--- a/src/ZF/Integ/Bin.ML Fri Sep 25 12:12:07 1998 +0200
+++ b/src/ZF/Integ/Bin.ML Fri Sep 25 13:18:07 1998 +0200
@@ -99,8 +99,8 @@
val bin_typechecks0 = bin_rec_type :: bin.intrs;
-Goalw [integ_of_def] "w: bin ==> integ_of(w) : integ";
-by (typechk_tac (bin_typechecks0@integ_typechecks@
+Goalw [integ_of_def] "w: bin ==> integ_of(w) : int";
+by (typechk_tac (bin_typechecks0@int_typechecks@
nat_typechecks@[bool_into_nat]));
qed "integ_of_type";
@@ -143,33 +143,13 @@
bin_rec_Pls, bin_rec_Min, bin_rec_Cons] @
bin_recs integ_of_def @ bin_typechecks);
-val typechecks = bin_typechecks @ integ_typechecks @ nat_typechecks @
+val typechecks = bin_typechecks @ int_typechecks @ nat_typechecks @
[bool_subset_nat RS subsetD];
(**** The carry/borrow functions, bin_succ and bin_pred ****)
-(** Lemmas **)
-
-goal Integ.thy
- "!!z v. [| z $+ v = z' $+ v'; \
-\ z: integ; z': integ; v: integ; v': integ; w: integ |] \
-\ ==> z $+ (v $+ w) = z' $+ (v' $+ w)";
-by (asm_simp_tac (simpset() addsimps [zadd_assoc RS sym]) 1);
-qed "zadd_assoc_cong";
-
-goal Integ.thy
- "!!z v w. [| z: integ; v: integ; w: integ |] \
-\ ==> z $+ (v $+ w) = v $+ (z $+ w)";
-by (REPEAT (ares_tac [zadd_commute RS zadd_assoc_cong] 1));
-qed "zadd_assoc_swap";
-
-(*Pushes 'constants' of the form $#m to the right -- LOOPS if two!*)
-bind_thm ("zadd_assoc_znat", (znat_type RS zadd_assoc_swap));
-
-
Addsimps (bin_recs bin_succ_def @ bin_recs bin_pred_def);
-
(*NCons preserves the integer value of its argument*)
Goal "[| w: bin; b: bool |] ==> \
\ integ_of(NCons(w,b)) = integ_of(Cons(w,b))";
@@ -346,7 +326,7 @@
(*** The computation simpset ***)
(*Adding the typechecking rules as rewrites is much slower, unfortunately...*)
-val bin_comp_ss = simpset_of Integ.thy
+val bin_comp_ss = simpset_of Int.thy
addsimps [integ_of_add RS sym, (*invoke bin_add*)
integ_of_minus RS sym, (*invoke bin_minus*)
integ_of_mult RS sym, (*invoke bin_mult*)
--- a/src/ZF/Integ/Bin.thy Fri Sep 25 12:12:07 1998 +0200
+++ b/src/ZF/Integ/Bin.thy Fri Sep 25 13:18:07 1998 +0200
@@ -18,7 +18,7 @@
Division is not defined yet!
*)
-Bin = Integ + Datatype +
+Bin = Int + Datatype +
consts
bin_rec :: [i, i, i, [i,i,i]=>i] => i
@@ -143,15 +143,15 @@
| bin_of (Const ("Cons", _) $ 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;
+ fun integ_of [] = 0
+ | integ_of (b :: bs) = b + 2 * integ_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));
+ val num = string_of_int (abs (integ_of rev_digs));
in
"#" ^ sign ^ implode (replicate zs "0") ^ num
end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/Integ/Int.ML Fri Sep 25 13:18:07 1998 +0200
@@ -0,0 +1,412 @@
+(* Title: ZF/Integ/Int.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ 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 $<
+*)
+
+AddSEs [quotientE];
+
+(*** Proving that intrel is an equivalence relation ***)
+
+(*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 2);
+by (rtac (add_left_commute RS trans) 1 THEN typechk_tac prems);
+by (stac eqb 1);
+by (rtac (add_left_commute RS trans) 1 THEN typechk_tac prems);
+by (stac eqa 1);
+by (rtac (add_left_commute) 1 THEN typechk_tac prems);
+qed "int_trans_lemma";
+
+(** Natural deduction for intrel **)
+
+Goalw [intrel_def]
+ "<<x1,y1>,<x2,y2>>: intrel <-> \
+\ x1: nat & y1: nat & x2: nat & y2: nat & x1#+y2 = x2#+y1";
+by (Fast_tac 1);
+qed "intrel_iff";
+
+Goalw [intrel_def]
+ "[| x1#+y2 = x2#+y1; x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
+\ <<x1,y1>,<x2,y2>>: intrel";
+by (fast_tac (claset() addIs prems) 1);
+qed "intrelI";
+
+(*intrelE is hard to derive because fast_tac tries hyp_subst_tac so soon*)
+Goalw [intrel_def]
+ "p: intrel --> (EX x1 y1 x2 y2. \
+\ p = <<x1,y1>,<x2,y2>> & x1#+y2 = x2#+y1 & \
+\ x1: nat & y1: nat & x2: nat & y2: nat)";
+by (Fast_tac 1);
+qed "intrelE_lemma";
+
+val [major,minor] = goal thy
+ "[| p: intrel; \
+\ !!x1 y1 x2 y2. [| p = <<x1,y1>,<x2,y2>>; x1#+y2 = x2#+y1; \
