author | clasohm |
Wed, 14 Dec 1994 11:41:49 +0100 | |
changeset 782 | 200a16083201 |
parent 760 | f0200e91b272 |
child 906 | 6cd9c397f36a |
permissions | -rw-r--r-- |
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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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qed "bin_rec_Plus"; |
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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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qed "bin_rec_Minus"; |
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goal Bin.thy "bin_rec(w$$x,a,b,h) = h(w, x, bin_rec(w,a,b,h))"; |
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by (rtac (bin_rec_def RS def_Vrec RS trans) 1); |
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by (rewrite_goals_tac bin.con_defs); |
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by (simp_tac rank_ss 1); |
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qed "bin_rec_Bcons"; |
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(*Type checking*) |
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val prems = goal Bin.thy |
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"[| w: bin; \ |
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\ a: C(Plus); b: C(Minus); \ |
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\ !!w x r. [| w: bin; x: bool; r: C(w) |] ==> h(w,x,r): C(w$$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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qed "bin_rec_type"; |
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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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qed "def_bin_rec_Plus"; |
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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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qed "def_bin_rec_Minus"; |
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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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qed "def_bin_rec_Bcons"; |
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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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qed "integ_of_bin_type"; |
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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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qed "bin_succ_type"; |
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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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qed "bin_pred_type"; |
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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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qed "bin_minus_type"; |
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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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qed "bin_add_type"; |
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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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qed "bin_mult_type"; |
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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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qed "zadd_assoc_cong"; |
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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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qed "zadd_assoc_swap"; |
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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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bind_thm ("zadd_assoc_znat", (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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qed "zadd_swap_pairs"; |
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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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qed "integ_of_bin_succ"; |
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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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qed "integ_of_bin_pred"; |
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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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qed "integ_of_bin_minus"; |
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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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qed "bin_add_Plus"; |
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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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qed "bin_add_Minus"; |
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goalw Bin.thy [bin_add_def] "bin_add(v$$x,Plus) = v$$x"; |
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by (simp_tac bin_ss 1); |
