src/ZF/Arith.ML
 author clasohm Thu, 16 Sep 1993 12:20:38 +0200 changeset 0 a5a9c433f639 child 6 8ce8c4d13d4d permissions -rw-r--r--
Initial revision
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(*  Title: 	ZF/arith.ML
ID:         \$Id\$
Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

For arith.thy.  Arithmetic operators and their definitions

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 Arith.thy "rec(0,a,b) = a";
by (rtac rec_trans 1);
by (rtac nat_case_0 1);
val rec_0 = result();

goal Arith.thy "rec(succ(m),a,b) = b(m, rec(m,a,b))";
val rec_ss = ZF_ss
by (rtac rec_trans 1);
by (SIMP_TAC rec_ss 1);
val rec_succ = result();

val major::prems = goal Arith.thy
"[| 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
val rec_type = result();

val prems = goalw Arith.thy [rec_def]
"[| n=n';  a=a';  !!m z. b(m,z)=b'(m,z)  \
\    |] ==> rec(n,a,b)=rec(n',a',b')";
val rec_cong = result();

val nat_typechecks = [rec_type,nat_0I,nat_1I,nat_succI,Ord_nat];

val nat_ss = ZF_ss addcongs [nat_case_cong,rec_cong]

"[| m:nat;  n:nat |] ==> m #+ n : nat"
(fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]);

"0 #+ n = n"
(fn _ => [ (rtac rec_0 1) ]);

"succ(m) #+ n = succ(m #+ n)"
(fn _=> [ (rtac rec_succ 1) ]);

(** Multiplication **)

val mult_type = prove_goalw Arith.thy [mult_def]
"[| m:nat;  n:nat |] ==> m #* n : nat"
(fn prems=>

val mult_0 = prove_goalw Arith.thy [mult_def]
"0 #* n = 0"
(fn _ => [ (rtac rec_0 1) ]);

val mult_succ = prove_goalw Arith.thy [mult_def]
"succ(m) #* n = n #+ (m #* n)"
(fn _ => [ (rtac rec_succ 1) ]);

(** Difference **)

val diff_type = prove_goalw Arith.thy [diff_def]
"[| m:nat;  n:nat |] ==> m #- n : nat"
(fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]);

val diff_0 = prove_goalw Arith.thy [diff_def]
"m #- 0 = m"
(fn _ => [ (rtac rec_0 1) ]);

val diff_0_eq_0 = prove_goalw Arith.thy [diff_def]
"n:nat ==> 0 #- n = 0"
(fn [prem]=>
[ (rtac (prem RS nat_induct) 1),
(ALLGOALS (ASM_SIMP_TAC nat_ss)) ]);

(*Must simplify BEFORE the induction!!  (Else we get a critical pair)
succ(m) #- succ(n)   rewrites to   pred(succ(m) #- n)  *)
val diff_succ_succ = prove_goalw Arith.thy [diff_def]
"[| m:nat;  n:nat |] ==> succ(m) #- succ(n) = m #- n"
(fn prems=>
[ (ASM_SIMP_TAC (nat_ss addrews prems) 1),
(nat_ind_tac "n" prems 1),
(ALLGOALS (ASM_SIMP_TAC (nat_ss addrews prems))) ]);

val prems = goal Arith.thy
"[| m:nat;  n:nat |] ==> m #- n : succ(m)";
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
by (resolve_tac prems 1);
by (resolve_tac prems 1);
by (etac succE 3);
by (ALLGOALS
(ASM_SIMP_TAC
val diff_leq = result();

(*** Simplification over add, mult, diff ***)

val arith_typechecks = [add_type, mult_type, diff_type];
mult_0, mult_succ,
diff_0, diff_0_eq_0, diff_succ_succ];

val arith_congs = mk_congs Arith.thy ["op #+", "op #-", "op #*"];

val arith_ss = nat_ss addcongs arith_congs

"m:nat ==> (m #+ n) #+ k = m #+ (n #+ k)"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]);

(*The following two lemmas are used for add_commute and sometimes
elsewhere, since they are safe for rewriting.*)
"m:nat ==> m #+ 0 = m"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]);

"m:nat ==> m #+ succ(n) = succ(m #+ n)"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]);

"[| m:nat;  n:nat |] ==> m #+ n = n #+ m"
(fn prems=>
[ (nat_ind_tac "n" prems 1),
(ALLGOALS
(ASM_SIMP_TAC

(*Cancellation law on the left*)
val [knat,eqn] = goal Arith.thy
"[| k:nat;  k #+ m = k #+ n |] ==> m=n";
by (rtac (eqn RS rev_mp) 1);
by (nat_ind_tac "k" [knat] 1);
by (ALLGOALS (SIMP_TAC arith_ss));
by (fast_tac ZF_cs 1);

(*** Multiplication ***)

(*right annihilation in product*)
val mult_0_right = prove_goal Arith.thy
"m:nat ==> m #* 0 = 0"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems)))  ]);

