--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Integ/Integ.ML Tue Feb 28 02:02:34 1995 +0100
@@ -0,0 +1,752 @@
+(* Title: Integ.ML
+ ID: $Id$
+ Authors: Riccardo Mattolini, Dip. Sistemi e Informatica
+ Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 Universita' di Firenze
+ Copyright 1993 University of Cambridge
+
+The integers as equivalence classes over nat*nat.
+
+Could also prove...
+"znegative(z) ==> $# zmagnitude(z) = $~ z"
+"~ znegative(z) ==> $# zmagnitude(z) = z"
+< is a linear ordering
++ and * are monotonic wrt <
+*)
+
+open Integ;
+
+
+(*** Proving that intrel is an equivalence relation ***)
+
+val eqa::eqb::prems = goal Arith.thy
+ "[| (x1::nat) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] ==> \
+\ x1 + y3 = x3 + y1";
+by (res_inst_tac [("k2","x2")] (add_left_cancel RS iffD1) 1);
+by (rtac (add_left_commute RS trans) 1);
+by (rtac (eqb RS ssubst) 1);
+by (rtac (add_left_commute RS trans) 1);
+by (rtac (eqa RS ssubst) 1);
+by (rtac (add_left_commute) 1);
+qed "integ_trans_lemma";
+
+(** Natural deduction for intrel **)
+
+val prems = goalw Integ.thy [intrel_def]
+ "[| x1+y2 = x2+y1|] ==> \
+\ <<x1,y1>,<x2,y2>>: intrel";
+by (fast_tac (rel_cs addIs prems) 1);
+qed "intrelI";
+
+(*intrelE is hard to derive because fast_tac tries hyp_subst_tac so soon*)
+goalw Integ.thy [intrel_def]
+ "p: intrel --> (EX x1 y1 x2 y2. \
+\ p = <<x1,y1>,<x2,y2>> & x1+y2 = x2+y1)";
+by (fast_tac rel_cs 1);
+qed "intrelE_lemma";
+
+val [major,minor] = goal Integ.thy
+ "[| p: intrel; \
+\ !!x1 y1 x2 y2. [| p = <<x1,y1>,<x2,y2>>; x1+y2 = x2+y1|] ==> Q |] \
+\ ==> Q";
+by (cut_facts_tac [major RS (intrelE_lemma RS mp)] 1);
+by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1));
+qed "intrelE";
+
+val intrel_cs = rel_cs addSIs [intrelI] addSEs [intrelE];
+
+goal Integ.thy "<<x1,y1>,<x2,y2>>: intrel = (x1+y2 = x2+y1)";
+by (fast_tac intrel_cs 1);
+qed "intrel_iff";
+
+goal Integ.thy "<x,x>: intrel";
+by (rtac (surjective_pairing RS ssubst) 1 THEN rtac (refl RS intrelI) 1);
+qed "intrel_refl";
+
+goalw Integ.thy [equiv_def, refl_def, sym_def, trans_def]
+ "equiv({x::(nat*nat).True}, intrel)";
+by (fast_tac (intrel_cs addSIs [intrel_refl]
+ addSEs [sym, integ_trans_lemma]) 1);
+qed "equiv_intrel";
+
+val equiv_intrel_iff =
+ [TrueI, TrueI] MRS
+ ([CollectI, CollectI] MRS
+ (equiv_intrel RS eq_equiv_class_iff));
+
+goalw Integ.thy [Integ_def,intrel_def,quotient_def] "intrel^^{<x,y>}:Integ";
+by (fast_tac set_cs 1);
+qed "intrel_in_integ";
+
+goal Integ.thy "inj_onto(Abs_Integ,Integ)";
+by (rtac inj_onto_inverseI 1);
+by (etac Abs_Integ_inverse 1);
+qed "inj_onto_Abs_Integ";
+
+val intrel_ss =
+ arith_ss addsimps [equiv_intrel_iff, inj_onto_Abs_Integ RS inj_onto_iff,
+ intrel_iff, intrel_in_integ, Abs_Integ_inverse];
+
+goal Integ.thy "inj(Rep_Integ)";
+by (rtac inj_inverseI 1);
+by (rtac Rep_Integ_inverse 1);
+qed "inj_Rep_Integ";
+
+
+
+
+(** znat: the injection from nat to Integ **)
+
+goal Integ.thy "inj(znat)";
+by (rtac injI 1);
+by (rewtac znat_def);
+by (dtac (inj_onto_Abs_Integ RS inj_ontoD) 1);
