--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Integ/Bin.ML Fri Mar 29 13:18:26 1996 +0100
@@ -0,0 +1,217 @@
+(* Title: HOL/Integ/Bin.ML
+ Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
+ David Spelt, University of Twente
+ Copyright 1994 University of Cambridge
+ Copyright 1996 University of Twente
+
+Arithmetic on binary integers.
+*)
+
+open Bin;
+
+(** extra rules for bin_succ, bin_pred, bin_add, bin_mult **)
+
+qed_goal "norm_Bcons_Plus_0" Bin.thy "norm_Bcons Plus False = Plus"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "norm_Bcons_Plus_1" Bin.thy "norm_Bcons Plus True = Bcons Plus True"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "norm_Bcons_Minus_0" Bin.thy "norm_Bcons Minus False = Bcons Minus False"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "norm_Bcons_Minus_1" Bin.thy "norm_Bcons Minus True = Minus"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "norm_Bcons_Bcons" Bin.thy "norm_Bcons (Bcons w x) b = Bcons (Bcons w x) b"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_succ_Bcons1" Bin.thy "bin_succ(Bcons w True) = Bcons (bin_succ w) False"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_succ_Bcons0" Bin.thy "bin_succ(Bcons w False) = norm_Bcons w True"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_pred_Bcons1" Bin.thy "bin_pred(Bcons w True) = norm_Bcons w False"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_pred_Bcons0" Bin.thy "bin_pred(Bcons w False) = Bcons (bin_pred w) True"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_minus_Bcons1" Bin.thy "bin_minus(Bcons w True) = bin_pred (Bcons(bin_minus w) False)"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_minus_Bcons0" Bin.thy "bin_minus(Bcons w False) = Bcons (bin_minus w) False"
+ (fn prem => [(Simp_tac 1)]);
+
+(*** bin_add: binary addition ***)
+
+qed_goal "bin_add_Bcons_Bcons11" Bin.thy "bin_add (Bcons v True) (Bcons w True) = norm_Bcons (bin_add v (bin_succ w)) False"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_add_Bcons_Bcons10" Bin.thy "bin_add (Bcons v True) (Bcons w False) = norm_Bcons (bin_add v w) True"
+ (fn prem => [(Simp_tac 1)]);
+
+(* SHOULD THIS THEOREM BE ADDED TO HOL_SS ? *)
+val my = prove_goal HOL.thy "(False = (~P)) = P"
+ (fn prem => [(fast_tac HOL_cs 1)]);
+
+qed_goal "bin_add_Bcons_Bcons0" Bin.thy "bin_add (Bcons v False) (Bcons w y) = norm_Bcons (bin_add v w) y"
+ (fn prem => [(simp_tac (!simpset addsimps [my]) 1)]);
+
+qed_goal "bin_add_Bcons_Plus" Bin.thy "bin_add (Bcons v x) Plus = Bcons v x"
+ (fn prems => [(Simp_tac 1)]);
+
+qed_goal "bin_add_Bcons_Minus" Bin.thy "bin_add (Bcons v x) Minus = bin_pred (Bcons v x)"
+ (fn prems => [(Simp_tac 1)]);
+
+qed_goal "bin_add_Bcons_Bcons" Bin.thy "bin_add (Bcons v x) (Bcons w y) = norm_Bcons(bin_add v (if x & y then (bin_succ w) else w)) (x~= y)"
+ (fn prems => [(Simp_tac 1)]);
+
+
+(*** bin_add: binary multiplication ***)
+
+qed_goal "bin_mult_Bcons1" Bin.thy "bin_mult (Bcons v True) w = bin_add (norm_Bcons (bin_mult v w) False) w"
+ (fn prem => [(Simp_tac 1)]);
+
+qed_goal "bin_mult_Bcons0" Bin.thy "bin_mult (Bcons v False) w = norm_Bcons (bin_mult v w) False"
+ (fn prem => [(Simp_tac 1)]);
+
+
