--- a/src/ZF/Arith.ML Thu Jan 07 11:08:29 1999 +0100
+++ b/src/ZF/Arith.ML Thu Jan 07 18:30:55 1999 +0100
@@ -6,50 +6,20 @@
Arithmetic operators and their definitions
Proofs about elementary arithmetic: addition, multiplication, etc.
-
-Could prove def_rec_0, def_rec_succ...
*)
-open Arith;
-
(*"Difference" is subtraction of natural numbers.
There are no negative numbers; we have
m #- n = 0 iff m<=n and m #- n = succ(k) iff m>n.
Also, rec(m, 0, %z w.z) is pred(m).
*)
-(** rec -- better than nat_rec; the succ case has no type requirement! **)
-
-val rec_trans = rec_def RS def_transrec RS trans;
-
-Goal "rec(0,a,b) = a";
-by (rtac rec_trans 1);
-by (rtac nat_case_0 1);
-qed "rec_0";
-
-Goal "rec(succ(m),a,b) = b(m, rec(m,a,b))";
-by (rtac rec_trans 1);
-by (Simp_tac 1);
-qed "rec_succ";
-
-Addsimps [rec_0, rec_succ];
-
-val major::prems = Goal
- "[| n: nat; \
-\ a: C(0); \
-\ !!m z. [| m: nat; z: C(m) |] ==> b(m,z): C(succ(m)) \
-\ |] ==> rec(n,a,b) : C(n)";
-by (rtac (major RS nat_induct) 1);
-by (ALLGOALS
- (asm_simp_tac (simpset() addsimps prems)));
-qed "rec_type";
-
Addsimps [rec_type, nat_0_le, nat_le_refl];
val nat_typechecks = [rec_type, nat_0I, nat_1I, nat_succI, Ord_nat];
Goal "[| 0<k; k: nat |] ==> EX j: nat. k = succ(j)";
by (etac rev_mp 1);
-by (etac nat_induct 1);
+by (induct_tac "k" 1);
by (Simp_tac 1);
by (Blast_tac 1);
val lemma = result();
@@ -60,63 +30,46 @@
(** Addition **)
-qed_goalw "add_type" Arith.thy [add_def]
- "[| m:nat; n:nat |] ==> m #+ n : nat"
- (fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]);
-
-qed_goalw "add_0" Arith.thy [add_def]
- "0 #+ n = n"
- (fn _ => [ (rtac rec_0 1) ]);
-
-qed_goalw "add_succ" Arith.thy [add_def]
- "succ(m) #+ n = succ(m #+ n)"
- (fn _=> [ (rtac rec_succ 1) ]);
-
-Addsimps [add_0, add_succ];
+Goal "[| m:nat; n:nat |] ==> m #+ n : nat";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "add_type";
+Addsimps [add_type];
(** Multiplication **)
-qed_goalw "mult_type" Arith.thy [mult_def]
- "[| m:nat; n:nat |] ==> m #* n : nat"
- (fn prems=>
- [ (typechk_tac (prems@[add_type]@nat_typechecks@ZF_typechecks)) ]);
+Goal "[| m:nat; n:nat |] ==> m #* n : nat";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "mult_type";
+Addsimps [mult_type];
-qed_goalw "mult_0" Arith.thy [mult_def]
- "0 #* n = 0"
- (fn _ => [ (rtac rec_0 1) ]);
-
-qed_goalw "mult_succ" Arith.thy [mult_def]
- "succ(m) #* n = n #+ (m #* n)"
- (fn _ => [ (rtac rec_succ 1) ]);
-
-Addsimps [mult_0, mult_succ];
(** Difference **)
-qed_goalw "diff_type" Arith.thy [diff_def]
- "[| m:nat; n:nat |] ==> m #- n : nat"
- (fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]);
+Goal "[| m:nat; n:nat |] ==> m #- n : nat";
+by (induct_tac "n" 1);
+by Auto_tac;
+by (fast_tac (claset() addIs [nat_case_type]) 1);
+qed "diff_type";
+Addsimps [diff_type];
-qed_goalw "diff_0" Arith.thy [diff_def]
- "m #- 0 = m"
- (fn _ => [ (rtac rec_0 1) ]);
+Goal "n:nat ==> 0 #- n = 0";
+by (induct_tac "n" 1);
+by Auto_tac;
+qed "diff_0_eq_0";
-qed_goalw "diff_0_eq_0" Arith.thy [diff_def]
- "n:nat ==> 0 #- n = 0"
- (fn [prem]=>
- [ (rtac (prem RS nat_induct) 1),
- (ALLGOALS (Asm_simp_tac)) ]);
+(*Must simplify BEFORE the induction: else we get a critical pair*)
+Goal "[| m:nat; n:nat |] ==> succ(m) #- succ(n) = m #- n";
+by (Asm_simp_tac 1);
+by (induct_tac "n" 1);
+by Auto_tac;
+qed "diff_succ_succ";
-(*Must simplify BEFORE the induction!! (Else we get a critical pair)
