--- a/src/ZF/ex/Integ.ML Thu Jun 23 17:38:12 1994 +0200
+++ b/src/ZF/ex/Integ.ML Thu Jun 23 17:52:58 1994 +0200
@@ -20,40 +20,34 @@
(*** Proving that intrel is an equivalence relation ***)
-val prems = goal Arith.thy
- "[| m #+ n = m' #+ n'; m: nat; m': nat |] \
-\ ==> m #+ (n #+ k) = m' #+ (n' #+ k)";
-by (asm_simp_tac (arith_ss addsimps ([add_assoc RS sym] @ prems)) 1);
-val add_assoc_cong = result();
-
-val prems = goal Arith.thy
- "[| m: nat; n: nat |] \
-\ ==> m #+ (n #+ k) = n #+ (m #+ k)";
-by (REPEAT (resolve_tac ([add_commute RS add_assoc_cong] @ prems) 1));
-val add_assoc_swap = result();
-
val add_kill = (refl RS add_cong);
-val add_assoc_swap_kill = add_kill RSN (3, add_assoc_swap RS trans);
+val add_left_commute_kill = add_kill RSN (3, add_left_commute RS trans);
(*By luck, requires no typing premises for y1, y2,y3*)
val eqa::eqb::prems = goal Arith.thy
"[| x1 #+ y2 = x2 #+ y1; x2 #+ y3 = x3 #+ y2; \
\ x1: nat; x2: nat; x3: nat |] ==> x1 #+ y3 = x3 #+ y1";
by (res_inst_tac [("k","x2")] add_left_cancel 1);
-by (resolve_tac prems 1);
-by (rtac (add_assoc_swap RS trans) 1 THEN typechk_tac prems);
+by (resolve_tac prems 2);
+by (rtac (add_left_commute RS trans) 1 THEN typechk_tac prems);
by (rtac (eqb RS ssubst) 1);
-by (rtac (add_assoc_swap RS trans) 1 THEN typechk_tac prems);
+by (rtac (add_left_commute RS trans) 1 THEN typechk_tac prems);
by (rtac (eqa RS ssubst) 1);
-by (rtac (add_assoc_swap) 1 THEN typechk_tac prems);
+by (rtac (add_left_commute) 1 THEN typechk_tac prems);
val integ_trans_lemma = result();
(** Natural deduction for intrel **)
-val prems = goalw Integ.thy [intrel_def]
- "[| x1#+y2 = x2#+y1; x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
-\ <<x1,y1>,<x2,y2>>: intrel";
+goalw Integ.thy [intrel_def]
+ "<<x1,y1>,<x2,y2>>: intrel <-> \
+\ x1: nat & y1: nat & x2: nat & y2: nat & x1#+y2 = x2#+y1";
+by (fast_tac ZF_cs 1);
+val intrel_iff = result();
+
+goalw Integ.thy [intrel_def]
+ "!!x1 x2. [| x1#+y2 = x2#+y1; x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
+\ <<x1,y1>,<x2,y2>>: intrel";
by (fast_tac (ZF_cs addIs prems) 1);
val intrelI = result();
@@ -76,50 +70,31 @@
val intrel_cs = ZF_cs addSIs [intrelI] addSEs [intrelE];
-goal Integ.thy
- "<<x1,y1>,<x2,y2>>: intrel <-> \
-\ x1#+y2 = x2#+y1 & x1: nat & y1: nat & x2: nat & y2: nat";
-by (fast_tac intrel_cs 1);
-val intrel_iff = result();
-
-val prems = goalw Integ.thy [equiv_def] "equiv(nat*nat, intrel)";
-by (safe_tac intrel_cs);
-by (rewtac refl_def);
-by (fast_tac intrel_cs 1);
-by (rewtac sym_def);
-by (fast_tac (intrel_cs addSEs [sym]) 1);
-by (rewtac trans_def);
-by (fast_tac (intrel_cs addSEs [integ_trans_lemma]) 1);
+goalw Integ.thy [equiv_def, refl_def, sym_def, trans_def]
+ "equiv(nat*nat, intrel)";
+by (fast_tac (intrel_cs addSEs [sym, integ_trans_lemma]) 1);
val equiv_intrel = result();
val intrel_ss =
- arith_ss addsimps [equiv_intrel RS eq_equiv_class_iff, intrel_iff];
+ arith_ss addsimps [equiv_intrel RS eq_equiv_class_iff, intrel_iff,
+ add_0_right, add_succ_right]
+ addcongs [conj_cong];
-(*Roughly twice as fast as simplifying with intrel_ss*)
-fun INTEG_SIMP_TAC ths =
