--- a/src/HOL/Integ/Bin.ML Tue Nov 26 14:28:17 1996 +0100
+++ b/src/HOL/Integ/Bin.ML Tue Nov 26 15:59:28 1996 +0100
@@ -1,6 +1,6 @@
-(* Title: HOL/Integ/Bin.ML
- Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
- David Spelt, University of Twente
+(* 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
@@ -11,132 +11,148 @@
(** 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_0" Bin.thy
+ "norm_Bcons Plus False = Plus"
+ (fn _ => [(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_Plus_1" Bin.thy
+ "norm_Bcons Plus True = Bcons Plus True"
+ (fn _ => [(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_0" Bin.thy
+ "norm_Bcons Minus False = Bcons Minus False"
+ (fn _ => [(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_Minus_1" Bin.thy
+ "norm_Bcons Minus True = Minus"
+ (fn _ => [(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 "norm_Bcons_Bcons" Bin.thy
+ "norm_Bcons (Bcons w x) b = Bcons (Bcons w x) b"
+ (fn _ => [(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_Bcons1" Bin.thy
+ "bin_succ(Bcons w True) = Bcons (bin_succ w) False"
+ (fn _ => [(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_succ_Bcons0" Bin.thy
+ "bin_succ(Bcons w False) = norm_Bcons w True"
+ (fn _ => [(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_Bcons1" Bin.thy
+ "bin_pred(Bcons w True) = norm_Bcons w False"
+ (fn _ => [(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_pred_Bcons0" Bin.thy
+ "bin_pred(Bcons w False) = Bcons (bin_pred w) True"
+ (fn _ => [(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_Bcons1" Bin.thy
+ "bin_minus(Bcons w True) = bin_pred (Bcons(bin_minus w) False)"
+ (fn _ => [(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)]);
+qed_goal "bin_minus_Bcons0" Bin.thy
+ "bin_minus(Bcons w False) = Bcons (bin_minus w) False"
+ (fn _ => [(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_Bcons11" Bin.thy
+ "bin_add (Bcons v True) (Bcons w True) = \
+\ norm_Bcons (bin_add v (bin_succ w)) False"
+ (fn _ => [(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)]);
+qed_goal "bin_add_Bcons_Bcons10" Bin.thy
+ "bin_add (Bcons v True) (Bcons w False) = norm_Bcons (bin_add v w) True"
+ (fn _ => [(Simp_tac 1)]);
-(* SHOULD THIS THEOREM BE ADDED TO HOL_SS ? *)
-val my = prove_goal HOL.thy "(False = (~P)) = P"
- (fn prem => [(Fast_tac 1)]);
+val lemma = prove_goal HOL.thy "(False = (~P)) = P"
+ (fn _ => [(Fast_tac 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_Bcons0" Bin.thy
+ "bin_add (Bcons v False) (Bcons w y) = norm_Bcons (bin_add v w) y"
+ (fn _ => [(simp_tac (!simpset addsimps [lemma]) 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_Plus" Bin.thy
+ "bin_add (Bcons v x) Plus = Bcons v x"
+ (fn _ => [(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_Minus" Bin.thy
+ "bin_add (Bcons v x) Minus = bin_pred (Bcons v x)"
+ (fn _ => [(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)]);