+by (REPEAT (rtac intrel_in_integ 1));
+by (dtac eq_equiv_class 1);
+by (rtac equiv_intrel 1);
+by (fast_tac set_cs 1);
+by (safe_tac intrel_cs);
+by (asm_full_simp_tac arith_ss 1);
+qed "inj_znat";
+
+
+(**** zminus: unary negation on Integ ****)
+
+goalw Integ.thy [congruent_def]
+ "congruent(intrel,%p. split(%x y. intrel^^{<y,x>},p))";
+by (safe_tac intrel_cs);
+by (asm_simp_tac (intrel_ss addsimps add_ac) 1);
+qed "zminus_congruent";
+
+
+(*Resolve th against the corresponding facts for zminus*)
+val zminus_ize = RSLIST [equiv_intrel, zminus_congruent];
+
+goalw Integ.thy [zminus_def]
+ "$~ Abs_Integ(intrel^^{<x,y>}) = Abs_Integ(intrel ^^ {<y,x>})";
+by (res_inst_tac [("f","Abs_Integ")] arg_cong 1);
+by (simp_tac (set_ss addsimps
+ [intrel_in_integ RS Abs_Integ_inverse,zminus_ize UN_equiv_class]) 1);
+by (rewtac split_def);
+by (simp_tac prod_ss 1);
+qed "zminus";
+
+(*by lcp*)
+val [prem] = goal Integ.thy
+ "(!!x y. z = Abs_Integ(intrel^^{<x,y>}) ==> P) ==> P";
+by (res_inst_tac [("x1","z")]
+ (rewrite_rule [Integ_def] Rep_Integ RS quotientE) 1);
+by (dres_inst_tac [("f","Abs_Integ")] arg_cong 1);
+by (res_inst_tac [("p","x")] PairE 1);
+by (rtac prem 1);
+by (asm_full_simp_tac (HOL_ss addsimps [Rep_Integ_inverse]) 1);
+qed "eq_Abs_Integ";
+
+goal Integ.thy "$~ ($~ z) = z";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (asm_simp_tac (HOL_ss addsimps [zminus]) 1);
+qed "zminus_zminus";
+
+goal Integ.thy "inj(zminus)";
+by (rtac injI 1);
+by (dres_inst_tac [("f","zminus")] arg_cong 1);
+by (asm_full_simp_tac (HOL_ss addsimps [zminus_zminus]) 1);
+qed "inj_zminus";
+
+goalw Integ.thy [znat_def] "$~ ($#0) = $#0";
+by (simp_tac (arith_ss addsimps [zminus]) 1);
+qed "zminus_0";
+
+
+(**** znegative: the test for negative integers ****)
+
+goal Arith.thy "!!m x n::nat. n+m=x ==> m<=x";
+by (dtac (disjI2 RS less_or_eq_imp_le) 1);
+by (asm_full_simp_tac (arith_ss addsimps add_ac) 1);
+by (dtac add_leD1 1);
+by (assume_tac 1);
+qed "not_znegative_znat_lemma";
+
+
+goalw Integ.thy [znegative_def, znat_def]
+ "~ znegative($# n)";
+by (simp_tac intrel_ss 1);
+by (safe_tac intrel_cs);
+by (rtac ccontr 1);
+by (etac notE 1);
+by (asm_full_simp_tac arith_ss 1);
+by (dtac not_znegative_znat_lemma 1);
+by (fast_tac (HOL_cs addDs [leD]) 1);
+qed "not_znegative_znat";
+
+goalw Integ.thy [znegative_def, znat_def] "znegative($~ $# Suc(n))";
+by (simp_tac (intrel_ss addsimps [zminus]) 1);
+by (REPEAT (ares_tac [exI, conjI] 1));
+by (rtac (intrelI RS ImageI) 2);
+by (rtac singletonI 3);
+by (simp_tac arith_ss 2);
+by (rtac less_add_Suc1 1);
+qed "znegative_zminus_znat";
+
+
+(**** zmagnitude: magnitide of an integer, as a natural number ****)
+
+goal Arith.thy "!!n::nat. n - Suc(n+m)=0";
+by (nat_ind_tac "n" 1);
+by (ALLGOALS(asm_simp_tac arith_ss));
+qed "diff_Suc_add_0";
+
+goal Arith.thy "Suc((n::nat)+m)-n=Suc(m)";
+by (nat_ind_tac "n" 1);
+by (ALLGOALS(asm_simp_tac arith_ss));
+qed "diff_Suc_add_inverse";
+
+goalw Integ.thy [congruent_def]
+ "congruent(intrel, split(%x y. intrel^^{<(y-x) + (x-(y::nat)),0>}))";
+by (safe_tac intrel_cs);