- let val ss = arith_ss addsimps ths
- in fn i =>
- EVERY [asm_simp_tac ss i,
- rtac (intrelI RS (equiv_intrel RS equiv_class_eq)) i,
- typechk_tac (ZF_typechecks@nat_typechecks@arith_typechecks),
- asm_simp_tac ss i]
- end;
-
+val eq_intrelD = equiv_intrel RSN (2,eq_equiv_class);
(** znat: the injection from nat to integ **)
-val prems = goalw Integ.thy [integ_def,quotient_def,znat_def]
- "m : nat ==> $#m : integ";
-by (fast_tac (ZF_cs addSIs (nat_0I::prems)) 1);
+goalw Integ.thy [integ_def,quotient_def,znat_def]
+ "!!m. m : nat ==> $#m : integ";
+by (fast_tac (ZF_cs addSIs [nat_0I]) 1);
val znat_type = result();
-val [major,nnat] = goalw Integ.thy [znat_def]
- "[| $#m = $#n; n: nat |] ==> m=n";
-by (rtac (make_elim (major RS eq_equiv_class)) 1);
-by (rtac equiv_intrel 1);
-by (typechk_tac [nat_0I,nnat,SigmaI]);
-by (safe_tac (intrel_cs addSEs [box_equals,add_0_right]));
+goalw Integ.thy [znat_def]
+ "!!m n. [| $#m = $#n; n: nat |] ==> m=n";
+by (dtac eq_intrelD 1);
+by (typechk_tac [nat_0I, SigmaI]);
+by (asm_full_simp_tac intrel_ss 1);
val znat_inject = result();
@@ -128,36 +103,30 @@
goalw Integ.thy [congruent_def]
"congruent(intrel, split(%x y. intrel``{<y,x>}))";
by (safe_tac intrel_cs);
-by (ALLGOALS (asm_simp_tac intrel_ss));
-by (etac (box_equals RS sym) 1);
-by (REPEAT (ares_tac [add_commute] 1));
+by (asm_full_simp_tac (intrel_ss addsimps add_ac) 1);
val zminus_congruent = result();
(*Resolve th against the corresponding facts for zminus*)
val zminus_ize = RSLIST [equiv_intrel, zminus_congruent];
-val [prem] = goalw Integ.thy [integ_def,zminus_def]
- "z : integ ==> $~z : integ";
-by (typechk_tac [split_type, SigmaI, prem, zminus_ize UN_equiv_class_type,
+goalw Integ.thy [integ_def,zminus_def]
+ "!!z. z : integ ==> $~z : integ";
+by (typechk_tac [split_type, SigmaI, zminus_ize UN_equiv_class_type,
quotientI]);
val zminus_type = result();
-val major::prems = goalw Integ.thy [integ_def,zminus_def]
- "[| $~z = $~w; z: integ; w: integ |] ==> z=w";
-by (rtac (major RS zminus_ize UN_equiv_class_inject) 1);
-by (REPEAT (ares_tac prems 1));
-by (REPEAT (etac SigmaE 1));
-by (etac rev_mp 1);
-by (asm_simp_tac ZF_ss 1);
-by (fast_tac (intrel_cs addSIs [SigmaI, equiv_intrel]
- addSEs [box_equals RS sym, add_commute,
- make_elim eq_equiv_class]) 1);
+goalw Integ.thy [integ_def,zminus_def]
+ "!!z w. [| $~z = $~w; z: integ; w: integ |] ==> z=w";
+by (etac (zminus_ize UN_equiv_class_inject) 1);
+by (safe_tac intrel_cs);
+(*The setloop is only needed because assumptions are in the wrong order!*)
+by (asm_full_simp_tac (intrel_ss addsimps add_ac
+ setloop dtac eq_intrelD) 1);
val zminus_inject = result();
-val prems = goalw Integ.thy [zminus_def]
- "[| x: nat; y: nat |] ==> $~ (intrel``{<x,y>}) = intrel `` {<y,x>}";
-by (asm_simp_tac
- (ZF_ss addsimps (prems@[zminus_ize UN_equiv_class, SigmaI])) 1);
+goalw Integ.thy [zminus_def]
+ "!!x y.[| x: nat; y: nat |] ==> $~ (intrel``{<x,y>}) = intrel `` {<y,x>}";
+by (asm_simp_tac (ZF_ss addsimps [zminus_ize UN_equiv_class, SigmaI]) 1);
val zminus = result();
goalw Integ.thy [integ_def] "!!z. z : integ ==> $~ ($~ z) = z";
@@ -172,13 +141,10 @@