+\ x1: nat; y1: nat; x2: nat; y2: nat |] ==> 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";
+
+AddSIs [intrelI];
+AddSEs [intrelE];
+
+Goalw [equiv_def, refl_def, sym_def, trans_def]
+ "equiv(nat*nat, intrel)";
+by (fast_tac (claset() addSEs [sym, int_trans_lemma]) 1);
+qed "equiv_intrel";
+
+
+Addsimps [equiv_intrel RS eq_equiv_class_iff, intrel_iff,
+ add_0_right, add_succ_right];
+Addcongs [conj_cong];
+
+val eq_intrelD = equiv_intrel RSN (2,eq_equiv_class);
+
+(** int_of: the injection from nat to int **)
+
+Goalw [int_def,quotient_def,int_of_def]
+ "m : nat ==> $#m : int";
+by (fast_tac (claset() addSIs [nat_0I]) 1);
+qed "int_of_type";
+
+Addsimps [int_of_type];
+
+Goalw [int_of_def] "[| $#m = $#n; m: nat |] ==> m=n";
+by (dtac (sym RS eq_intrelD) 1);
+by (typechk_tac [nat_0I, SigmaI]);
+by (Asm_full_simp_tac 1);
+qed "int_of_inject";
+
+AddSDs [int_of_inject];
+
+Goal "m: nat ==> ($# m = $# n) <-> (m = n)";
+by (Blast_tac 1);
+qed "int_of_eq";
+Addsimps [int_of_eq];
+
+(**** zminus: unary negation on int ****)
+
+Goalw [congruent_def] "congruent(intrel, %<x,y>. intrel``{<y,x>})";
+by Safe_tac;
+by (asm_full_simp_tac (simpset() addsimps add_ac) 1);
+qed "zminus_congruent";
+
+(*Resolve th against the corresponding facts for zminus*)
+val zminus_ize = RSLIST [equiv_intrel, zminus_congruent];
+
+Goalw [int_def,zminus_def] "z : int ==> $~z : int";
+by (typechk_tac [split_type, SigmaI, zminus_ize UN_equiv_class_type,
+ quotientI]);
+qed "zminus_type";
+
+Goalw [int_def,zminus_def] "[| $~z = $~w; z: int; w: int |] ==> z=w";
+by (etac (zminus_ize UN_equiv_class_inject) 1);
+by Safe_tac;
+(*The setloop is only needed because assumptions are in the wrong order!*)
+by (asm_full_simp_tac (simpset() addsimps add_ac
+ setloop dtac eq_intrelD) 1);
+qed "zminus_inject";
+
+Goalw [zminus_def]
+ "[| x: nat; y: nat |] ==> $~ (intrel``{<x,y>}) = intrel `` {<y,x>}";
+by (asm_simp_tac (simpset() addsimps [zminus_ize UN_equiv_class, SigmaI]) 1);
+qed "zminus";
+
+Goalw [int_def] "z : int ==> $~ ($~ z) = z";
+by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zminus]) 1);
+qed "zminus_zminus";
+
+Goalw [int_def, int_of_def] "$~ ($#0) = $#0";
+by (simp_tac (simpset() addsimps [zminus]) 1);
+qed "zminus_0";
+
+Addsimps [zminus_zminus, zminus_0];
+
+
+(**** znegative: the test for negative integers ****)
+
+(*No natural number is negative!*)
+Goalw [znegative_def, int_of_def] "~ znegative($# n)";
+by Safe_tac;
+by (dres_inst_tac [("psi", "?lhs=?rhs")] asm_rl 1);
+by (dres_inst_tac [("psi", "?lhs<?rhs")] asm_rl 1);
+by (force_tac (claset(),
+ simpset() addsimps [add_le_self2 RS le_imp_not_lt]) 1);
+qed "not_znegative_int_of";
+
+Addsimps [not_znegative_int_of];
+AddSEs [not_znegative_int_of RS notE];
+
+Goalw [znegative_def, int_of_def] "n: nat ==> znegative($~ $# succ(n))";
+by (asm_simp_tac (simpset() addsimps [zminus]) 1);
+by (blast_tac (claset() addIs [nat_0_le]) 1);
+qed "znegative_zminus_int_of";
+
+Addsimps [znegative_zminus_int_of];
+
+Goalw [znegative_def, int_of_def] "[| n: nat; ~ znegative($~ $# n) |] ==> n=0";
+by (asm_full_simp_tac (simpset() addsimps [zminus, image_singleton_iff]) 1);
+be natE 1;
+by (dres_inst_tac [("x","0")] spec 2);
+by Auto_tac;
+qed "not_znegative_imp_zero";
+
+(**** zmagnitude: magnitide of an integer, as a natural number ****)
+
+Goalw [zmagnitude_def] "n: nat ==> zmagnitude($# n) = n";
+by Auto_tac;
+qed "zmagnitude_int_of";
+
+Goalw [zmagnitude_def] "n: nat ==> zmagnitude($~ $# n) = n";
+by (auto_tac(claset() addDs [not_znegative_imp_zero], simpset()));
+qed "zmagnitude_zminus_int_of";
+
+Addsimps [zmagnitude_int_of, zmagnitude_zminus_int_of];
+
+Goalw [zmagnitude_def] "zmagnitude(z) : nat";
+br theI2 1;
+by Auto_tac;
+qed "zmagnitude_type";
+
+Goalw [int_def, znegative_def, int_of_def]
+ "[| z: int; ~ znegative(z) |] ==> EX n:nat. z = $# n";
+by (auto_tac(claset() , simpset() addsimps [image_singleton_iff]));
