(binary_of_int 1234, binary_of_int 5678);
Goal "#1359 $+ #~2468 = #~1109";
by (simp_tac bin_comp_ss 1); (*1.2 secs*)
result();
bin_add(binary_of_int 1359, binary_of_int ~2468);
Goal "#93746 $+ #~46375 = #47371";
by (simp_tac bin_comp_ss 1); (*1.9 secs*)
result();
bin_add(binary_of_int 93746, binary_of_int ~46375);
Goal "$~ #65745 = #~65745";
by (simp_tac bin_comp_ss 1); (*0.4 secs*)
result();
bin_minus(binary_of_int 65745);
(* negation of ~54321 *)
Goal "$~ #~54321 = #54321";
by (simp_tac bin_comp_ss 1); (*0.5 secs*)
result();
bin_minus(binary_of_int ~54321);
Goal "#13 $* #19 = #247";
by (simp_tac bin_comp_ss 1); (*0.7 secs*)
result();
bin_mult(binary_of_int 13, binary_of_int 19);
Goal "#~84 $* #51 = #~4284";
by (simp_tac bin_comp_ss 1); (*1.3 secs*)
result();
bin_mult(binary_of_int ~84, binary_of_int 51);
(*The worst case for 8-bit operands *)
Goal "#255 $* #255 = #65025";
by (simp_tac bin_comp_ss 1); (*4.3 secs*)
result();
bin_mult(binary_of_int 255, binary_of_int 255);
Goal "#1359 $* #~2468 = #~3354012";
by (simp_tac bin_comp_ss 1); (*6.1 secs*)
result();
bin_mult(binary_of_int 1359, binary_of_int ~2468);