(**** znegative: the test for negative integers ****)
-goalw Integ.thy [znegative_def, znat_def]
- "~ znegative($# n)";
-by (safe_tac intrel_cs);
-by (rtac (add_le_self2 RS le_imp_not_lt RS notE) 1);
-by (etac ssubst 3);
-by (asm_simp_tac (arith_ss addsimps [add_0_right]) 3);
-by (REPEAT (assume_tac 1));
+goalw Integ.thy [znegative_def, znat_def] "~ znegative($# n)";
+by (asm_full_simp_tac (intrel_ss setloop K(safe_tac intrel_cs)) 1);
+be rev_mp 1;
+by (asm_simp_tac (arith_ss addsimps [add_le_self2 RS le_imp_not_lt]) 1);
val not_znegative_znat = result();
goalw Integ.thy [znegative_def, znat_def]
@@ -197,33 +163,33 @@
by (safe_tac intrel_cs);
by (ALLGOALS (asm_simp_tac intrel_ss));
by (etac rev_mp 1);
-by (res_inst_tac [("m","x1"),("n","y1")] diff_induct 1);
-by (REPEAT (assume_tac 1));
-by (asm_simp_tac (arith_ss addsimps [add_succ_right,succ_inject_iff]) 3);
-by (asm_simp_tac
+by (res_inst_tac [("m","x1"),("n","y1")] diff_induct 1 THEN
+ REPEAT (assume_tac 1));
+by (asm_simp_tac (intrel_ss addsimps [succ_inject_iff]) 3);
+by (asm_simp_tac (*this one's very sensitive to order of rewrites*)
(arith_ss addsimps [diff_add_inverse,diff_add_0,add_0_right]) 2);
-by (asm_simp_tac (arith_ss addsimps [add_0_right]) 1);
+by (asm_simp_tac intrel_ss 1);
by (rtac impI 1);
by (etac subst 1);
-by (res_inst_tac [("m1","x")] (add_commute RS ssubst) 1);
-by (REPEAT (assume_tac 1));
-by (asm_simp_tac (arith_ss addsimps [diff_add_inverse,diff_add_0]) 1);
+by (res_inst_tac [("m1","x")] (add_commute RS ssubst) 1 THEN
+ REPEAT (assume_tac 1));
+by (asm_simp_tac (arith_ss addsimps [diff_add_inverse, diff_add_0]) 1);
val zmagnitude_congruent = result();
+
(*Resolve th against the corresponding facts for zmagnitude*)
val zmagnitude_ize = RSLIST [equiv_intrel, zmagnitude_congruent];
-val [prem] = goalw Integ.thy [integ_def,zmagnitude_def]
- "z : integ ==> zmagnitude(z) : nat";
-by (typechk_tac [split_type, prem, zmagnitude_ize UN_equiv_class_type,
+goalw Integ.thy [integ_def,zmagnitude_def]
+ "!!z. z : integ ==> zmagnitude(z) : nat";
+by (typechk_tac [split_type, zmagnitude_ize UN_equiv_class_type,
add_type, diff_type]);
val zmagnitude_type = result();
-val prems = goalw Integ.thy [zmagnitude_def]
- "[| x: nat; y: nat |] ==> \
-\ zmagnitude (intrel``{<x,y>}) = (y #- x) #+ (x #- y)";
-by (asm_simp_tac
- (ZF_ss addsimps (prems@[zmagnitude_ize UN_equiv_class, SigmaI])) 1);
+goalw Integ.thy [zmagnitude_def]
+ "!!x y. [| x: nat; y: nat |] ==> \
+\ zmagnitude (intrel``{<x,y>}) = (y #- x) #+ (x #- y)";
+by (asm_simp_tac (ZF_ss addsimps [zmagnitude_ize UN_equiv_class, SigmaI]) 1);
val zmagnitude = result();
goalw Integ.thy [znat_def]
@@ -233,7 +199,7 @@
goalw Integ.thy [znat_def]
"!!n. n: nat ==> zmagnitude($~ $# n) = n";
-by (asm_simp_tac (intrel_ss addsimps [zmagnitude, zminus ,add_0_right]) 1);
+by (asm_simp_tac (intrel_ss addsimps [zmagnitude, zminus]) 1);
val zmagnitude_zminus_znat = result();
@@ -246,76 +212,75 @@
\ 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 (INTEG_SIMP_TAC [add_assoc] 1);
-(*The rest should be trivial, but rearranging terms is hard*)
-by (res_inst_tac [("m1","x1a")] (add_assoc_swap RS ssubst) 1);