+by (rename_tac "i j" 1);
+by (dres_inst_tac [("x", "i")] spec 1);
+by (dres_inst_tac [("x", "j")] spec 1);
+br bexI 1;
+br (add_diff_inverse2 RS sym) 1;
+by Auto_tac;
+by (asm_full_simp_tac (simpset() addsimps [nat_into_Ord, not_lt_iff_le]) 1);
+qed "not_zneg_int_of";
+
+Goal "[| z: int; ~ znegative(z) |] ==> $# (zmagnitude(z)) = z";
+bd not_zneg_int_of 1;
+by Auto_tac;
+qed "not_zneg_mag";
+
+Addsimps [not_zneg_mag];
+
+
+Goalw [int_def, znegative_def, int_of_def]
+ "[| z: int; znegative(z) |] ==> EX n:nat. z = $~ ($# succ(n))";
+by (auto_tac(claset() addSDs [less_imp_Suc_add],
+ simpset() addsimps [zminus, image_singleton_iff]));
+by (rename_tac "m n j k" 1);
+by (subgoal_tac "j #+ succ(m #+ k) = j #+ n" 1);
+by (rotate_tac ~2 2);
+by (asm_full_simp_tac (simpset() addsimps add_ac) 2);
+by (blast_tac (claset() addSDs [add_left_cancel]) 1);
+qed "zneg_int_of";
+
+Goal "[| z: int; znegative(z) |] ==> $# (zmagnitude(z)) = $~ z";
+bd zneg_int_of 1;
+by Auto_tac;
+qed "zneg_mag";
+
+Addsimps [zneg_mag];
+
+
+(**** zadd: addition on int ****)
+
+(** Congruence property for addition **)
+
+Goalw [congruent2_def]
+ "congruent2(intrel, %z1 z2. \
+\ let <x1,y1>=z1; <x2,y2>=z2 \
+\ in intrel``{<x1#+x2, y1#+y2>})";
+(*Proof via congruent2_commuteI seems longer*)
+by Safe_tac;
+by (asm_simp_tac (simpset() addsimps [add_assoc, Let_def]) 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 (simpset() addsimps [add_assoc RS sym]) 1);
+qed "zadd_congruent2";
+
+(*Resolve th against the corresponding facts for zadd*)
+val zadd_ize = RSLIST [equiv_intrel, zadd_congruent2];
+
+Goalw [int_def,zadd_def] "[| z: int; w: int |] ==> z $+ w : int";
+by (rtac (zadd_ize UN_equiv_class_type2) 1);
+by (simp_tac (simpset() addsimps [Let_def]) 3);
+by (REPEAT (ares_tac [split_type, add_type, quotientI, SigmaI] 1));
+qed "zadd_type";
+
+Goalw [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 (simpset() addsimps [zadd_ize UN_equiv_class2, SigmaI]) 1);
+by (simp_tac (simpset() addsimps [Let_def]) 1);
+qed "zadd";
+
+Goalw [int_def,int_of_def] "z : int ==> $#0 $+ z = z";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zadd]) 1);
+qed "zadd_0";
+
+Goalw [int_def] "[| z: int; w: int |] ==> $~ (z $+ w) = $~ z $+ $~ w";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zminus,zadd]) 1);
+qed "zminus_zadd_distrib";
+
+Goalw [int_def] "[| z: int; w: int |] ==> z $+ w = w $+ z";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps add_ac @ [zadd]) 1);
+qed "zadd_commute";
+
+Goalw [int_def]
+ "[| z1: int; z2: int; z3: int |] \
+\ ==> (z1 $+ z2) $+ z3 = z1 $+ (z2 $+ z3)";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+(*rewriting is much faster without intrel_iff, etc.*)
+by (asm_simp_tac (simpset() addsimps [zadd, add_assoc]) 1);
+qed "zadd_assoc";
+
+(*For AC rewriting*)
+Goal "[| z1:int; z2:int; z3: int |] ==> z1$+(z2$+z3) = z2$+(z1$+z3)";
+by (asm_simp_tac (simpset() addsimps [zadd_assoc RS sym, zadd_commute]) 1);
+qed "zadd_left_commute";
+
+(*Integer addition is an AC operator*)
+val zadd_ac = [zadd_assoc, zadd_commute, zadd_left_commute];
+
+Goalw [int_of_def]
+ "[| m: nat; n: nat |] ==> $# (m #+ n) = ($#m) $+ ($#n)";
+by (asm_simp_tac (simpset() addsimps [zadd]) 1);
+qed "int_of_add";
+
+Goalw [int_def,int_of_def] "z : int ==> z $+ ($~ z) = $#0";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zminus, zadd, add_commute]) 1);
+qed "zadd_zminus_inverse";
+
+Goal "z : int ==> ($~ z) $+ z = $#0";
+by (asm_simp_tac
+ (simpset() addsimps [zadd_commute, zminus_type, zadd_zminus_inverse]) 1);
+qed "zadd_zminus_inverse2";
+
+Goal "z:int ==> z $+ $#0 = z";
+by (rtac (zadd_commute RS trans) 1);
+by (REPEAT (ares_tac [int_of_type, nat_0I, zadd_0] 1));
+qed "zadd_0_right";
+
+Addsimps [zadd_0, zadd_0_right, zadd_zminus_inverse, zadd_zminus_inverse2];
+
+
+(*Need properties of $- ??? Or use $- just as an abbreviation?