+by (asm_simp_tac intrel_ss 1);
+by (etac rev_mp 1);
+by (res_inst_tac [("m","x1"),("n","y1")] diff_induct 1);
+by (asm_simp_tac (arith_ss addsimps [inj_Suc RS inj_eq]) 3);
+by (asm_simp_tac (arith_ss addsimps [diff_add_inverse,diff_add_0]) 2);
+by (asm_simp_tac arith_ss 1);
+by (rtac impI 1);
+by (etac subst 1);
+by (res_inst_tac [("m1","x")] (add_commute RS ssubst) 1);
+by (asm_simp_tac (arith_ss addsimps [diff_add_inverse,diff_add_0]) 1);
+by (rtac impI 1);
+by (asm_simp_tac (arith_ss addsimps
+ [diff_add_inverse, diff_add_0, diff_Suc_add_0,
+ diff_Suc_add_inverse]) 1);
+qed "zmagnitude_congruent";
+
+(*Resolve th against the corresponding facts for zmagnitude*)
+val zmagnitude_ize = RSLIST [equiv_intrel, zmagnitude_congruent];
+
+
+goalw Integ.thy [zmagnitude_def]
+ "zmagnitude (Abs_Integ(intrel^^{<x,y>})) = \
+\ Abs_Integ(intrel^^{<(y - x) + (x - y),0>})";
+by (res_inst_tac [("f","Abs_Integ")] arg_cong 1);
+by (asm_simp_tac (intrel_ss addsimps [zmagnitude_ize UN_equiv_class]) 1);
+qed "zmagnitude";
+
+goalw Integ.thy [znat_def] "zmagnitude($# n) = $#n";
+by (asm_simp_tac (intrel_ss addsimps [zmagnitude]) 1);
+qed "zmagnitude_znat";
+
+goalw Integ.thy [znat_def] "zmagnitude($~ $# n) = $#n";
+by (asm_simp_tac (intrel_ss addsimps [zmagnitude, zminus]) 1);
+qed "zmagnitude_zminus_znat";
+
+
+(**** zadd: addition on Integ ****)
+
+(** Congruence property for addition **)
+
+goalw Integ.thy [congruent2_def]
+ "congruent2(intrel, %p1 p2. \
+\ split(%x1 y1. split(%x2 y2. intrel^^{<x1+x2, y1+y2>},p2),p1))";
+(*Proof via congruent2_commuteI seems longer*)
+by (safe_tac intrel_cs);
+by (asm_simp_tac (intrel_ss addsimps [add_assoc]) 1);
+(*The rest should be trivial, but rearranging terms is hard*)
+by (res_inst_tac [("x1","x1a")] (add_left_commute RS ssubst) 1);
+by (asm_simp_tac (arith_ss addsimps [add_assoc RS sym]) 1);
+by (asm_simp_tac (arith_ss addsimps add_ac) 1);
+qed "zadd_congruent2";
+
+(*Resolve th against the corresponding facts for zadd*)
+val zadd_ize = RSLIST [equiv_intrel, zadd_congruent2];
+
+goalw Integ.thy [zadd_def]
+ "Abs_Integ(intrel^^{<x1,y1>}) + Abs_Integ(intrel^^{<x2,y2>}) = \
+\ Abs_Integ(intrel^^{<x1+x2, y1+y2>})";
+by (asm_simp_tac
+ (intrel_ss addsimps [zadd_ize UN_equiv_class2]) 1);
+qed "zadd";
+
+goalw Integ.thy [znat_def] "$#0 + z = z";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (asm_simp_tac (arith_ss addsimps [zadd]) 1);
+qed "zadd_0";
+
+goal Integ.thy "$~ (z + w) = $~ z + $~ w";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (asm_simp_tac (arith_ss addsimps [zminus,zadd]) 1);
+qed "zminus_zadd_distrib";
+
+goal Integ.thy "(z::int) + w = w + z";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (asm_simp_tac (intrel_ss addsimps (add_ac @ [zadd])) 1);
+qed "zadd_commute";
+
+goal Integ.thy "((z1::int) + z2) + z3 = z1 + (z2 + z3)";
+by (res_inst_tac [("z","z1")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","z2")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","z3")] eq_Abs_Integ 1);
+by (asm_simp_tac (arith_ss addsimps [zadd, add_assoc]) 1);
+qed "zadd_assoc";
+