+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 _ => [(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_Bcons1" Bin.thy
+ "bin_mult (Bcons v True) w = bin_add (norm_Bcons (bin_mult v w) False) w"
+ (fn _ => [(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)]);
+qed_goal "bin_mult_Bcons0" Bin.thy
+ "bin_mult (Bcons v False) w = norm_Bcons (bin_mult v w) False"
+ (fn _ => [(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]) ;
+val if_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_norm_Bcons" Bin.thy
+ "integ_of_bin(norm_Bcons w b) = integ_of_bin(Bcons w b)"
+ (fn _ =>[(bin.induct_tac "w" 1),
+ (ALLGOALS(simp_tac if_ss)) ]);
-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_succ" Bin.thy
+ "integ_of_bin(bin_succ w) = $#1 + integ_of_bin w"
+ (fn _ =>[(rtac bin.induct 1),
+ (ALLGOALS(asm_simp_tac
+ (if_ss addsimps (integ_of_bin_norm_Bcons::zadd_ac)))) ]);
-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_pred" Bin.thy
+ "integ_of_bin(bin_pred w) = $~ ($#1) + integ_of_bin w"
+ (fn _ =>[(rtac bin.induct 1),
+ (ALLGOALS(asm_simp_tac
+ (if_ss addsimps (integ_of_bin_norm_Bcons::zadd_ac)))) ]);
-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)]);
+qed_goal "integ_of_bin_minus" Bin.thy
+ "integ_of_bin(bin_minus w) = $~ (integ_of_bin w)"
+ (fn _ =>[(rtac bin.induct 1),
+ (Simp_tac 1),
+ (Simp_tac 1),
+ (asm_simp_tac (if_ss
+ delsimps [pred_Plus,pred_Minus,pred_Bcons]
+ addsimps [integ_of_bin_succ,integ_of_bin_pred,
+ zadd_assoc]) 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_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";
+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 (simp_tac (if_ss addsimps bin_add_simps) 1);
+by (simp_tac (if_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 (asm_simp_tac (if_ss addsimps (bin_add_simps)) 1);
+by (asm_simp_tac (if_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);
+by (asm_simp_tac (if_ss addsimps (bin_add_simps @ zadd_ac)) 1);
+by (asm_simp_tac (if_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";
@@ -144,35 +160,129 @@
by (Fast_tac 1);
qed "integ_of_bin_add";
-val bin_mult_simps = [integ_of_bin_minus, integ_of_bin_add,integ_of_bin_norm_Bcons];
+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 (simp_tac (if_ss addsimps bin_mult_simps) 1);
+by (simp_tac (if_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);
+by (asm_simp_tac (if_ss addsimps (bin_mult_simps @ [zadd_zmult_distrib])) 2);
+by (asm_simp_tac (if_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];
+ 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];
+ 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];
+
+
+(** Simplification rules for comparison of binary numbers (Norbert Völker) **)
+
+Addsimps [zadd_assoc];
+
+goal Bin.thy
+ "(integ_of_bin x = integ_of_bin y) \
+\ = (integ_of_bin (bin_add x (bin_minus y)) = $# 0)";
+ by (simp_tac (HOL_ss addsimps