- succ(m) #- succ(n) rewrites to pred(succ(m) #- n) *)
-qed_goalw "diff_succ_succ" Arith.thy [diff_def]
- "[| m:nat; n:nat |] ==> succ(m) #- succ(n) = m #- n"
- (fn prems=>
- [ (asm_simp_tac (simpset() addsimps prems) 1),
- (nat_ind_tac "n" prems 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps prems))) ]);
+Addsimps [diff_0_eq_0, diff_succ_succ];
-Addsimps [diff_0, diff_0_eq_0, diff_succ_succ];
+(*This defining property is no longer wanted*)
+Delsimps [diff_SUCC];
val prems = goal Arith.thy
"[| m:nat; n:nat |] ==> m #- n le m";
@@ -129,72 +82,64 @@
(*** Simplification over add, mult, diff ***)
val arith_typechecks = [add_type, mult_type, diff_type];
-Addsimps arith_typechecks;
(*** Addition ***)
(*Associative law for addition*)
-qed_goal "add_assoc" Arith.thy
- "m:nat ==> (m #+ n) #+ k = m #+ (n #+ k)"
- (fn prems=>
- [ (nat_ind_tac "m" prems 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps prems))) ]);
+Goal "m:nat ==> (m #+ n) #+ k = m #+ (n #+ k)";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "add_assoc";
(*The following two lemmas are used for add_commute and sometimes
elsewhere, since they are safe for rewriting.*)
-qed_goal "add_0_right" Arith.thy
- "m:nat ==> m #+ 0 = m"
- (fn prems=>
- [ (nat_ind_tac "m" prems 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps prems))) ]);
+Goal "m:nat ==> m #+ 0 = m";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "add_0_right";
-qed_goal "add_succ_right" Arith.thy
- "m:nat ==> m #+ succ(n) = succ(m #+ n)"
- (fn prems=>
- [ (nat_ind_tac "m" prems 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps prems))) ]);
+Goal "m:nat ==> m #+ succ(n) = succ(m #+ n)";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "add_succ_right";
Addsimps [add_0_right, add_succ_right];
(*Commutative law for addition*)
-qed_goal "add_commute" Arith.thy
- "!!m n. [| m:nat; n:nat |] ==> m #+ n = n #+ m"
- (fn _ =>
- [ (nat_ind_tac "n" [] 1),
- (ALLGOALS Asm_simp_tac) ]);
+Goal "[| m:nat; n:nat |] ==> m #+ n = n #+ m";
+by (induct_tac "n" 1);
+by Auto_tac;
+qed "add_commute";
(*for a/c rewriting*)
-qed_goal "add_left_commute" Arith.thy
- "!!m n k. [| m:nat; n:nat |] ==> m#+(n#+k)=n#+(m#+k)"
- (fn _ => [asm_simp_tac(simpset() addsimps [add_assoc RS sym, add_commute]) 1]);
+Goal "[| m:nat; n:nat |] ==> m#+(n#+k)=n#+(m#+k)";
+by (asm_simp_tac (simpset() addsimps [add_assoc RS sym, add_commute]) 1);
+qed "add_left_commute";
(*Addition is an AC-operator*)
val add_ac = [add_assoc, add_commute, add_left_commute];
(*Cancellation law on the left*)
-val [eqn,knat] = goal Arith.thy
- "[| k #+ m = k #+ n; k:nat |] ==> m=n";
-by (rtac (eqn RS rev_mp) 1);
-by (nat_ind_tac "k" [knat] 1);
-by (ALLGOALS (Simp_tac));
+Goal "[| k #+ m = k #+ n; k:nat |] ==> m=n";
+by (etac rev_mp 1);
+by (induct_tac "k" 1);
+by Auto_tac;
qed "add_left_cancel";
(*** Multiplication ***)
(*right annihilation in product*)
-qed_goal "mult_0_right" Arith.thy
- "!!m. m:nat ==> m #* 0 = 0"
- (fn _=>
- [ (nat_ind_tac "m" [] 1),
- (ALLGOALS (Asm_simp_tac)) ]);
+Goal "m:nat ==> m #* 0 = 0";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "mult_0_right";
(*right successor law for multiplication*)
-qed_goal "mult_succ_right" Arith.thy
- "!!m n. [| m:nat; n:nat |] ==> m #* succ(n) = m #+ (m #* n)"
- (fn _ =>
- [ (nat_ind_tac "m" [] 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps add_ac))) ]);
+Goal "[| m:nat; n:nat |] ==> m #* succ(n) = m #+ (m #* n)";
+by (induct_tac "m" 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps add_ac)));