+(*For AC rewriting*)
+goal Integ.thy "(x::int)+(y+z)=y+(x+z)";
+by (rtac (zadd_commute RS trans) 1);
+by (rtac (zadd_assoc RS trans) 1);
+by (rtac (zadd_commute RS arg_cong) 1);
+qed "zadd_left_commute";
+
+(*Integer addition is an AC operator*)
+val zadd_ac = [zadd_assoc,zadd_commute,zadd_left_commute];
+
+goalw Integ.thy [znat_def] "$# (m + n) = ($#m) + ($#n)";
+by (asm_simp_tac (arith_ss addsimps [zadd]) 1);
+qed "znat_add";
+
+goalw Integ.thy [znat_def] "z + ($~ z) = $#0";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (asm_simp_tac (intrel_ss addsimps [zminus, zadd, add_commute]) 1);
+qed "zadd_zminus_inverse";
+
+goal Integ.thy "($~ z) + z = $#0";
+by (rtac (zadd_commute RS trans) 1);
+by (rtac zadd_zminus_inverse 1);
+qed "zadd_zminus_inverse2";
+
+goal Integ.thy "z + $#0 = z";
+by (rtac (zadd_commute RS trans) 1);
+by (rtac zadd_0 1);
+qed "zadd_0_right";
+
+
+(*Need properties of subtraction? Or use $- just as an abbreviation!*)
+
+(**** zmult: multiplication on Integ ****)
+
+(** Congruence property for multiplication **)
+
+goal Integ.thy "((k::nat) + l) + (m + n) = (k + m) + (n + l)";
+by (simp_tac (arith_ss addsimps add_ac) 1);
+qed "zmult_congruent_lemma";
+
+goal Integ.thy
+ "congruent2(intrel, %p1 p2. \
+\ split(%x1 y1. split(%x2 y2. \
+\ intrel^^{<x1*x2 + y1*y2, x1*y2 + y1*x2>}, p2), p1))";
+by (rtac (equiv_intrel RS congruent2_commuteI) 1);
+by (safe_tac intrel_cs);
+by (rewtac split_def);
+by (simp_tac (arith_ss addsimps add_ac@mult_ac) 1);
+by (asm_simp_tac (arith_ss addsimps add_ac@mult_ac) 1);
+by (rtac (intrelI RS(equiv_intrel RS equiv_class_eq)) 1);
+by (rtac (zmult_congruent_lemma RS trans) 1);
+by (rtac (zmult_congruent_lemma RS trans RS sym) 1);
+by (rtac (zmult_congruent_lemma RS trans RS sym) 1);
+by (rtac (zmult_congruent_lemma RS trans RS sym) 1);
+by (asm_simp_tac (HOL_ss addsimps [add_mult_distrib RS sym]) 1);
+by (asm_simp_tac (arith_ss addsimps add_ac@mult_ac) 1);
+qed "zmult_congruent2";
+
+(*Resolve th against the corresponding facts for zmult*)
+val zmult_ize = RSLIST [equiv_intrel, zmult_congruent2];
+
+goalw Integ.thy [zmult_def]
+ "Abs_Integ((intrel^^{<x1,y1>})) * Abs_Integ((intrel^^{<x2,y2>})) = \
+\ Abs_Integ(intrel ^^ {<x1*x2 + y1*y2, x1*y2 + y1*x2>})";
+by (simp_tac (intrel_ss addsimps [zmult_ize UN_equiv_class2]) 1);
+qed "zmult";
+
+goalw Integ.thy [znat_def] "$#0 * z = $#0";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (asm_simp_tac (arith_ss addsimps [zmult]) 1);
+qed "zmult_0";
+
+goalw Integ.thy [znat_def] "$#Suc(0) * z = z";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (asm_simp_tac (arith_ss addsimps [zmult, add_0_right]) 1);
+qed "zmult_1";
+
+goal Integ.thy "($~ z) * w = $~ (z * w)";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (asm_simp_tac (intrel_ss addsimps ([zminus, zmult] @ add_ac)) 1);
+qed "zmult_zminus";
+
+
+goal Integ.thy "($~ z) * ($~ w) = (z * w)";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (asm_simp_tac (intrel_ss addsimps ([zminus, zmult] @ add_ac)) 1);
+qed "zmult_zminus_zminus";
+