+ [| m: nat; n: nat; m>=n |] ==> $# (m #- n) = ($#m) $- ($#n)
+*)
+
+(**** zmult: multiplication on int ****)
+
+(** Congruence property for multiplication **)
+
+Goal "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;
+by (ALLGOALS Asm_simp_tac);
+(*Proof that zmult is congruent in one argument*)
+by (asm_simp_tac
+ (simpset() addsimps add_ac @ [add_mult_distrib_left RS sym]) 2);
+by (asm_simp_tac
+ (simpset() addsimps [add_assoc RS sym, add_mult_distrib_left RS sym]) 2);
+(*Proof that zmult is commutative on representatives*)
+by (asm_simp_tac (simpset() addsimps mult_ac@add_ac) 1);
+qed "zmult_congruent2";
+
+
+(*Resolve th against the corresponding facts for zmult*)
+val zmult_ize = RSLIST [equiv_intrel, zmult_congruent2];
+
+Goalw [int_def,zmult_def] "[| z: int; w: int |] ==> z $* w : int";
+by (REPEAT (ares_tac [zmult_ize UN_equiv_class_type2,
+ split_type, add_type, mult_type,
+ quotientI, SigmaI] 1));
+qed "zmult_type";
+
+Goalw [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 (simpset() addsimps [zmult_ize UN_equiv_class2, SigmaI]) 1);
+qed "zmult";
+
+Goalw [int_def,int_of_def] "z : int ==> $#0 $* z = $#0";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zmult]) 1);
+qed "zmult_0";
+
+Goalw [int_def,int_of_def] "z : int ==> $#1 $* z = z";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zmult, add_0_right]) 1);
+qed "zmult_1";
+
+Goalw [int_def] "[| z: int; w: int |] ==> ($~ z) $* w = $~ (z $* w)";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zminus, zmult] @ add_ac) 1);
+qed "zmult_zminus";
+
+Addsimps [zmult_0, zmult_1, zmult_zminus];
+
+Goalw [int_def] "[| z: int; w: int |] ==> ($~ z) $* ($~ w) = (z $* w)";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zminus, zmult] @ add_ac) 1);
+qed "zmult_zminus_zminus";
+
+Goalw [int_def] "[| z: int; w: int |] ==> z $* w = w $* z";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zmult] @ add_ac @ mult_ac) 1);
+qed "zmult_commute";
+
+Goalw [int_def]
+ "[| z1: int; z2: int; z3: int |] \
+\ ==> (z1 $* z2) $* z3 = z1 $* (z2 $* z3)";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac
+ (simpset() addsimps [zmult, add_mult_distrib_left,
+ add_mult_distrib] @ add_ac @ mult_ac) 1);
+qed "zmult_assoc";
+
+(*For AC rewriting*)
+Goal "[| z1:int; z2:int; z3: int |] ==> z1$*(z2$*z3) = z2$*(z1$*z3)";
+by (asm_simp_tac (simpset() addsimps [zmult_assoc RS sym, zmult_commute]) 1);
+qed "zmult_left_commute";
+
+(*Integer multiplication is an AC operator*)
+val zmult_ac = [zmult_assoc, zmult_commute, zmult_left_commute];
+
+Goalw [int_def]
+ "[| z1: int; z2: int; w: int |] ==> \
+\ (z1 $+ z2) $* w = (z1 $* w) $+ (z2 $* w)";
+by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
+by (asm_simp_tac (simpset() addsimps [zadd, zmult, add_mult_distrib]) 1);
+by (asm_simp_tac (simpset() addsimps add_ac @ mult_ac) 1);
+qed "zadd_zmult_distrib";
+
+val int_typechecks =
+ [int_of_type, zminus_type, zmagnitude_type, zadd_type, zmult_type];
+
+Addsimps int_typechecks;
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/Integ/Int.thy Fri Sep 25 13:18:07 1998 +0200
@@ -0,0 +1,60 @@
+(* Title: ZF/Integ/Int.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1993 University of Cambridge
+
+The integers as equivalence classes over nat*nat.
+*)
+
+Int = EquivClass + Arith +
+consts
+ intrel,int:: i
+ int_of :: i=>i ("$# _" [80] 80)
+ zminus :: i=>i ("$~ _" [80] 80)
+ znegative :: i=>o
+ zmagnitude :: i=>i
+ "$*" :: [i,i]=>i (infixl 70)
+ "$'/" :: [i,i]=>i (infixl 70)
+ "$'/'/" :: [i,i]=>i (infixl 70)
+ "$+" :: [i,i]=>i (infixl 65)
+ "$-" :: [i,i]=>i (infixl 65)
+ "$<" :: [i,i]=>o (infixl 50)
+
+defs
+
+ intrel_def
+ "intrel == {p:(nat*nat)*(nat*nat).