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qed "bin_add_Bcons_Plus"; |
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goalw Bin.thy [bin_add_def] "bin_add(v$$x,Minus) = bin_pred(v$$x)"; |
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by (simp_tac bin_ss 1); |
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qed "bin_add_Bcons_Minus"; |
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goalw Bin.thy [bin_add_def] |
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"!!w y. [| w: bin; y: bool |] ==> \ |
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\ bin_add(v$$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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qed "bin_add_Bcons_Bcons"; |
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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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qed "integ_of_bin_add_lemma"; |
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val integ_of_bin_add = integ_of_bin_add_lemma RS bspec; |
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255 |
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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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qed "integ_of_bin_mult"; |
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(**** Computations ****) |
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280 |
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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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qed "bin_succ_Bcons1"; |
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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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qed "bin_succ_Bcons0"; |
515 | 293 |
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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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qed "bin_pred_Bcons1"; |
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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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qed "bin_pred_Bcons0"; |
515 | 301 |
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(** extra rules for bin_minus **) |
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303 |
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304 |
val [bin_minus_Plus, bin_minus_Minus, _] = bin_recs bin_minus_def; |
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305 |
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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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qed "bin_minus_Bcons1"; |
515 | 309 |
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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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qed "bin_minus_Bcons0"; |
515 | 313 |
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(** extra rules for bin_add **) |
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315 |
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316 |
goal Bin.thy |
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"!!w. w: bin ==> bin_add(v$$1, w$$1) = bin_add(v, bin_succ(w)) $$ 0"; |
|
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by (asm_simp_tac bin_add_ss 1); |
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319 |
qed "bin_add_Bcons_Bcons11"; |
515 | 320 |
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321 |
goal Bin.thy |
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322 |
"!!w. w: bin ==> bin_add(v$$1, w$$0) = bin_add(v,w) $$ 1"; |
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by (asm_simp_tac bin_add_ss 1); |
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200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset
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324 |
qed "bin_add_Bcons_Bcons10"; |
515 | 325 |
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326 |
goal Bin.thy |
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327 |
"!!w y.[| w: bin; y: bool |] ==> bin_add(v$$0, w$$y) = bin_add(v,w) $$ y"; |
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328 |
by (asm_simp_tac bin_add_ss 1); |
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200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset
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329 |
qed "bin_add_Bcons_Bcons0"; |
515 | 330 |
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331 |
(** extra rules for bin_mult **) |
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332 |
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333 |
val [bin_mult_Plus, bin_mult_Minus, _] = bin_recs bin_mult_def; |
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334 |
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335 |
goal Bin.thy "bin_mult(v$$1, w) = bin_add(bin_mult(v,w)$$0, w)"; |
|
336 |
by (simp_tac bin_mult_ss 1); |
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782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset
|
337 |
qed "bin_mult_Bcons1"; |
515 | 338 |
|
339 |
goal Bin.thy "bin_mult(v$$0, w) = bin_mult(v,w)$$0"; |
|
340 |
by (simp_tac bin_mult_ss 1); |
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782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset
|
341 |
qed "bin_mult_Bcons0"; |
515 | 342 |
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343 |
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344 |
(*** The computation simpset ***) |
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345 |
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346 |
val bin_comp_ss = integ_ss |
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347 |
addsimps [bin_succ_Plus, bin_succ_Minus, |
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348 |
bin_succ_Bcons1, bin_succ_Bcons0, |
|
349 |
bin_pred_Plus, bin_pred_Minus, |
|
350 |
bin_pred_Bcons1, bin_pred_Bcons0, |
|
351 |
bin_minus_Plus, bin_minus_Minus, |
|
352 |
bin_minus_Bcons1, bin_minus_Bcons0, |
|
353 |
bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus, |
|
354 |
bin_add_Bcons_Minus, bin_add_Bcons_Bcons0, |
|
355 |
bin_add_Bcons_Bcons10, bin_add_Bcons_Bcons11, |
|
356 |
bin_mult_Plus, bin_mult_Minus, |
|
357 |
bin_mult_Bcons1, bin_mult_Bcons0] |
|
358 |
setsolver (type_auto_tac ([bool_1I, bool_0I] @ bin_typechecks0)); |
|
359 |
||
360 |
(*** Examples of performing binary arithmetic by simplification ***) |
|
361 |
||
362 |
proof_timing := true; |
|
363 |
(*All runtimes below are on a SPARCserver 10*) |
|
364 |
||
365 |
(* 13+19 = 32 *) |
|
366 |
goal Bin.thy |
|
367 |
"bin_add(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = Plus$$1$$0$$0$$0$$0$$0"; |
|
368 |
by (simp_tac bin_comp_ss 1); (*0.6 secs*) |
|
369 |
result(); |
|
370 |
||
371 |
bin_add(binary_of_int 13, binary_of_int 19); |
|
372 |
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373 |
(* 1234+5678 = 6912 *) |
|
374 |
goal Bin.thy |
|
375 |
"bin_add(Plus$$1$$0$$0$$1$$1$$0$$1$$0$$0$$1$$0, \ |
|
376 |
\ Plus$$1$$0$$1$$1$$0$$0$$0$$1$$0$$1$$1$$1$$0) = \ |
|
377 |
\ Plus$$1$$1$$0$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0"; |
|
378 |
by (simp_tac bin_comp_ss 1); (*2.6 secs*) |
|
379 |
result(); |
|
380 |
||
381 |
bin_add(binary_of_int 1234, binary_of_int 5678); |
|
382 |
||
383 |
(* 1359-2468 = ~1109 *) |
|
384 |
goal Bin.thy |
|
385 |
"bin_add(Plus$$1$$0$$1$$0$$1$$0$$0$$1$$1$$1$$1, \ |
|
386 |
\ Minus$$0$$1$$1$$0$$0$$1$$0$$1$$1$$1$$0$$0) = \ |
|
387 |
\ Minus$$1$$0$$1$$1$$1$$0$$1$$0$$1$$0$$1$$1"; |
|
388 |
by (simp_tac bin_comp_ss 1); (*2.3 secs*) |
|
389 |
result(); |
|
390 |
||
391 |
bin_add(binary_of_int 1359, binary_of_int ~2468); |
|
392 |
||
393 |
(* 93746-46375 = 47371 *) |
|
394 |
goal Bin.thy |
|
395 |
"bin_add(Plus$$1$$0$$1$$1$$0$$1$$1$$1$$0$$0$$0$$1$$1$$0$$0$$1$$0, \ |
|
396 |
\ Minus$$0$$1$$0$$0$$1$$0$$1$$0$$1$$1$$0$$1$$1$$0$$0$$1) = \ |
|
397 |
\ Plus$$0$$1$$0$$1$$1$$1$$0$$0$$1$$0$$0$$0$$0$$1$$0$$1$$1"; |
|
398 |
by (simp_tac bin_comp_ss 1); (*3.9 secs*) |
|
399 |
result(); |
|
400 |
||
401 |
bin_add(binary_of_int 93746, binary_of_int ~46375); |
|
402 |
||
403 |
(* negation of 65745 *) |
|
404 |
goal Bin.thy |
|
405 |
"bin_minus(Plus$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1$$1$$0$$1$$0$$0$$0$$1) = \ |
|
406 |
\ Minus$$0$$1$$1$$1$$1$$1$$1$$1$$1$$0$$0$$1$$0$$1$$1$$1$$1"; |
|
407 |
by (simp_tac bin_comp_ss 1); (*0.6 secs*) |
|
408 |
result(); |
|
409 |
||
410 |
bin_minus(binary_of_int 65745); |
|
411 |
||
412 |
(* negation of ~54321 *) |
|
413 |
goal Bin.thy |
|
414 |
"bin_minus(Minus$$0$$0$$1$$0$$1$$0$$1$$1$$1$$1$$0$$0$$1$$1$$1$$1) = \ |
|
415 |
\ Plus$$0$$1$$1$$0$$1$$0$$1$$0$$0$$0$$0$$1$$1$$0$$0$$0$$1"; |
|
416 |
by (simp_tac bin_comp_ss 1); (*0.7 secs*) |
|
417 |
result(); |
|
418 |
||
419 |
bin_minus(binary_of_int ~54321); |
|
420 |
||
421 |
(* 13*19 = 247 *) |
|
422 |
goal Bin.thy "bin_mult(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = \ |
|
423 |
\ Plus$$1$$1$$1$$1$$0$$1$$1$$1"; |
|
424 |
by (simp_tac bin_comp_ss 1); (*1.5 secs*) |
|
425 |
result(); |
|
426 |
||
427 |
bin_mult(binary_of_int 13, binary_of_int 19); |
|
428 |
||
429 |
(* ~84 * 51 = ~4284 *) |
|
430 |
goal Bin.thy |
|
431 |
"bin_mult(Minus$$0$$1$$0$$1$$1$$0$$0, Plus$$1$$1$$0$$0$$1$$1) = \ |
|
432 |
\ Minus$$0$$1$$1$$1$$1$$0$$1$$0$$0$$0$$1$$0$$0"; |
|
433 |
by (simp_tac bin_comp_ss 1); (*2.6 secs*) |
|
434 |
result(); |
|
435 |
||
436 |
bin_mult(binary_of_int ~84, binary_of_int 51); |
|
437 |
||
438 |
(* 255*255 = 65025; the worst case for 8-bit operands *) |
|
439 |
goal Bin.thy |
|
440 |
"bin_mult(Plus$$1$$1$$1$$1$$1$$1$$1$$1, \ |
|
441 |
\ Plus$$1$$1$$1$$1$$1$$1$$1$$1) = \ |
|
442 |
\ Plus$$1$$1$$1$$1$$1$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1"; |
|
443 |
by (simp_tac bin_comp_ss 1); (*9.8 secs*) |
|
444 |
result(); |
|
445 |
||
446 |
bin_mult(binary_of_int 255, binary_of_int 255); |
|
447 |
||
448 |
(* 1359 * ~2468 = ~3354012 *) |
|
449 |
goal Bin.thy |
|
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); |