(*right successor law for multiplication*)
val mult_succ_right = prove_goal Arith.thy
"[| m:nat;  n:nat |] ==> m #* succ(n) = m #+ (m #* n)"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(*The final goal requires the commutative law for addition*)

(*Commutative law for multiplication*)
val mult_commute = prove_goal Arith.thy
"[| m:nat;  n:nat |] ==> m #* n = n #* m"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (ASM_SIMP_TAC

"[| m:nat;  k:nat |] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)"
(fn prems=>
[ (nat_ind_tac "m" prems 1),

(*Distributive law on the left; requires an extra typing premise*)
"[| m:nat;  n:nat;  k:nat |] ==> k #* (m #+ n) = (k #* m) #+ (k #* n)"
(fn prems=>
let val mult_commute' = read_instantiate [("m","k")] mult_commute
in [ (SIMP_TAC ss 1) ]
end);

(*Associative law for multiplication*)
val mult_assoc = prove_goal Arith.thy
"[| m:nat;  n:nat;  k:nat |] ==> (m #* n) #* k = m #* (n #* k)"
(fn prems=>
[ (nat_ind_tac "m" prems 1),

(*** Difference ***)

val diff_self_eq_0 = prove_goal Arith.thy
"m:nat ==> m #- m = 0"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]);

(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
val notless::prems = goal Arith.thy
"[| ~m:n;  m:nat;  n:nat |] ==> n #+ (m#-n) = m";
by (rtac (notless RS rev_mp) 1);
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
by (resolve_tac prems 1);
by (resolve_tac prems 1);
by (ALLGOALS (ASM_SIMP_TAC
naturals_are_ordinals]))));

(*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 (arith_ss addrews [mnat])));

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 (arith_ss addrews [mnat])));

(*** Remainder ***)

(*In ordinary notation: if 0<n and n<=m then m-n < m *)
val prems = goal Arith.thy
"[| 0:n; ~ m:n;  m:nat;  n:nat |] ==> m #- n : m";
by (cut_facts_tac prems 1);
by (etac rev_mp 1);
by (etac rev_mp 1);
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
by (resolve_tac prems 1);
by (resolve_tac prems 1);
by (ALLGOALS (ASM_SIMP_TAC
val div_termination = result();

val div_rls =
[Ord_transrec_type, apply_type, div_termination, if_type] @
nat_typechecks;

(*Type checking depends upon termination!*)
val prems = goalw Arith.thy [mod_def]
"[| 0:n;  m:nat;  n:nat |] ==> m mod n : nat";
by (REPEAT (ares_tac (prems @ div_rls) 1 ORELSE etac Ord_trans 1));
val mod_type = result();

val div_ss = ZF_ss addrews [naturals_are_ordinals,div_termination];

val prems = goal Arith.thy "[| 0:n;  m:n;  m:nat;  n:nat |] ==> m mod n = m";
by (rtac (mod_def RS def_transrec RS trans) 1);
by (SIMP_TAC (div_ss addrews prems) 1);
val mod_less = result();

val prems = goal Arith.thy
"[| 0:n;  ~m:n;  m:nat;  n:nat |] ==> m mod n = (m#-n) mod n";
by (rtac (mod_def RS def_transrec RS trans) 1);
by (SIMP_TAC (div_ss addrews prems) 1);
val mod_geq = result();

(*** Quotient ***)

(*Type checking depends upon termination!*)
val prems = goalw Arith.thy [div_def]
"[| 0:n;  m:nat;  n:nat |] ==> m div n : nat";
by (REPEAT (ares_tac (prems @ div_rls) 1 ORELSE etac Ord_trans 1));
val div_type = result();

val prems = goal Arith.thy
"[| 0:n;  m:n;  m:nat;  n:nat |] ==> m div n = 0";
by (rtac (div_def RS def_transrec RS trans) 1);
by (SIMP_TAC (div_ss addrews prems) 1);
val div_less = result();

val prems = goal Arith.thy
"[| 0:n;  ~m:n;  m:nat;  n:nat |] ==> m div n = succ((m#-n) div n)";
by (rtac (div_def RS def_transrec RS trans) 1);
by (SIMP_TAC (div_ss addrews prems) 1);
val div_geq = result();

(*Main Result.*)
val prems = goal Arith.thy
"[| 0:n;  m:nat;  n:nat |] ==> (m div n)#*n #+ m mod n = m";
by (res_inst_tac [("i","m")] complete_induct 1);
by (resolve_tac prems 1);
by (res_inst_tac [("Q","x:n")] (excluded_middle RS disjE) 1);
by (ALLGOALS
(ASM_SIMP_TAC
(arith_ss addrews ([mod_type,div_type] @ prems @
[mod_less,mod_geq, div_less, div_geq,
val mod_div_equality = result();

val [mnat,nnat] = goal Arith.thy
"[| m:nat;  n:nat |] ==> ~ (m #+ n) : n";
by (rtac (mnat RS nat_induct) 1);
by (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews [mem_not_refl])));
by (rtac notI 1);
by (etac notE 1);
by (etac (succI1 RS Ord_trans) 1);
by (rtac (nnat RS naturals_are_ordinals) 1);