+ [integ_of_bin_add, integ_of_bin_minus,zdiff_def]) 1);
+ by (rtac iffI 1);
+ by (etac ssubst 1);
+ by (rtac zadd_zminus_inverse 1);
+ by (dres_inst_tac [("f","(% z. z + integ_of_bin y)")] arg_cong 1);
+ by (asm_full_simp_tac
+ (HOL_ss addsimps[zadd_assoc,zadd_0,zadd_0_right,
+ zadd_zminus_inverse2]) 1);
+val iob_eq_eq_diff_0 = result();
+
+goal Bin.thy "(integ_of_bin Plus = $# 0) = True";
+ by (stac iob_Plus 1); by (Simp_tac 1);
+val iob_Plus_eq_0 = result();
+
+goal Bin.thy "(integ_of_bin Minus = $# 0) = False";
+ by (stac iob_Minus 1);
+ by (Simp_tac 1);
+val iob_Minus_not_eq_0 = result();
+
+goal Bin.thy
+ "(integ_of_bin (Bcons w x) = $# 0) = (~x & integ_of_bin w = $# 0)";
+ by (stac iob_Bcons 1);
+ by (case_tac "x" 1);
+ by (ALLGOALS Asm_simp_tac);
+ by (ALLGOALS(asm_simp_tac (HOL_ss addsimps [integ_of_bin_add])));
+ by (ALLGOALS(int_case_tac "integ_of_bin w"));
+ by (ALLGOALS(asm_simp_tac
+ (!simpset addsimps[zminus_zadd_distrib RS sym,
+ znat_add RS sym])));
+ by (stac eq_False_conv 1);
+ by (rtac notI 1);
+ by (dres_inst_tac [("f","(% z. z + $# Suc (Suc (n + n)))")] arg_cong 1);
+ by (Asm_full_simp_tac 1);
+val iob_Bcons_eq_0 = result();
+
+goalw Bin.thy [zless_def,zdiff_def]
+ "integ_of_bin x < integ_of_bin y \
+\ = (integ_of_bin (bin_add x (bin_minus y)) < $# 0)";
+ by (Simp_tac 1);
+val iob_less_eq_diff_lt_0 = result();
+
+goal Bin.thy "(integ_of_bin Plus < $# 0) = False";
+ by (stac iob_Plus 1); by (Simp_tac 1);
+val iob_Plus_not_lt_0 = result();
+
+goal Bin.thy "(integ_of_bin Minus < $# 0) = True";
+ by (stac iob_Minus 1); by (Simp_tac 1);
+val iob_Minus_lt_0 = result();
+
+goal Bin.thy
+ "(integ_of_bin (Bcons w x) < $# 0) = (integ_of_bin w < $# 0)";
+ by (stac iob_Bcons 1);
+ by (case_tac "x" 1);
+ by (ALLGOALS Asm_simp_tac);
+ by (ALLGOALS(asm_simp_tac (HOL_ss addsimps [integ_of_bin_add])));
+ by (ALLGOALS(int_case_tac "integ_of_bin w"));
+ by (ALLGOALS(asm_simp_tac
+ (!simpset addsimps[zminus_zadd_distrib RS sym,
+ znat_add RS sym])));
+ by (stac (zadd_zminus_inverse RS sym) 1);
+ by (rtac zadd_zless_mono1 1);
+ by (Simp_tac 1);
+val iob_Bcons_lt_0 = result();
+
+goal Bin.thy
+ "integ_of_bin x <= integ_of_bin y \
+\ = ( integ_of_bin (bin_add x (bin_minus y)) < $# 0 \
+\ | integ_of_bin (bin_add x (bin_minus y)) = $# 0)";
+by (simp_tac (HOL_ss addsimps
+ [zle_eq_zless_or_eq,iob_less_eq_diff_lt_0,zdiff_def
+ ,iob_eq_eq_diff_0,integ_of_bin_minus,integ_of_bin_add]) 1);
+val iob_le_diff_lt_0_or_eq_0 = result();
+
+Delsimps [zless_eq_less, zle_eq_le, zadd_assoc, znegative_zminus_znat,
+ not_znegative_znat, zero_zless_Suc_pos, negative_zless_0,
+ negative_zle_0, not_zle_0_negative, not_znat_zless_negative,
+ zminus_zless_zminus, zminus_zle_zminus, negative_eq_positive];
+
+Addsimps [zdiff_def, iob_eq_eq_diff_0, iob_less_eq_diff_lt_0,
+ iob_le_diff_lt_0_or_eq_0, iob_Plus_eq_0, iob_Minus_not_eq_0,
+ iob_Bcons_eq_0, iob_Plus_not_lt_0, iob_Minus_lt_0, iob_Bcons_lt_0];