+(**** The carry/borrow functions, bin_succ and bin_pred ****)
+
+(** Lemmas **)
+
+qed_goal "zadd_assoc_cong" Integ.thy "(z::int) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
+ (fn prems => [(asm_simp_tac (!simpset addsimps (prems @ [zadd_assoc RS sym])) 1)]);
+
+qed_goal "zadd_assoc_swap" Integ.thy "(z::int) + (v + w) = v + (z + w)"
+ (fn prems => [(REPEAT (ares_tac [zadd_commute RS zadd_assoc_cong] 1))]);
+
+
+val my_ss = !simpset setloop (split_tac [expand_if]) ;
+
+
+(**** integ_of_bin ****)
+
+
+qed_goal "integ_of_bin_norm_Bcons" Bin.thy "integ_of_bin(norm_Bcons w b) = integ_of_bin(Bcons w b)"
+ (fn prems=>[ (bin.induct_tac "w" 1),
+ (REPEAT(simp_tac (!simpset setloop (split_tac [expand_if])) 1)) ]);
+
+qed_goal "integ_of_bin_succ" Bin.thy "integ_of_bin(bin_succ w) = $#1 + integ_of_bin w"
+ (fn prems=>[ (rtac bin.induct 1),
+ (REPEAT(asm_simp_tac (!simpset addsimps (integ_of_bin_norm_Bcons::zadd_ac)
+ setloop (split_tac [expand_if])) 1)) ]);
+
+qed_goal "integ_of_bin_pred" Bin.thy "integ_of_bin(bin_pred w) = $~ ($#1) + integ_of_bin w"
+ (fn prems=>[ (rtac bin.induct 1),
+ (REPEAT(asm_simp_tac (!simpset addsimps (integ_of_bin_norm_Bcons::zadd_ac)
+ setloop (split_tac [expand_if])) 1)) ]);
+
+qed_goal "integ_of_bin_minus" Bin.thy "integ_of_bin(bin_minus w) = $~ (integ_of_bin w)"
+ (fn prems=>[ (rtac bin.induct 1),
+ (Simp_tac 1),
+ (Simp_tac 1),
+ (asm_simp_tac (!simpset
+ delsimps [pred_Plus,pred_Minus,pred_Bcons]
+ addsimps [integ_of_bin_succ,integ_of_bin_pred,
+ zadd_assoc]
+ setloop (split_tac [expand_if])) 1)]);
+
+
+val bin_add_simps = [add_Plus,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];
+val bin_simps = [iob_Plus,iob_Minus,iob_Bcons];
+
+goal Bin.thy "! w. integ_of_bin(bin_add v w) = integ_of_bin v + integ_of_bin w";
+by (bin.induct_tac "v" 1);
+by (simp_tac (my_ss addsimps bin_add_simps) 1);
+by (simp_tac (my_ss addsimps bin_add_simps) 1);
+by (rtac allI 1);
+by (bin.induct_tac "w" 1);
+by (asm_simp_tac (my_ss addsimps (bin_add_simps)) 1);
+by (asm_simp_tac (my_ss addsimps (bin_add_simps @ zadd_ac)) 1);
+by (cut_inst_tac [("P","bool")] True_or_False 1);
+by (etac disjE 1);
+by (asm_simp_tac (my_ss addsimps (bin_add_simps @ zadd_ac)) 1);
+by (asm_simp_tac (my_ss addsimps (bin_add_simps @ zadd_ac)) 1);
+val integ_of_bin_add_lemma = result();
+
+goal Bin.thy "integ_of_bin(bin_add v w) = integ_of_bin v + integ_of_bin w";
+by (cut_facts_tac [integ_of_bin_add_lemma] 1);
+by (fast_tac HOL_cs 1);
+qed "integ_of_bin_add";
+
+val bin_mult_simps = [integ_of_bin_minus, integ_of_bin_add,integ_of_bin_norm_Bcons];
+
+goal Bin.thy "integ_of_bin(bin_mult v w) = integ_of_bin v * integ_of_bin w";
+by (bin.induct_tac "v" 1);
+by (simp_tac (my_ss addsimps bin_mult_simps) 1);
+by (simp_tac (my_ss addsimps bin_mult_simps) 1);
+by (cut_inst_tac [("P","bool")] True_or_False 1);
+by (etac disjE 1);
+by (asm_simp_tac (my_ss addsimps (bin_mult_simps @ [zadd_zmult_distrib])) 2);
+by (asm_simp_tac (my_ss addsimps (bin_mult_simps @ [zadd_zmult_distrib] @ zadd_ac)) 1);
+qed "integ_of_bin_mult";
+
+
+Delsimps [succ_Bcons,pred_Bcons,min_Bcons,add_Bcons,mult_Bcons,
+ iob_Plus,iob_Minus,iob_Bcons,
+ norm_Plus,norm_Minus,norm_Bcons];
+
+Addsimps [integ_of_bin_add RS sym,