+qed "mult_succ_right";
Addsimps [mult_0_right, mult_succ_right];
@@ -206,53 +151,49 @@
by (Asm_simp_tac 1);
qed "mult_1_right";
+Addsimps [mult_1, mult_1_right];
+
(*Commutative law for multiplication*)
-qed_goal "mult_commute" Arith.thy
- "!!m n. [| m:nat; n:nat |] ==> m #* n = n #* m"
- (fn _=>
- [ (nat_ind_tac "m" [] 1),
- (ALLGOALS Asm_simp_tac) ]);
+Goal "[| m:nat; n:nat |] ==> m #* n = n #* m";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "mult_commute";
(*addition distributes over multiplication*)
-qed_goal "add_mult_distrib" Arith.thy
- "!!m n. [| m:nat; k:nat |] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)"
- (fn _=>
- [ (etac nat_induct 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym]))) ]);
+Goal "[| m:nat; k:nat |] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)";
+by (induct_tac "m" 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
+qed "add_mult_distrib";
(*Distributive law on the left; requires an extra typing premise*)
-qed_goal "add_mult_distrib_left" Arith.thy
- "!!m. [| m:nat; n:nat; k:nat |] ==> k #* (m #+ n) = (k #* m) #+ (k #* n)"
- (fn prems=>
- [ (nat_ind_tac "m" [] 1),
- (Asm_simp_tac 1),
- (asm_simp_tac (simpset() addsimps add_ac) 1) ]);
+Goal "[| m:nat; n:nat; k:nat |] ==> k #* (m #+ n) = (k #* m) #+ (k #* n)";
+by (induct_tac "m" 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps add_ac)));
+qed "add_mult_distrib_left";
(*Associative law for multiplication*)
-qed_goal "mult_assoc" Arith.thy
- "!!m n k. [| m:nat; n:nat; k:nat |] ==> (m #* n) #* k = m #* (n #* k)"
- (fn _=>
- [ (etac nat_induct 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))) ]);
+Goal "[| m:nat; n:nat; k:nat |] ==> (m #* n) #* k = m #* (n #* k)";
+by (induct_tac "m" 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib])));
+qed "mult_assoc";
(*for a/c rewriting*)
-qed_goal "mult_left_commute" Arith.thy
- "!!m n k. [| m:nat; n:nat; k:nat |] ==> m #* (n #* k) = n #* (m #* k)"
- (fn _ => [rtac (mult_commute RS trans) 1,
- rtac (mult_assoc RS trans) 3,
- rtac (mult_commute RS subst_context) 6,
- REPEAT (ares_tac [mult_type] 1)]);
+Goal "[| m:nat; n:nat; k:nat |] ==> m #* (n #* k) = n #* (m #* k)";
+by (rtac (mult_commute RS trans) 1);
+by (rtac (mult_assoc RS trans) 3);
+by (rtac (mult_commute RS subst_context) 6);
+by (REPEAT (ares_tac [mult_type] 1));
+qed "mult_left_commute";
val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
(*** Difference ***)
-qed_goal "diff_self_eq_0" Arith.thy
- "m:nat ==> m #- m = 0"
- (fn prems=>
- [ (nat_ind_tac "m" prems 1),
- (ALLGOALS (asm_simp_tac (simpset() addsimps prems))) ]);
+Goal "m:nat ==> m #- m = 0";
+by (induct_tac "m" 1);
+by Auto_tac;
+qed "diff_self_eq_0";
(*Addition is the inverse of subtraction*)
Goal "[| n le m; m:nat |] ==> n #+ (m#-n) = m";
@@ -287,10 +228,9 @@
(** Subtraction is the inverse of addition. **)
-val [mnat,nnat] = goal Arith.thy
- "[| m:nat; n:nat |] ==> (n#+m) #- n = m";
-by (rtac (nnat RS nat_induct) 1);
-by (ALLGOALS (asm_simp_tac (simpset() addsimps [mnat])));
+Goal "[| m:nat; n:nat |] ==> (n#+m) #- n = m";
+by (induct_tac "n" 1);
+by Auto_tac;
qed "diff_add_inverse";
Goal "[| m:nat; n:nat |] ==> (m#+n) #- n = m";
@@ -299,7 +239,7 @@
qed "diff_add_inverse2";
Goal "[| k:nat; m: nat; n: nat |] ==> (k#+m) #- (k#+n) = m #- n";
-by (nat_ind_tac "k" [] 1);
+by (induct_tac "k" 1);
by (ALLGOALS Asm_simp_tac);
qed "diff_cancel";
@@ -308,10 +248,9 @@