+
(*** Examples of performing binary arithmetic by simplification ***)
@@ -215,3 +325,27 @@
goal Bin.thy "#1359 * #~2468 = #~3354012";
by (Simp_tac 1);
result();
+
+goal Bin.thy "#89 * #10 ~= #889";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#13 < #18 - #4";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#~345 < #~242 + #~100";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#13557456 < #18678654";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#999999 <= (#1000001 + #1)-#2";
+by (Simp_tac 1);
+result();
+
+goal Bin.thy "#1234567 <= #1234567";
+by (Simp_tac 1);
+result();
--- a/src/HOL/Integ/Integ.ML Tue Nov 26 14:28:17 1996 +0100
+++ b/src/HOL/Integ/Integ.ML Tue Nov 26 15:59:28 1996 +0100
@@ -318,6 +318,16 @@
qed "zadd_0_right";
+(** Lemmas **)
+
+qed_goal "zadd_assoc_cong" Integ.thy
+ "!!z. (z::int) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
+ (fn _ => [(asm_simp_tac (!simpset addsimps [zadd_assoc RS sym]) 1)]);
+
+qed_goal "zadd_assoc_swap" Integ.thy "(z::int) + (v + w) = v + (z + w)"
+ (fn _ => [(REPEAT (ares_tac [zadd_commute RS zadd_assoc_cong] 1))]);
+
+
(*Need properties of subtraction? Or use $- just as an abbreviation!*)
(**** zmult: multiplication on Integ ****)
@@ -745,3 +755,139 @@
by (dres_inst_tac [("z", "w")] zadd_zle_mono1 1);
by (asm_full_simp_tac (!simpset addsimps [zadd_commute]) 1);
qed "zadd_zle_self";
+
+
+(**** Comparisons: lemmas and proofs by Norbert Völker ****)
+
+(** One auxiliary theorem...**)
+
+goal HOL.thy "(x = False) = (~ x)";
+ by (fast_tac HOL_cs 1);
+qed "eq_False_conv";
+
+(** Additional theorems for Integ.thy **)
+
+Addsimps [zless_eq_less, zle_eq_le,
+ znegative_zminus_znat, not_znegative_znat];
+
+goal Integ.thy "!! x. (x::int) = y ==> x <= y";
+ by (etac subst 1); by (rtac zle_refl 1);
+val zequalD1 = result();
+
+goal Integ.thy "($~ x < $~ y) = (y < x)";
+ by (rewrite_goals_tac [zless_def,zdiff_def]);
+ by (simp_tac (!simpset addsimps zadd_ac ) 1);
+val zminus_zless_zminus = result();
+
+goal Integ.thy "($~ x <= $~ y) = (y <= x)";
+ by (simp_tac (HOL_ss addsimps[zle_def, zminus_zless_zminus]) 1);
+val zminus_zle_zminus = result();
+
+goal Integ.thy "(x < $~ y) = (y < $~ x)";
+ by (rewrite_goals_tac [zless_def,zdiff_def]);
+ by (simp_tac (!simpset addsimps zadd_ac ) 1);
+val zless_zminus = result();
+
+goal Integ.thy "($~ x < y) = ($~ y < x)";
+ by (rewrite_goals_tac [zless_def,zdiff_def]);
+ by (simp_tac (!simpset addsimps zadd_ac ) 1);
+val zminus_zless = result();
+
+goal Integ.thy "(x <= $~ y) = (y <= $~ x)";
+ by (simp_tac (HOL_ss addsimps[zle_def, zminus_zless]) 1);
+val zle_zminus = result();
+
+goal Integ.thy "($~ x <= y) = ($~ y <= x)";
+ by (simp_tac (HOL_ss addsimps[zle_def, zless_zminus]) 1);
+val zminus_zle = result();
+
+goal Integ.thy " $#0 < $# Suc n";
+ by (rtac (zero_less_Suc RS (zless_eq_less RS iffD2)) 1);