+ integ_of_bin_minus RS sym,
+ integ_of_bin_mult RS sym,
+ bin_succ_Bcons1,bin_succ_Bcons0,
+ bin_pred_Bcons1,bin_pred_Bcons0,
+ bin_minus_Bcons1,bin_minus_Bcons0,
+ bin_add_Bcons_Plus,bin_add_Bcons_Minus,
+ bin_add_Bcons_Bcons0,bin_add_Bcons_Bcons10,bin_add_Bcons_Bcons11,
+ bin_mult_Bcons1,bin_mult_Bcons0,
+ norm_Bcons_Plus_0,norm_Bcons_Plus_1,
+ norm_Bcons_Minus_0,norm_Bcons_Minus_1,
+ norm_Bcons_Bcons];
+
+(*** Examples of performing binary arithmetic by simplification ***)
+
+goal Bin.thy "#13 + #19 = #32";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#1234 + #5678 = #6912";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#1359 + #~2468 = #~1109";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#93746 + #~46375 = #47371";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "$~ #65745 = #~65745";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "$~ #~54321 = #54321";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#13 * #19 = #247";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#~84 * #51 = #~4284";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#255 * #255 = #65025";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#1359 * #~2468 = #~3354012";
+by (Simp_tac 1);
+result();
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Integ/Bin.thy Fri Mar 29 13:18:26 1996 +0100
@@ -0,0 +1,163 @@
+(* Title: HOL/Integ/Bin.thy
+ Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
+ David Spelt, University of Twente
+ Copyright 1994 University of Cambridge
+ Copyright 1996 University of Twente
+
+Arithmetic on binary integers.
+
+ The sign Plus stands for an infinite string of leading F's.
+ The sign Minus stands for an infinite string of leading T's.
+
+A number can have multiple representations, namely leading F's with sign
+Plus and leading T's with sign Minus. See twos-compl.ML/int_of_binary for
+the numerical interpretation.
+
+The representation expects that (m mod 2) is 0 or 1, even if m is negative;
+For instance, ~5 div 2 = ~3 and ~5 mod 2 = 1; thus ~5 = (~3)*2 + 1
+
+Division is not defined yet!
+*)
+
+Bin = Integ +
+
+syntax
+ "_Int" :: xnum => int ("_")
+
+datatype
+ bin = Plus
+ | Minus
+ | Bcons bin bool
+
+consts
+ integ_of_bin :: bin=>int
+ norm_Bcons :: [bin,bool]=>bin
+ bin_succ :: bin=>bin
+ bin_pred :: bin=>bin
+ bin_minus :: bin=>bin
+ bin_add,bin_mult :: [bin,bin]=>bin
+ h_bin :: [bin,bool,bin]=>bin
+
+(*norm_Bcons adds a bit, suppressing leading 0s and 1s*)
+
+primrec norm_Bcons bin
+ norm_Plus "norm_Bcons Plus b = (if b then (Bcons Plus b) else Plus)"
+ norm_Minus "norm_Bcons Minus b = (if b then Minus else (Bcons Minus b))"
+ norm_Bcons "norm_Bcons (Bcons w' x') b = Bcons (Bcons w' x') b"
+
+primrec integ_of_bin bin
+ iob_Plus "integ_of_bin Plus = $#0"
+ iob_Minus "integ_of_bin Minus = $~($#1)"
+ iob_Bcons "integ_of_bin(Bcons w x) = (if x then $#1 else $#0) + (integ_of_bin w) + (integ_of_bin w)"
+
+primrec bin_succ bin
+ succ_Plus "bin_succ Plus = Bcons Plus True"
+ succ_Minus "bin_succ Minus = Plus"
+ succ_Bcons "bin_succ(Bcons w x) = (if x then (Bcons (bin_succ w) False) else (norm_Bcons w True))"
+
+primrec bin_pred bin
+ pred_Plus "bin_pred(Plus) = Minus"