+goal Integ.thy "(z::int) * w = w * z";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (asm_simp_tac (intrel_ss addsimps ([zmult] @ add_ac @ mult_ac)) 1);
+qed "zmult_commute";
+
+goal Integ.thy "z * $# 0 = $#0";
+by (rtac ([zmult_commute, zmult_0] MRS trans) 1);
+qed "zmult_0_right";
+
+goal Integ.thy "z * $#Suc(0) = z";
+by (rtac ([zmult_commute, zmult_1] MRS trans) 1);
+qed "zmult_1_right";
+
+goal Integ.thy "((z1::int) * z2) * z3 = z1 * (z2 * z3)";
+by (res_inst_tac [("z","z1")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","z2")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","z3")] eq_Abs_Integ 1);
+by (asm_simp_tac (intrel_ss addsimps ([zmult] @ add_ac @ mult_ac)) 1);
+qed "zmult_assoc";
+
+(*For AC rewriting*)
+qed_goal "zmult_left_commute" Integ.thy
+ "(z1::int)*(z2*z3) = z2*(z1*z3)"
+ (fn _ => [rtac (zmult_commute RS trans) 1, rtac (zmult_assoc RS trans) 1,
+ rtac (zmult_commute RS arg_cong) 1]);
+
+(*Integer multiplication is an AC operator*)
+val zmult_ac = [zmult_assoc, zmult_commute, zmult_left_commute];
+
+goal Integ.thy "((z1::int) + z2) * w = (z1 * w) + (z2 * w)";
+by (res_inst_tac [("z","z1")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","z2")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (asm_simp_tac
+ (intrel_ss addsimps ([zadd, zmult, add_mult_distrib] @
+ add_ac @ mult_ac)) 1);
+qed "zadd_zmult_distrib";
+
+val zmult_commute'= read_instantiate [("z","w")] zmult_commute;
+
+goal Integ.thy "w * ($~ z) = $~ (w * z)";
+by (simp_tac (HOL_ss addsimps [zmult_commute', zmult_zminus]) 1);
+qed "zmult_zminus_right";
+
+goal Integ.thy "(w::int) * (z1 + z2) = (w * z1) + (w * z2)";
+by (simp_tac (HOL_ss addsimps [zmult_commute',zadd_zmult_distrib]) 1);
+qed "zadd_zmult_distrib2";
+
+val zadd_simps =
+ [zadd_0, zadd_0_right, zadd_zminus_inverse, zadd_zminus_inverse2];
+
+val zminus_simps = [zminus_zminus, zminus_0, zminus_zadd_distrib];
+
+val zmult_simps = [zmult_0, zmult_1, zmult_0_right, zmult_1_right,
+ zmult_zminus, zmult_zminus_right];
+
+val integ_ss =
+ arith_ss addsimps (zadd_simps @ zminus_simps @ zmult_simps @
+ [zmagnitude_znat, zmagnitude_zminus_znat]);
+
+
+(**** Additional Theorems (by Mattolini; proofs mainly by lcp) ****)
+
+(* Some Theorems about zsuc and zpred *)
+goalw Integ.thy [zsuc_def] "$#(Suc(n)) = zsuc($# n)";
+by (simp_tac (arith_ss addsimps [znat_add RS sym]) 1);
+qed "znat_Suc";
+
+goalw Integ.thy [zpred_def,zsuc_def,zdiff_def] "$~ zsuc(z) = zpred($~ z)";
+by (simp_tac integ_ss 1);
+qed "zminus_zsuc";
+
+goalw Integ.thy [zpred_def,zsuc_def,zdiff_def] "$~ zpred(z) = zsuc($~ z)";
+by (simp_tac integ_ss 1);
+qed "zminus_zpred";
+
+goalw Integ.thy [zsuc_def,zpred_def,zdiff_def]
+ "zpred(zsuc(z)) = z";
+by (simp_tac (integ_ss addsimps [zadd_assoc]) 1);
+qed "zpred_zsuc";
+
+goalw Integ.thy [zsuc_def,zpred_def,zdiff_def]
+ "zsuc(zpred(z)) = z";
+by (simp_tac (integ_ss addsimps [zadd_assoc]) 1);
+qed "zsuc_zpred";
+
+goal Integ.thy "(zpred(z)=w) = (z=zsuc(w))";
+by (safe_tac HOL_cs);
+by (rtac (zsuc_zpred RS sym) 1);
+by (rtac zpred_zsuc 1);
+qed "zpred_to_zsuc";
+
+goal Integ.thy "(zsuc(z)=w)=(z=zpred(w))";