+ EX x1 y1 x2 y2. p=<<x1,y1>,<x2,y2>> & x1#+y2 = x2#+y1}"
+
+ int_def "int == (nat*nat)/intrel"
+
+ int_of_def "$# m == intrel `` {<m,0>}"
+
+ zminus_def "$~ Z == UN <x,y>:Z. intrel``{<y,x>}"
+
+ znegative_def
+ "znegative(Z) == EX x y. x<y & y:nat & <x,y>:Z"
+
+ zmagnitude_def
+ "zmagnitude(Z) ==
+ THE m. m : nat & ((~ znegative(Z) & Z = $# m) |
+ (znegative(Z) & $~ Z = $# m))"
+
+ (*Cannot use UN<x1,y2> here or in zmult because of the form of congruent2.
+ Perhaps a "curried" or even polymorphic congruent predicate would be
+ better.*)
+ zadd_def
+ "Z1 $+ Z2 ==
+ UN z1:Z1. UN z2:Z2. let <x1,y1>=z1; <x2,y2>=z2
+ in intrel``{<x1#+x2, y1#+y2>}"
+
+ zdiff_def "Z1 $- Z2 == Z1 $+ zminus(Z2)"
+ zless_def "Z1 $< Z2 == znegative(Z1 $- Z2)"
+
+ (*This illustrates the primitive form of definitions (no patterns)*)
+ zmult_def
+ "Z1 $* Z2 ==
+ UN p1:Z1. UN p2:Z2. split(%x1 y1. split(%x2 y2.
+ intrel``{<x1#*x2 #+ y1#*y2, x1#*y2 #+ y1#*x2>}, p2), p1)"
+
+ end
--- a/src/ZF/Integ/Integ.ML Fri Sep 25 12:12:07 1998 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,412 +0,0 @@
-(* Title: ZF/ex/integ.ML
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- 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 $<
-*)
-
-AddSEs [quotientE];
-
-(*** Proving that intrel is an equivalence relation ***)
-
-(*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 2);
-by (rtac (add_left_commute RS trans) 1 THEN typechk_tac prems);
-by (stac eqb 1);
-by (rtac (add_left_commute RS trans) 1 THEN typechk_tac prems);
-by (stac eqa 1);
-by (rtac (add_left_commute) 1 THEN typechk_tac prems);
-qed "integ_trans_lemma";
-
-(** Natural deduction for intrel **)
-
-Goalw [intrel_def]
- "<<x1,y1>,<x2,y2>>: intrel <-> \
-\ x1: nat & y1: nat & x2: nat & y2: nat & x1#+y2 = x2#+y1";
-by (Fast_tac 1);
-qed "intrel_iff";
-
-Goalw [intrel_def]
- "[| x1#+y2 = x2#+y1; x1: nat; y1: nat; x2: nat; y2: nat |] ==> \
-\ <<x1,y1>,<x2,y2>>: intrel";
-by (fast_tac (claset() addIs prems) 1);
-qed "intrelI";
-
-(*intrelE is hard to derive because fast_tac tries hyp_subst_tac so soon*)
-Goalw [intrel_def]
- "p: intrel --> (EX x1 y1 x2 y2. \
-\ p = <<x1,y1>,<x2,y2>> & x1#+y2 = x2#+y1 & \
-\ x1: nat & y1: nat & x2: nat & y2: nat)";
-by (Fast_tac 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; \
-\ x1: nat; y1: nat; x2: nat; y2: nat |] ==> 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";
-
-AddSIs [intrelI];
-AddSEs [intrelE];
-
-Goalw [equiv_def, refl_def, sym_def, trans_def]
- "equiv(nat*nat, intrel)";
-by (fast_tac (claset() addSEs [sym, integ_trans_lemma]) 1);
-qed "equiv_intrel";
-
-
-Addsimps [equiv_intrel RS eq_equiv_class_iff, intrel_iff,
- add_0_right, add_succ_right];
-Addcongs [conj_cong];
-
-val eq_intrelD = equiv_intrel RSN (2,eq_equiv_class);
-
-(** znat: the injection from nat to integ **)
-
-Goalw [integ_def,quotient_def,znat_def]
- "m : nat ==> $#m : integ";
-by (fast_tac (claset() addSIs [nat_0I]) 1);
-qed "znat_type";
-
-Addsimps [znat_type];
-
-Goalw [znat_def] "[| $#m = $#n; m: nat |] ==> m=n";
-by (dtac (sym RS eq_intrelD) 1);
-by (typechk_tac [nat_0I, SigmaI]);
-by (Asm_full_simp_tac 1);
-qed "znat_inject";
-
-AddSDs [znat_inject];
-
-Goal "m: nat ==> ($# m = $# n) <-> (m = n)";
-by (Blast_tac 1);
-qed "znat_znat_eq";
-Addsimps [znat_znat_eq];
-
-(**** zminus: unary negation on integ ****)
-
-Goalw [congruent_def] "congruent(intrel, %<x,y>. intrel``{<y,x>})";
-by Safe_tac;
-by (asm_full_simp_tac (simpset() 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_def,zminus_def] "z : integ ==> $~z : integ";
-by (typechk_tac [split_type, SigmaI, zminus_ize UN_equiv_class_type,
- quotientI]);
-qed "zminus_type";
-
-Goalw [integ_def,zminus_def] "[| $~z = $~w; z: integ; w: integ |] ==> z=w";
-by (etac (zminus_ize UN_equiv_class_inject) 1);
-by Safe_tac;