+ pred_Minus "bin_pred(Minus) = Bcons Minus False"
+ pred_Bcons "bin_pred(Bcons w x) = (if x then (norm_Bcons w False) else (Bcons (bin_pred w) True))"
+
+primrec bin_minus bin
+ min_Plus "bin_minus Plus = Plus"
+ min_Minus "bin_minus Minus = Bcons Plus True"
+ min_Bcons "bin_minus(Bcons w x) = (if x then (bin_pred (Bcons (bin_minus w) False)) else (Bcons (bin_minus w) False))"
+
+primrec bin_add bin
+ add_Plus "bin_add Plus w = w"
+ add_Minus "bin_add Minus w = bin_pred w"
+ add_Bcons "bin_add (Bcons v x) w = h_bin v x w"
+
+primrec bin_mult bin
+ mult_Plus "bin_mult Plus w = Plus"
+ mult_Minus "bin_mult Minus w = bin_minus w"
+ mult_Bcons "bin_mult (Bcons v x) w = (if x then (bin_add (norm_Bcons (bin_mult v w) False) w) else (norm_Bcons (bin_mult v w) False))"
+
+primrec h_bin bin
+ h_Plus "h_bin v x Plus = Bcons v x"
+ h_Minus "h_bin v x Minus = bin_pred (Bcons v x)"
+ h_BCons "h_bin v x (Bcons w y) = norm_Bcons (bin_add v (if (x & y) then bin_succ w else w)) (x~=y)"
+
+end
+
+ML
+
+(** Concrete syntax for integers **)
+
+local
+ open Syntax;
+
+ (* Bits *)
+
+ fun mk_bit 0 = const "False"
+ | mk_bit 1 = const "True"
+ | mk_bit _ = sys_error "mk_bit";
+
+ fun dest_bit (Const ("False", _)) = 0
+ | dest_bit (Const ("True", _)) = 1
+ | dest_bit _ = raise Match;
+
+
+ (* Bit strings *) (*we try to handle superfluous leading digits nicely*)
+
+ fun prefix_len _ [] = 0
+ | prefix_len pred (x :: xs) =
+ if pred x then 1 + prefix_len pred xs else 0;
+
+ fun mk_bin str =
+ let
+ val (sign, digs) =
+ (case explode str of
+ "#" :: "~" :: cs => (~1, cs)
+ | "#" :: cs => (1, cs)
+ | _ => raise ERROR);
+
+ val zs = prefix_len (equal "0") digs;
+
+ fun bin_of 0 = replicate zs 0
+ | bin_of ~1 = replicate zs 1 @ [~1]
+ | bin_of n = (n mod 2) :: bin_of (n div 2);
+
+ fun term_of [] = const "Plus"
+ | term_of [~1] = const "Minus"
+ | term_of (b :: bs) = const "Bcons" $ term_of bs $ mk_bit b;
+ in
+ term_of (bin_of (sign * (#1 (scan_int digs))))
+ end;
+
+ fun dest_bin tm =
+ let
+ fun bin_of (Const ("Plus", _)) = []
+ | bin_of (Const ("Minus", _)) = [~1]
+ | bin_of (Const ("Bcons", _) $ bs $ b) = dest_bit b :: bin_of bs
+ | bin_of _ = raise Match;
+
+ fun int_of [] = 0
+ | int_of (b :: bs) = b + 2 * int_of bs;
+
+ val rev_digs = bin_of tm;
+ val (sign, zs) =
+ (case rev rev_digs of
+ ~1 :: bs => ("~", prefix_len (equal 1) bs)
+ | bs => ("", prefix_len (equal 0) bs));
+ val num = string_of_int (abs (int_of rev_digs));
+ in
+ "#" ^ sign ^ implode (replicate zs "0") ^ num
+ end;
+
+
+ (* translation of integer constant tokens to and from binary *)
+
+ fun int_tr (*"_Int"*) [t as Free (str, _)] =
+ (const "integ_of_bin" $
+ (mk_bin str handle ERROR => raise_term "int_tr" [t]))
+ | int_tr (*"_Int"*) ts = raise_term "int_tr" ts;
+
+ fun int_tr' (*"integ_of"*) [t] = const "_Int" $ free (dest_bin t)
+ | int_tr' (*"integ_of"*) _ = raise Match;
+in
+ val parse_translation = [("_Int", int_tr)];
+ val print_translation = [("integ_of_bin", int_tr')];
+end;
--- a/src/HOL/Integ/ROOT.ML Fri Mar 29 13:16:38 1996 +0100
+++ b/src/HOL/Integ/ROOT.ML Fri Mar 29 13:18:26 1996 +0100
@@ -8,4 +8,4 @@
HOL_build_completed; (*Cause examples to fail if HOL did*)
-time_use_thy "Integ";
+time_use_thy "Bin";