-by (res_inst_tac [("m1","x2a")] (add_assoc_swap RS ssubst) 3);
+by (asm_simp_tac (intrel_ss addsimps [add_assoc]) 1);
+(*The rest should be trivial, but rearranging terms is hard;
+ add_ac does not help rewriting with the assumptions.*)
+by (res_inst_tac [("m1","x1a")] (add_left_commute RS ssubst) 1);
+by (res_inst_tac [("m1","x2a")] (add_left_commute RS ssubst) 3);
by (typechk_tac [add_type]);
by (asm_simp_tac (arith_ss addsimps [add_assoc RS sym]) 1);
val zadd_congruent2 = result();
+
(*Resolve th against the corresponding facts for zadd*)
val zadd_ize = RSLIST [equiv_intrel, zadd_congruent2];
-val prems = goalw Integ.thy [integ_def,zadd_def]
- "[| z: integ; w: integ |] ==> z $+ w : integ";
-by (REPEAT (ares_tac (prems@[zadd_ize UN_equiv_class_type2,
- split_type, add_type, quotientI, SigmaI]) 1));
+goalw Integ.thy [integ_def,zadd_def]
+ "!!z w. [| z: integ; w: integ |] ==> z $+ w : integ";
+by (REPEAT (ares_tac [zadd_ize UN_equiv_class_type2,
+ split_type, add_type, quotientI, SigmaI] 1));
val zadd_type = result();
-val prems = goalw Integ.thy [zadd_def]
- "[| x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
-\ (intrel``{<x1,y1>}) $+ (intrel``{<x2,y2>}) = intrel `` {<x1#+x2, y1#+y2>}";
-by (asm_simp_tac (ZF_ss addsimps
- (prems @ [zadd_ize UN_equiv_class2, SigmaI])) 1);
+goalw Integ.thy [zadd_def]
+ "!!x1 y1. [| x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
+\ (intrel``{<x1,y1>}) $+ (intrel``{<x2,y2>}) = \
+\ intrel `` {<x1#+x2, y1#+y2>}";
+by (asm_simp_tac (ZF_ss addsimps [zadd_ize UN_equiv_class2, SigmaI]) 1);
val zadd = result();
goalw Integ.thy [integ_def,znat_def] "!!z. z : integ ==> $#0 $+ z = z";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
by (asm_simp_tac (arith_ss addsimps [zadd]) 1);
val zadd_0 = result();
goalw Integ.thy [integ_def]
"!!z w. [| z: integ; w: integ |] ==> $~ (z $+ w) = $~ z $+ $~ w";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
by (asm_simp_tac (arith_ss addsimps [zminus,zadd]) 1);
val zminus_zadd_distrib = result();
goalw Integ.thy [integ_def]
"!!z w. [| z: integ; w: integ |] ==> z $+ w = w $+ z";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (INTEG_SIMP_TAC [zadd] 1);
-by (REPEAT (ares_tac [add_commute,add_cong] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (intrel_ss addsimps (add_ac @ [zadd])) 1);
val zadd_commute = result();
goalw Integ.thy [integ_def]
"!!z1 z2 z3. [| z1: integ; z2: integ; z3: integ |] ==> \
\ (z1 $+ z2) $+ z3 = z1 $+ (z2 $+ z3)";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
(*rewriting is much faster without intrel_iff, etc.*)
-by (asm_simp_tac (arith_ss addsimps [zadd,add_assoc]) 1);
+by (asm_simp_tac (arith_ss addsimps [zadd, add_assoc]) 1);
val zadd_assoc = result();
-val prems = goalw Integ.thy [znat_def]
- "[| m: nat; n: nat |] ==> $# (m #+ n) = ($#m) $+ ($#n)";
-by (asm_simp_tac (arith_ss addsimps (zadd::prems)) 1);
+goalw Integ.thy [znat_def]
+ "!!m n. [| m: nat; n: nat |] ==> $# (m #+ n) = ($#m) $+ ($#n)";
+by (asm_simp_tac (arith_ss addsimps [zadd]) 1);
val znat_add = result();