by (asm_simp_tac (simpset() addsimps [add_commute_k, diff_cancel]) 1);
qed "diff_cancel2";
-val [mnat,nnat] = goal Arith.thy
- "[| m:nat; n:nat |] ==> n #- (n#+m) = 0";
-by (rtac (nnat RS nat_induct) 1);
-by (ALLGOALS (asm_simp_tac (simpset() addsimps [mnat])));
+Goal "[| m:nat; n:nat |] ==> n #- (n#+m) = 0";
+by (induct_tac "n" 1);
+by Auto_tac;
qed "diff_add_0";
(** Difference distributes over multiplication **)
@@ -334,7 +273,7 @@
by (etac rev_mp 1);
by (etac rev_mp 1);
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_le_self,diff_succ_succ])));
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_le_self])));
qed "div_termination";
val div_rls = (*for mod and div*)
@@ -342,8 +281,8 @@
[Ord_transrec_type, apply_type, div_termination RS ltD, if_type,
nat_into_Ord, not_lt_iff_le RS iffD1];
-val div_ss = (simpset()) addsimps [nat_into_Ord, div_termination RS ltD,
- not_lt_iff_le RS iffD2];
+val div_ss = simpset() addsimps [nat_into_Ord, div_termination RS ltD,
+ not_lt_iff_le RS iffD2];
(*Type checking depends upon termination!*)
Goalw [mod_def] "[| 0<n; m:nat; n:nat |] ==> m mod n : nat";
@@ -442,7 +381,7 @@
qed "mod2_succ_succ";
Goal "m:nat ==> (m#+m) mod 2 = 0";
-by (etac nat_induct 1);
+by (induct_tac "m" 1);
by (simp_tac (simpset() addsimps [mod_less]) 1);
by (asm_simp_tac (simpset() addsimps [mod2_succ_succ, add_succ_right]) 1);
qed "mod2_add_self";
@@ -451,7 +390,7 @@
(**** Additional theorems about "le" ****)
Goal "[| m:nat; n:nat |] ==> m le m #+ n";
-by (etac nat_induct 1);
+by (induct_tac "m" 1);
by (ALLGOALS Asm_simp_tac);
qed "add_le_self";
@@ -510,7 +449,7 @@
Goal "[| i le j; j:nat; k:nat |] ==> i#*k le j#*k";
by (forward_tac [lt_nat_in_nat] 1);
-by (nat_ind_tac "k" [] 2);
+by (induct_tac "k" 2);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
qed "mult_le_mono1";
@@ -529,7 +468,7 @@
by (etac zero_lt_natE 1);
by (forward_tac [lt_nat_in_nat] 2);
by (ALLGOALS Asm_simp_tac);
-by (nat_ind_tac "x" [] 1);
+by (induct_tac "x" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_lt_mono])));
qed "mult_lt_mono2";
@@ -571,38 +510,28 @@
div_termination RS ltD]) 1);
qed "mult_mod_distrib";
-(** Lemma for gcd **)
-
-val mono_lemma = (nat_into_Ord RS Ord_0_lt) RSN (2,mult_lt_mono2);
-
+(*Lemma for gcd*)
Goal "[| m = m#*n; m: nat; n: nat |] ==> n=1 | m=0";
by (rtac disjCI 1);
by (dtac sym 1);
by (rtac Ord_linear_lt 1 THEN REPEAT_SOME (ares_tac [nat_into_Ord,nat_1I]));
-by (fast_tac (claset() addss (simpset())) 1);
-by (fast_tac (le_cs addDs [mono_lemma]
- addss (simpset() addsimps [mult_1_right])) 1);
+by (dtac (nat_into_Ord RS Ord_0_lt RSN (2,mult_lt_mono2)) 2);
+by Auto_tac;
qed "mult_eq_self_implies_10";
-
(*Thanks to Sten Agerholm*)
Goal "[|m#+n le m#+k; m:nat; n:nat; k:nat|] ==> n le k";
by (etac rev_mp 1);
-by (eres_inst_tac [("n","n")] nat_induct 1);
+by (induct_tac "m" 1);
by (Asm_simp_tac 1);
by Safe_tac;
by (asm_full_simp_tac (simpset() addsimps [not_le_iff_lt,nat_into_Ord]) 1);
-by (etac lt_asym 1);
-by (assume_tac 1);
-by (Asm_full_simp_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [le_iff, nat_into_Ord]) 1);
-by (Blast_tac 1);
qed "add_le_elim1";
Goal "[| m<n; n: nat |] ==> EX k: nat. n = succ(m#+k)";
by (forward_tac [lt_nat_in_nat] 1 THEN assume_tac 1);
be rev_mp 1;
-by (etac nat_induct 1);
+by (induct_tac "n" 1);
by (ALLGOALS (simp_tac (simpset() addsimps [le_iff])));
by (blast_tac (claset() addSEs [leE]
addSIs [add_0_right RS sym, add_succ_right RS sym]) 1);