-(*The setloop is only needed because assumptions are in the wrong order!*)
-by (asm_full_simp_tac (simpset() addsimps add_ac
- setloop dtac eq_intrelD) 1);
-qed "zminus_inject";
-
-Goalw [zminus_def]
- "[| x: nat; y: nat |] ==> $~ (intrel``{<x,y>}) = intrel `` {<y,x>}";
-by (asm_simp_tac (simpset() addsimps [zminus_ize UN_equiv_class, SigmaI]) 1);
-qed "zminus";
-
-Goalw [integ_def] "z : integ ==> $~ ($~ z) = z";
-by (REPEAT (eresolve_tac [quotientE,SigmaE,ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zminus]) 1);
-qed "zminus_zminus";
-
-Goalw [integ_def, znat_def] "$~ ($#0) = $#0";
-by (simp_tac (simpset() addsimps [zminus]) 1);
-qed "zminus_0";
-
-Addsimps [zminus_zminus, zminus_0];
-
-
-(**** znegative: the test for negative integers ****)
-
-(*No natural number is negative!*)
-Goalw [znegative_def, znat_def] "~ znegative($# n)";
-by Safe_tac;
-by (dres_inst_tac [("psi", "?lhs=?rhs")] asm_rl 1);
-by (dres_inst_tac [("psi", "?lhs<?rhs")] asm_rl 1);
-by (force_tac (claset(),
- simpset() addsimps [add_le_self2 RS le_imp_not_lt]) 1);
-qed "not_znegative_znat";
-
-Addsimps [not_znegative_znat];
-AddSEs [not_znegative_znat RS notE];
-
-Goalw [znegative_def, znat_def] "n: nat ==> znegative($~ $# succ(n))";
-by (asm_simp_tac (simpset() addsimps [zminus]) 1);
-by (blast_tac (claset() addIs [nat_0_le]) 1);
-qed "znegative_zminus_znat";
-
-Addsimps [znegative_zminus_znat];
-
-Goalw [znegative_def, znat_def] "[| n: nat; ~ znegative($~ $# n) |] ==> n=0";
-by (asm_full_simp_tac (simpset() addsimps [zminus, image_singleton_iff]) 1);
-be natE 1;
-by (dres_inst_tac [("x","0")] spec 2);
-by Auto_tac;
-qed "not_znegative_imp_zero";
-
-(**** zmagnitude: magnitide of an integer, as a natural number ****)
-
-Goalw [zmagnitude_def] "n: nat ==> zmagnitude($# n) = n";
-by Auto_tac;
-qed "zmagnitude_znat";
-
-Goalw [zmagnitude_def] "n: nat ==> zmagnitude($~ $# n) = n";
-by (auto_tac(claset() addDs [not_znegative_imp_zero], simpset()));
-qed "zmagnitude_zminus_znat";
-
-Addsimps [zmagnitude_znat, zmagnitude_zminus_znat];
-
-Goalw [zmagnitude_def] "zmagnitude(z) : nat";
-br theI2 1;
-by Auto_tac;
-qed "zmagnitude_type";
-
-Goalw [integ_def, znegative_def, znat_def]
- "[| z: integ; ~ znegative(z) |] ==> EX n:nat. z = $# n";
-by (auto_tac(claset() , simpset() addsimps [image_singleton_iff]));
-by (rename_tac "i j" 1);
-by (dres_inst_tac [("x", "i")] spec 1);
-by (dres_inst_tac [("x", "j")] spec 1);
-br bexI 1;
-br (add_diff_inverse2 RS sym) 1;
-by Auto_tac;
-by (asm_full_simp_tac (simpset() addsimps [nat_into_Ord, not_lt_iff_le]) 1);
-qed "not_zneg_znat";
-
-Goal "[| z: integ; ~ znegative(z) |] ==> $# (zmagnitude(z)) = z";
-bd not_zneg_znat 1;
-by Auto_tac;
-qed "not_zneg_mag";
-
-Addsimps [not_zneg_mag];
-
-
-Goalw [integ_def, znegative_def, znat_def]
- "[| z: integ; znegative(z) |] ==> EX n:nat. z = $~ ($# succ(n))";
-by (auto_tac(claset() addSDs [less_imp_Suc_add],
- simpset() addsimps [zminus, image_singleton_iff]));
-by (rename_tac "m n j k" 1);
-by (subgoal_tac "j #+ succ(m #+ k) = j #+ n" 1);
-by (rotate_tac ~2 2);
-by (asm_full_simp_tac (simpset() addsimps add_ac) 2);
-by (blast_tac (claset() addSDs [add_left_cancel]) 1);
-qed "zneg_znat";
-
-Goal "[| z: integ; znegative(z) |] ==> $# (zmagnitude(z)) = $~ z";
-bd zneg_znat 1;
-by Auto_tac;
-qed "zneg_mag";
-
-Addsimps [zneg_mag];
-
-
-(**** zadd: addition on integ ****)
-
-(** Congruence property for addition **)
-
-Goalw [congruent2_def]
- "congruent2(intrel, %z1 z2. \
-\ let <x1,y1>=z1; <x2,y2>=z2 \
-\ in intrel``{<x1#+x2, y1#+y2>})";
-(*Proof via congruent2_commuteI seems longer*)
-by Safe_tac;
-by (asm_simp_tac (simpset() addsimps [add_assoc, Let_def]) 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 (simpset() addsimps [add_assoc RS sym]) 1);
-qed "zadd_congruent2";
-