goalw Integ.thy [integ_def,znat_def] "!!z. z : integ ==> z $+ ($~ z) = $#0";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (asm_simp_tac (intrel_ss addsimps [zminus,zadd,add_0_right]) 1);
-by (REPEAT (ares_tac [add_commute] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (intrel_ss addsimps [zminus, zadd, add_commute]) 1);
val zadd_zminus_inverse = result();
-val prems = goal Integ.thy
- "z : integ ==> ($~ z) $+ z = $#0";
-by (rtac (zadd_commute RS trans) 1);
-by (REPEAT (resolve_tac (prems@[zminus_type, zadd_zminus_inverse]) 1));
+goal Integ.thy "!!z. z : integ ==> ($~ z) $+ z = $#0";
+by (asm_simp_tac
+ (ZF_ss addsimps [zadd_commute, zminus_type, zadd_zminus_inverse]) 1);
val zadd_zminus_inverse2 = result();
-val prems = goal Integ.thy "z:integ ==> z $+ $#0 = z";
+goal Integ.thy "!!z. z:integ ==> z $+ $#0 = z";
by (rtac (zadd_commute RS trans) 1);
-by (REPEAT (resolve_tac (prems@[znat_type,nat_0I,zadd_0]) 1));
+by (REPEAT (ares_tac [znat_type, nat_0I, zadd_0] 1));
val zadd_0_right = result();
@@ -327,102 +292,85 @@
(** Congruence property for multiplication **)
-val prems = goalw Integ.thy [znat_def]
- "[| k: nat; l: nat; m: nat; n: nat |] ==> \
-\ (k #+ l) #+ (m #+ n) = (k #+ m) #+ (n #+ l)";
-val add_commute' = read_instantiate [("m","l")] add_commute;
-by (simp_tac (arith_ss addsimps ([add_commute',add_assoc]@prems)) 1);
-val zmult_congruent_lemma = result();
-
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 (ALLGOALS (INTEG_SIMP_TAC []));
+by (ALLGOALS (asm_simp_tac intrel_ss));
(*Proof that zmult is congruent in one argument*)
-by (rtac (zmult_congruent_lemma RS trans) 2);
-by (rtac (zmult_congruent_lemma RS trans RS sym) 6);
-by (typechk_tac [mult_type]);
-by (asm_simp_tac (arith_ss addsimps [add_mult_distrib_left RS sym]) 2);
+by (asm_simp_tac
+ (arith_ss addsimps (add_ac @ [add_mult_distrib_left RS sym])) 2);
+by (asm_simp_tac
+ (arith_ss addsimps ([add_assoc RS sym, add_mult_distrib_left RS sym])) 2);
(*Proof that zmult is commutative on representatives*)
-by (rtac add_cong 1);
-by (rtac (add_commute RS trans) 2);
-by (REPEAT (ares_tac [mult_commute,add_type,mult_type,add_cong] 1));
+by (asm_simp_tac (arith_ss addsimps (mult_ac@add_ac)) 1);
val zmult_congruent2 = result();
+
(*Resolve th against the corresponding facts for zmult*)
val zmult_ize = RSLIST [equiv_intrel, zmult_congruent2];
-val prems = goalw Integ.thy [integ_def,zmult_def]
- "[| z: integ; w: integ |] ==> z $* w : integ";
-by (REPEAT (ares_tac (prems@[zmult_ize UN_equiv_class_type2,
- split_type, add_type, mult_type,
- quotientI, SigmaI]) 1));
+goalw Integ.thy [integ_def,zmult_def]
+ "!!z w. [| z: integ; w: integ |] ==> z $* w : integ";
+by (REPEAT (ares_tac [zmult_ize UN_equiv_class_type2,
+ split_type, add_type, mult_type,
+ quotientI, SigmaI] 1));
val zmult_type = result();
-
-val prems = goalw Integ.thy [zmult_def]
- "[| x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
-\ (intrel``{<x1,y1>}) $* (intrel``{<x2,y2>}) = \
-\ intrel `` {<x1#*x2 #+ y1#*y2, x1#*y2 #+ y1#*x2>}";
-by (asm_simp_tac (ZF_ss addsimps
- (prems @ [zmult_ize UN_equiv_class2, SigmaI])) 1);
+goalw Integ.thy [zmult_def]
+ "!!x1 x2. [| x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