+by (safe_tac HOL_cs);
+by (rtac (zpred_zsuc RS sym) 1);
+by (rtac zsuc_zpred 1);
+qed "zsuc_to_zpred";
+
+goal Integ.thy "($~ z = w) = (z = $~ w)";
+by (safe_tac HOL_cs);
+by (rtac (zminus_zminus RS sym) 1);
+by (rtac zminus_zminus 1);
+qed "zminus_exchange";
+
+goal Integ.thy"(zsuc(z)=zsuc(w)) = (z=w)";
+by (safe_tac intrel_cs);
+by (dres_inst_tac [("f","zpred")] arg_cong 1);
+by (asm_full_simp_tac (HOL_ss addsimps [zpred_zsuc]) 1);
+qed "bijective_zsuc";
+
+goal Integ.thy"(zpred(z)=zpred(w)) = (z=w)";
+by (safe_tac intrel_cs);
+by (dres_inst_tac [("f","zsuc")] arg_cong 1);
+by (asm_full_simp_tac (HOL_ss addsimps [zsuc_zpred]) 1);
+qed "bijective_zpred";
+
+(* Additional Theorems about zadd *)
+
+goalw Integ.thy [zsuc_def] "zsuc(z) + w = zsuc(z+w)";
+by (simp_tac (arith_ss addsimps zadd_ac) 1);
+qed "zadd_zsuc";
+
+goalw Integ.thy [zsuc_def] "w + zsuc(z) = zsuc(w+z)";
+by (simp_tac (arith_ss addsimps zadd_ac) 1);
+qed "zadd_zsuc_right";
+
+goalw Integ.thy [zpred_def,zdiff_def] "zpred(z) + w = zpred(z+w)";
+by (simp_tac (arith_ss addsimps zadd_ac) 1);
+qed "zadd_zpred";
+
+goalw Integ.thy [zpred_def,zdiff_def] "w + zpred(z) = zpred(w+z)";
+by (simp_tac (arith_ss addsimps zadd_ac) 1);
+qed "zadd_zpred_right";
+
+
+(* Additional Theorems about zmult *)
+
+goalw Integ.thy [zsuc_def] "zsuc(w) * z = z + w * z";
+by (simp_tac (integ_ss addsimps [zadd_zmult_distrib, zadd_commute]) 1);
+qed "zmult_zsuc";
+
+goalw Integ.thy [zsuc_def] "z * zsuc(w) = z + w * z";
+by (simp_tac
+ (integ_ss addsimps [zadd_zmult_distrib2, zadd_commute, zmult_commute]) 1);
+qed "zmult_zsuc_right";
+
+goalw Integ.thy [zpred_def, zdiff_def] "zpred(w) * z = w * z - z";
+by (simp_tac (integ_ss addsimps [zadd_zmult_distrib]) 1);
+qed "zmult_zpred";
+
+goalw Integ.thy [zpred_def, zdiff_def] "z * zpred(w) = w * z - z";
+by (simp_tac (integ_ss addsimps [zadd_zmult_distrib2, zmult_commute]) 1);
+qed "zmult_zpred_right";
+
+(* Further Theorems about zsuc and zpred *)
+goal Integ.thy "$#Suc(m) ~= $#0";
+by (simp_tac (integ_ss addsimps [inj_znat RS inj_eq]) 1);
+qed "znat_Suc_not_znat_Zero";
+
+bind_thm ("znat_Zero_not_znat_Suc", (znat_Suc_not_znat_Zero RS not_sym));
+
+
+goalw Integ.thy [zsuc_def,znat_def] "w ~= zsuc(w)";
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (asm_full_simp_tac (intrel_ss addsimps [zadd]) 1);
+qed "n_not_zsuc_n";
+
+val zsuc_n_not_n = n_not_zsuc_n RS not_sym;
+
+goal Integ.thy "w ~= zpred(w)";
+by (safe_tac HOL_cs);
+by (dres_inst_tac [("x","w"),("f","zsuc")] arg_cong 1);
+by (asm_full_simp_tac (HOL_ss addsimps [zsuc_zpred,zsuc_n_not_n]) 1);
+qed "n_not_zpred_n";
+
+val zpred_n_not_n = n_not_zpred_n RS not_sym;
+
+
+(* Theorems about less and less_equal *)
+
+goalw Integ.thy [zless_def, znegative_def, zdiff_def, znat_def]
+ "!!w. w<z ==> ? n. z = w + $#(Suc(n))";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (safe_tac intrel_cs);
+by (asm_full_simp_tac (intrel_ss addsimps [zadd, zminus]) 1);
+by (safe_tac (intrel_cs addSDs [less_eq_Suc_add]));
+by (res_inst_tac [("x","k")] exI 1);