-(*Resolve th against the corresponding facts for zadd*)
-val zadd_ize = RSLIST [equiv_intrel, zadd_congruent2];
-
-Goalw [integ_def,zadd_def] "[| z: integ; w: integ |] ==> z $+ w : integ";
-by (rtac (zadd_ize UN_equiv_class_type2) 1);
-by (simp_tac (simpset() addsimps [Let_def]) 3);
-by (REPEAT (ares_tac [split_type, add_type, quotientI, SigmaI] 1));
-qed "zadd_type";
-
-Goalw [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 (simpset() addsimps [zadd_ize UN_equiv_class2, SigmaI]) 1);
-by (simp_tac (simpset() addsimps [Let_def]) 1);
-qed "zadd";
-
-Goalw [integ_def,znat_def] "z : integ ==> $#0 $+ z = z";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zadd]) 1);
-qed "zadd_0";
-
-Goalw [integ_def] "[| z: integ; w: integ |] ==> $~ (z $+ w) = $~ z $+ $~ w";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zminus,zadd]) 1);
-qed "zminus_zadd_distrib";
-
-Goalw [integ_def] "[| z: integ; w: integ |] ==> z $+ w = w $+ z";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps add_ac @ [zadd]) 1);
-qed "zadd_commute";
-
-Goalw [integ_def]
- "[| z1: integ; z2: integ; z3: integ |] \
-\ ==> (z1 $+ z2) $+ z3 = z1 $+ (z2 $+ z3)";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-(*rewriting is much faster without intrel_iff, etc.*)
-by (asm_simp_tac (simpset() addsimps [zadd, add_assoc]) 1);
-qed "zadd_assoc";
-
-(*For AC rewriting*)
-Goal "[| z1:integ; z2:integ; z3: integ |] ==> z1$+(z2$+z3) = z2$+(z1$+z3)";
-by (asm_simp_tac (simpset() addsimps [zadd_assoc RS sym, zadd_commute]) 1);
-qed "zadd_left_commute";
-
-(*Integer addition is an AC operator*)
-val zadd_ac = [zadd_assoc, zadd_commute, zadd_left_commute];
-
-Goalw [znat_def]
- "[| m: nat; n: nat |] ==> $# (m #+ n) = ($#m) $+ ($#n)";
-by (asm_simp_tac (simpset() addsimps [zadd]) 1);
-qed "znat_add";
-
-Goalw [integ_def,znat_def] "z : integ ==> z $+ ($~ z) = $#0";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zminus, zadd, add_commute]) 1);
-qed "zadd_zminus_inverse";
-
-Goal "z : integ ==> ($~ z) $+ z = $#0";
-by (asm_simp_tac
- (simpset() addsimps [zadd_commute, zminus_type, zadd_zminus_inverse]) 1);
-qed "zadd_zminus_inverse2";
-
-Goal "z:integ ==> z $+ $#0 = z";
-by (rtac (zadd_commute RS trans) 1);
-by (REPEAT (ares_tac [znat_type, nat_0I, zadd_0] 1));
-qed "zadd_0_right";
-
-Addsimps [zadd_0, zadd_0_right, zadd_zminus_inverse, zadd_zminus_inverse2];
-
-
-(*Need properties of $- ??? Or use $- just as an abbreviation?
- [| m: nat; n: nat; m>=n |] ==> $# (m #- n) = ($#m) $- ($#n)
-*)
-
-(**** zmult: multiplication on integ ****)
-
-(** Congruence property for multiplication **)
-
-Goal "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;
-by (ALLGOALS Asm_simp_tac);
-(*Proof that zmult is congruent in one argument*)
-by (asm_simp_tac
- (simpset() addsimps add_ac @ [add_mult_distrib_left RS sym]) 2);
-by (asm_simp_tac
- (simpset() addsimps [add_assoc RS sym, add_mult_distrib_left RS sym]) 2);
-(*Proof that zmult is commutative on representatives*)
-by (asm_simp_tac (simpset() addsimps mult_ac@add_ac) 1);
-qed "zmult_congruent2";
-
-
-(*Resolve th against the corresponding facts for zmult*)
-val zmult_ize = RSLIST [equiv_intrel, zmult_congruent2];
-
-Goalw [integ_def,zmult_def] "[| 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));
-qed "zmult_type";
-
-Goalw [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 (simpset() addsimps [zmult_ize UN_equiv_class2, SigmaI]) 1);
-qed "zmult";
-
-Goalw [integ_def,znat_def] "z : integ ==> $#0 $* z = $#0";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zmult]) 1);
-qed "zmult_0";
-
-Goalw [integ_def,znat_def] "z : integ ==> $#1 $* z = z";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zmult, add_0_right]) 1);
-qed "zmult_1";
-
-Goalw [integ_def] "[| z: integ; w: integ |] ==> ($~ z) $* w = $~ (z $* w)";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zminus, zmult] @ add_ac) 1);