+\ (intrel``{<x1,y1>}) $* (intrel``{<x2,y2>}) = \
+\ intrel `` {<x1#*x2 #+ y1#*y2, x1#*y2 #+ y1#*x2>}";
+by (asm_simp_tac (ZF_ss addsimps [zmult_ize UN_equiv_class2, SigmaI]) 1);
val zmult = result();
goalw Integ.thy [integ_def,znat_def] "!!z. z : integ ==> $#0 $* z = $#0";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
by (asm_simp_tac (arith_ss addsimps [zmult]) 1);
val zmult_0 = result();
goalw Integ.thy [integ_def,znat_def]
"!!z. z : integ ==> $#1 $* z = z";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (asm_simp_tac (arith_ss addsimps [zmult,add_0_right]) 1);
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (arith_ss addsimps [zmult, add_0_right]) 1);
val zmult_1 = result();
goalw Integ.thy [integ_def]
"!!z w. [| z: integ; w: integ |] ==> ($~ z) $* w = $~ (z $* w)";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (INTEG_SIMP_TAC [zminus,zmult] 1);
-by (REPEAT (ares_tac [mult_type,add_commute,add_cong] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (intrel_ss addsimps ([zminus, zmult] @ add_ac)) 1);
val zmult_zminus = result();
goalw Integ.thy [integ_def]
"!!z w. [| z: integ; w: integ |] ==> ($~ z) $* ($~ w) = (z $* w)";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (INTEG_SIMP_TAC [zminus,zmult] 1);
-by (REPEAT (ares_tac [mult_type,add_commute,add_cong] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (intrel_ss addsimps ([zminus, zmult] @ add_ac)) 1);
val zmult_zminus_zminus = result();
goalw Integ.thy [integ_def]
"!!z w. [| z: integ; w: integ |] ==> z $* w = w $* z";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (INTEG_SIMP_TAC [zmult] 1);
-by (res_inst_tac [("m1","xc #* y")] (add_commute RS ssubst) 1);
-by (REPEAT (ares_tac [mult_type,mult_commute,add_cong] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (intrel_ss addsimps ([zmult] @ add_ac @ mult_ac)) 1);
val zmult_commute = result();
goalw Integ.thy [integ_def]
"!!z1 z2 z3. [| z1: integ; z2: integ; z3: integ |] ==> \
\ (z1 $* z2) $* z3 = z1 $* (z2 $* z3)";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (INTEG_SIMP_TAC [zmult, add_mult_distrib_left,
- add_mult_distrib, add_assoc, mult_assoc] 1);
-(*takes 54 seconds due to wasteful type-checking*)
-by (REPEAT (ares_tac [add_type, mult_type, add_commute, add_kill,
- add_assoc_swap_kill, add_assoc_swap_kill RS sym] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac
+ (intrel_ss addsimps ([zmult, add_mult_distrib_left,
+ add_mult_distrib] @ add_ac @ mult_ac)) 1);
val zmult_assoc = result();
goalw Integ.thy [integ_def]
"!!z1 z2 z3. [| z1: integ; z2: integ; w: integ |] ==> \
\ (z1 $+ z2) $* w = (z1 $* w) $+ (z2 $* w)";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (INTEG_SIMP_TAC [zadd, zmult, add_mult_distrib, add_assoc] 1);
-(*takes 30 seconds due to wasteful type-checking*)
-by (REPEAT (ares_tac [add_type, mult_type, refl, add_commute, add_kill,
- add_assoc_swap_kill, add_assoc_swap_kill RS sym] 1));
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac
+ (intrel_ss addsimps ([zadd, zmult, add_mult_distrib] @
+ add_ac @ mult_ac)) 1);
val zadd_zmult_distrib = result();
val integ_typechecks =