+val zero_zless_Suc_pos = result();
+
+goal Integ.thy "($# n= $# m) = (n = m)";
+ by (fast_tac (HOL_cs addSEs[inj_znat RS injD]) 1);
+val znat_znat_eq = result();
+AddIffs[znat_znat_eq];
+
+goal Integ.thy "$~ $# Suc n < $#0";
+ by (stac (zminus_0 RS sym) 1);
+ by (rtac (zminus_zless_zminus RS iffD2) 1);
+ by (rtac (zero_less_Suc RS (zless_eq_less RS iffD2)) 1);
+val negative_zless_0 = result();
+Addsimps [zero_zless_Suc_pos, negative_zless_0];
+
+goal Integ.thy "$~ $# n <= $#0";
+ by (rtac zless_or_eq_imp_zle 1);
+ by (nat_ind_tac "n" 1);
+ by (ALLGOALS Asm_simp_tac);
+val negative_zle_0 = result();
+Addsimps[negative_zle_0];
+
+goal Integ.thy "~($#0 <= $~ $# Suc n)";
+ by (stac zle_zminus 1);
+ by (Simp_tac 1);
+val not_zle_0_negative = result();
+Addsimps[not_zle_0_negative];
+
+goal Integ.thy "($# n <= $~ $# m) = (n = 0 & m = 0)";
+ by (safe_tac HOL_cs);
+ by (Simp_tac 3);
+ by (dtac (zle_zminus RS iffD1) 2);
+ by (ALLGOALS(dtac (negative_zle_0 RSN(2,zle_trans))));
+ by (ALLGOALS Asm_full_simp_tac);
+val znat_zle_znegative = result();
+
+goal Integ.thy "~($# n < $~ $# Suc m)";
+ by (rtac notI 1); by (forward_tac [zless_imp_zle] 1);
+ by (dtac (znat_zle_znegative RS iffD1) 1);
+ by (safe_tac HOL_cs);
+ by (dtac (zless_zminus RS iffD1) 1);
+ by (Asm_full_simp_tac 1);
+val not_znat_zless_negative = result();
+
+goal Integ.thy "($~ $# n = $# m) = (n = 0 & m = 0)";
+ by (rtac iffI 1);
+ by (rtac (znat_zle_znegative RS iffD1) 1);
+ by (dtac sym 1);
+ by (ALLGOALS Asm_simp_tac);
+val negative_eq_positive = result();
+
+Addsimps [zminus_zless_zminus, zminus_zle_zminus,
+ negative_eq_positive, not_znat_zless_negative];
+
+goalw Integ.thy [zdiff_def,zless_def] "!! x. znegative x = (x < $# 0)";
+ by (Auto_tac());
+val znegative_less_0 = result();
+
+goalw Integ.thy [zdiff_def,zless_def] "!! x. (~znegative x) = ($# 0 <= x)";
+ by (stac znegative_less_0 1);
+ by (safe_tac (HOL_cs addSDs[zleD,not_zleE,zleI]) );
+val not_znegative_ge_0 = result();
+
+goal Integ.thy "!! x. znegative x ==> ? n. x = $~ $# Suc n";
+ by (dtac (znegative_less_0 RS iffD1 RS zless_eq_zadd_Suc) 1);
+ by (etac exE 1);
+ by (rtac exI 1);
+ by (dres_inst_tac [("f","(% z. z + $~ $# Suc n )")] arg_cong 1);
+ by (auto_tac(!claset, !simpset addsimps [zadd_assoc]));
+val znegativeD = result();
+
+goal Integ.thy "!! x. ~znegative x ==> ? n. x = $# n";
+ by (dtac (not_znegative_ge_0 RS iffD1) 1);
+ by (dtac zle_imp_zless_or_eq 1);
+ by (etac disjE 1);
+ by (dtac zless_eq_zadd_Suc 1);
+ by (Auto_tac());
+val not_znegativeD = result();
+
+(* a case theorem distinguishing positive and negative int *)
+
+val prems = goal Integ.thy
+ "[|!! n. P ($# n); !! n. P ($~ $# Suc n) |] ==> P z";
+ by (cut_inst_tac [("P","znegative z")] excluded_middle 1);
+ by (fast_tac (HOL_cs addSDs[znegativeD,not_znegativeD] addSIs prems) 1);
+val int_cases = result();
+
+fun int_case_tac x = res_inst_tac [("z",x)] int_cases;
+