-qed "zmult_zminus";
-
-Addsimps [zmult_0, zmult_1, zmult_zminus];
-
-Goalw [integ_def] "[| z: integ; w: integ |] ==> ($~ z) $* ($~ w) = (z $* w)";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zminus, zmult] @ add_ac) 1);
-qed "zmult_zminus_zminus";
-
-Goalw [integ_def] "[| z: integ; w: integ |] ==> z $* w = w $* z";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zmult] @ add_ac @ mult_ac) 1);
-qed "zmult_commute";
-
-Goalw [integ_def]
- "[| z1: integ; z2: integ; z3: integ |] \
-\ ==> (z1 $* z2) $* z3 = z1 $* (z2 $* z3)";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac
- (simpset() addsimps [zmult, add_mult_distrib_left,
- add_mult_distrib] @ add_ac @ mult_ac) 1);
-qed "zmult_assoc";
-
-(*For AC rewriting*)
-Goal "[| z1:integ; z2:integ; z3: integ |] ==> z1$*(z2$*z3) = z2$*(z1$*z3)";
-by (asm_simp_tac (simpset() addsimps [zmult_assoc RS sym, zmult_commute]) 1);
-qed "zmult_left_commute";
-
-(*Integer multiplication is an AC operator*)
-val zmult_ac = [zmult_assoc, zmult_commute, zmult_left_commute];
-
-Goalw [integ_def]
- "[| z1: integ; z2: integ; w: integ |] ==> \
-\ (z1 $+ z2) $* w = (z1 $* w) $+ (z2 $* w)";
-by (REPEAT (eresolve_tac [quotientE, SigmaE, ssubst] 1));
-by (asm_simp_tac (simpset() addsimps [zadd, zmult, add_mult_distrib]) 1);
-by (asm_simp_tac (simpset() addsimps add_ac @ mult_ac) 1);
-qed "zadd_zmult_distrib";
-
-val integ_typechecks =
- [znat_type, zminus_type, zmagnitude_type, zadd_type, zmult_type];
-
-Addsimps integ_typechecks;
-
-
-
--- a/src/ZF/Integ/Integ.thy Fri Sep 25 12:12:07 1998 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,60 +0,0 @@
-(* Title: ZF/ex/integ.thy
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1993 University of Cambridge
-
-The integers as equivalence classes over nat*nat.
-*)
-
-Integ = EquivClass + Arith +
-consts
- intrel,integ:: i
- znat :: i=>i ("$# _" [80] 80)
- zminus :: i=>i ("$~ _" [80] 80)
- znegative :: i=>o
- zmagnitude :: i=>i
- "$*" :: [i,i]=>i (infixl 70)
- "$'/" :: [i,i]=>i (infixl 70)
- "$'/'/" :: [i,i]=>i (infixl 70)
- "$+" :: [i,i]=>i (infixl 65)
- "$-" :: [i,i]=>i (infixl 65)
- "$<" :: [i,i]=>o (infixl 50)
-
-defs
-
- intrel_def
- "intrel == {p:(nat*nat)*(nat*nat).
- EX x1 y1 x2 y2. p=<<x1,y1>,<x2,y2>> & x1#+y2 = x2#+y1}"
-
- integ_def "integ == (nat*nat)/intrel"
-
- znat_def "$# m == intrel `` {<m,0>}"
-
- zminus_def "$~ Z == UN <x,y>:Z. intrel``{<y,x>}"
-
- znegative_def
- "znegative(Z) == EX x y. x<y & y:nat & <x,y>:Z"
-
- zmagnitude_def
- "zmagnitude(Z) ==
- THE m. m : nat & ((~ znegative(Z) & Z = $# m) |
- (znegative(Z) & $~ Z = $# m))"
-
- (*Cannot use UN<x1,y2> here or in zmult because of the form of congruent2.
- Perhaps a "curried" or even polymorphic congruent predicate would be
- better.*)
- zadd_def
- "Z1 $+ Z2 ==
- UN z1:Z1. UN z2:Z2. let <x1,y1>=z1; <x2,y2>=z2
- in intrel``{<x1#+x2, y1#+y2>}"
-
- zdiff_def "Z1 $- Z2 == Z1 $+ zminus(Z2)"
- zless_def "Z1 $< Z2 == znegative(Z1 $- Z2)"
-
- (*This illustrates the primitive form of definitions (no patterns)*)
- zmult_def
- "Z1 $* Z2 ==
- UN p1:Z1. UN p2:Z2. split(%x1 y1. split(%x2 y2.
- intrel``{<x1#*x2 #+ y1#*y2, x1#*y2 #+ y1#*x2>}, p2), p1)"
-
- end
--- a/src/ZF/IsaMakefile Fri Sep 25 12:12:07 1998 +0200
+++ b/src/ZF/IsaMakefile Fri Sep 25 13:18:07 1998 +0200
@@ -44,7 +44,7 @@
ind_syntax.ML ind_syntax.thy indrule.ML indrule.thy intr_elim.ML \
intr_elim.thy mono.ML mono.thy pair.ML pair.thy simpdata.ML subset.ML \
subset.thy thy_syntax.ML typechk.ML upair.ML upair.thy \
- Integ/EquivClass.ML Integ/EquivClass.thy Integ/Integ.ML Integ/Integ.thy \
+ Integ/EquivClass.ML Integ/EquivClass.thy Integ/Int.ML Integ/Int.thy \
Integ/twos_compl.ML Integ/Bin.ML Integ/Bin.thy
@$(ISATOOL) usedir -b $(OUT)/FOL ZF