+by (asm_full_simp_tac (HOL_ss addsimps ([add_Suc RS sym] @ add_ac)) 1);
+(*To cancel x2, rename it to be first!*)
+by (rename_tac "a b c" 1);
+by (asm_full_simp_tac (HOL_ss addsimps (add_left_cancel::add_ac)) 1);
+qed "zless_eq_zadd_Suc";
+
+goalw Integ.thy [zless_def, znegative_def, zdiff_def, znat_def]
+ "z < z + $#(Suc(n))";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (safe_tac intrel_cs);
+by (simp_tac (intrel_ss addsimps [zadd, zminus]) 1);
+by (REPEAT_SOME (ares_tac [refl, exI, singletonI, ImageI, conjI, intrelI]));
+by (rtac le_less_trans 1);
+by (rtac lessI 2);
+by (asm_simp_tac (arith_ss addsimps ([le_add1,add_left_cancel_le]@add_ac)) 1);
+qed "zless_zadd_Suc";
+
+goal Integ.thy "!!z1 z2 z3. [| z1<z2; z2<z3 |] ==> z1 < (z3::int)";
+by (safe_tac (HOL_cs addSDs [zless_eq_zadd_Suc]));
+by (simp_tac
+ (arith_ss addsimps [zadd_assoc, zless_zadd_Suc, znat_add RS sym]) 1);
+qed "zless_trans";
+
+goalw Integ.thy [zsuc_def] "z<zsuc(z)";
+by (rtac zless_zadd_Suc 1);
+qed "zlessI";
+
+val zless_zsucI = zlessI RSN (2,zless_trans);
+
+goal Integ.thy "!!z w::int. z<w ==> ~w<z";
+by (safe_tac (HOL_cs addSDs [zless_eq_zadd_Suc]));
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (safe_tac intrel_cs);
+by (asm_full_simp_tac (intrel_ss addsimps ([znat_def, zadd])) 1);
+by (asm_full_simp_tac
+ (HOL_ss addsimps [add_left_cancel, add_assoc, add_Suc_right RS sym]) 1);
+by (resolve_tac [less_not_refl2 RS notE] 1);
+by (etac sym 2);
+by (REPEAT (resolve_tac [lessI, trans_less_add2, less_SucI] 1));
+qed "zless_not_sym";
+
+(* [| n<m; m<n |] ==> R *)
+bind_thm ("zless_asym", (zless_not_sym RS notE));
+
+goal Integ.thy "!!z::int. ~ z<z";
+by (resolve_tac [zless_asym RS notI] 1);
+by (REPEAT (assume_tac 1));
+qed "zless_not_refl";
+
+(* z<z ==> R *)
+bind_thm ("zless_anti_refl", (zless_not_refl RS notE));
+
+goal Integ.thy "!!w. z<w ==> w ~= (z::int)";
+by(fast_tac (HOL_cs addEs [zless_anti_refl]) 1);
+qed "zless_not_refl2";
+
+
+(*"Less than" is a linear ordering*)
+goalw Integ.thy [zless_def, znegative_def, zdiff_def]
+ "z<w | z=w | w<(z::int)";
+by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
+by (res_inst_tac [("z","w")] eq_Abs_Integ 1);
+by (safe_tac intrel_cs);
+by (asm_full_simp_tac
+ (intrel_ss addsimps [zadd, zminus, Image_iff, Bex_def]) 1);
+by (res_inst_tac [("m1", "x+ya"), ("n1", "xa+y")] (less_linear RS disjE) 1);
+by (etac disjE 2);
+by (assume_tac 2);
+by (DEPTH_SOLVE
+ (swap_res_tac [exI] 1 THEN
+ swap_res_tac [exI] 1 THEN
+ etac conjI 1 THEN
+ simp_tac (arith_ss addsimps add_ac) 1));
+qed "zless_linear";
+
+
+(*** Properties of <= ***)
+
+goalw Integ.thy [zless_def, znegative_def, zdiff_def, znat_def]
+ "($#m < $#n) = (m<n)";
+by (simp_tac
+ (intrel_ss addsimps [zadd, zminus, Image_iff, Bex_def]) 1);
+by (fast_tac (HOL_cs addIs [add_commute] addSEs [less_add_eq_less]) 1);
+qed "zless_eq_less";
+
+goalw Integ.thy [zle_def, le_def] "($#m <= $#n) = (m<=n)";
+by (simp_tac (HOL_ss addsimps [zless_eq_less]) 1);
+qed "zle_eq_le";
+
+goalw Integ.thy [zle_def] "!!w. ~(w<z) ==> z<=(w::int)";
+by (assume_tac 1);
+qed "zleI";
+
+goalw Integ.thy [zle_def] "!!w. z<=w ==> ~(w<(z::int))";
+by (assume_tac 1);
+qed "zleD";
+
+val zleE = make_elim zleD;
+
+goalw Integ.thy [zle_def] "!!z. ~ z <= w ==> w<(z::int)";
+by (fast_tac HOL_cs 1);
+qed "not_zleE";
+
+goalw Integ.thy [zle_def] "!!z. z < w ==> z <= (w::int)";
+by (fast_tac (HOL_cs addEs [zless_asym]) 1);
+qed "zless_imp_zle";
+
+goalw Integ.thy [zle_def] "!!z. z <= w ==> z < w | z=(w::int)";
+by (cut_facts_tac [zless_linear] 1);
+by (fast_tac (HOL_cs addEs [zless_anti_refl,zless_asym]) 1);
+qed "zle_imp_zless_or_eq";
+
+goalw Integ.thy [zle_def] "!!z. z<w | z=w ==> z <=(w::int)";
+by (cut_facts_tac [zless_linear] 1);
+by (fast_tac (HOL_cs addEs [zless_anti_refl,zless_asym]) 1);
+qed "zless_or_eq_imp_zle";
+
+goal Integ.thy "(x <= (y::int)) = (x < y | x=y)";
+by (REPEAT(ares_tac [iffI, zless_or_eq_imp_zle, zle_imp_zless_or_eq] 1));
+qed "zle_eq_zless_or_eq";
+
+goal Integ.thy "w <= (w::int)";
+by (simp_tac (HOL_ss addsimps [zle_eq_zless_or_eq]) 1);
+qed "zle_refl";
+
+val prems = goal Integ.thy "!!i. [| i <= j; j < k |] ==> i < (k::int)";
+by (dtac zle_imp_zless_or_eq 1);
+by (fast_tac (HOL_cs addIs [zless_trans]) 1);
+qed "zle_zless_trans";
+
+goal Integ.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::int)";
+by (EVERY1 [dtac zle_imp_zless_or_eq, dtac zle_imp_zless_or_eq,
+ rtac zless_or_eq_imp_zle, fast_tac (HOL_cs addIs [zless_trans])]);
+qed "zle_trans";
+
+goal Integ.thy "!!z. [| z <= w; w <= z |] ==> z = (w::int)";
+by (EVERY1 [dtac zle_imp_zless_or_eq, dtac zle_imp_zless_or_eq,
+ fast_tac (HOL_cs addEs [zless_anti_refl,zless_asym])]);
+qed "zle_anti_sym";
+
+
+goal Integ.thy "!!w w' z::int. z + w' = z + w ==> w' = w";
+by (dres_inst_tac [("f", "%x. x + $~z")] arg_cong 1);
+by (asm_full_simp_tac (integ_ss addsimps zadd_ac) 1);
+qed "zadd_left_cancel";
+
+
+(*** Monotonicity results ***)
+
+goal Integ.thy "!!v w z::int. v < w ==> v + z < w + z";
+by (safe_tac (HOL_cs addSDs [zless_eq_zadd_Suc]));
+by (simp_tac (HOL_ss addsimps zadd_ac) 1);
+by (simp_tac (HOL_ss addsimps [zadd_assoc RS sym, zless_zadd_Suc]) 1);
+qed "zadd_zless_mono1";
+
+goal Integ.thy "!!v w z::int. (v+z < w+z) = (v < w)";
+by (safe_tac (HOL_cs addSEs [zadd_zless_mono1]));
+by (dres_inst_tac [("z", "$~z")] zadd_zless_mono1 1);
+by (asm_full_simp_tac (integ_ss addsimps [zadd_assoc]) 1);
+qed "zadd_left_cancel_zless";
+
+goal Integ.thy "!!v w z::int. (v+z <= w+z) = (v <= w)";
+by (asm_full_simp_tac
+ (integ_ss addsimps [zle_def, zadd_left_cancel_zless]) 1);
+qed "zadd_left_cancel_zle";
+
+(*"v<=w ==> v+z <= w+z"*)
+bind_thm ("zadd_zle_mono1", zadd_left_cancel_zle RS iffD2);
+
+
+goal Integ.thy "!!z' z::int. [| w'<=w; z'<=z |] ==> w' + z' <= w + z";
+by (etac (zadd_zle_mono1 RS zle_trans) 1);
+by (simp_tac (HOL_ss addsimps [zadd_commute]) 1);
+(*w moves to the end because it is free while z', z are bound*)
+by (etac zadd_zle_mono1 1);
+qed "zadd_zle_mono";
+
+goal Integ.thy "!!w z::int. z<=$#0 ==> w+z <= w";
+by (dres_inst_tac [("z", "w")] zadd_zle_mono1 1);
+by (asm_full_simp_tac (integ_ss addsimps [zadd_commute]) 1);
+qed "zadd_zle_self";