(* Title: ZF/arith.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
For arith.thy. 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 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))";
by (rtac rec_trans 1);
by (simp_tac (ZF_ss addsimps [nat_case_succ, nat_succI]) 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
(asm_simp_tac (ZF_ss addsimps (prems@[rec_0,rec_succ]))));
val rec_type = result();
val nat_le_refl = naturals_are_ordinals RS le_refl;
val nat_typechecks = [rec_type, nat_0I, nat_1I, nat_succI, Ord_nat];
val nat_simps = [rec_0, rec_succ, not_lt0, nat_0_le, le0_iff, succ_le_iff,
nat_le_refl];
val nat_ss = ZF_ss addsimps (nat_simps @ nat_typechecks);
(** Addition **)
val add_type = prove_goalw Arith.thy [add_def]
"[| m:nat; n:nat |] ==> m #+ n : nat"
(fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]);
val add_0 = prove_goalw Arith.thy [add_def]
"0 #+ n = n"
(fn _ => [ (rtac rec_0 1) ]);
val add_succ = prove_goalw Arith.thy [add_def]
"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=>
[ (typechk_tac (prems@[add_type]@nat_typechecks@ZF_typechecks)) ]);
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 addsimps prems) 1),
(nat_ind_tac "n" prems 1),
(ALLGOALS (asm_simp_tac (nat_ss addsimps prems))) ]);
val prems = goal Arith.thy
"[| m:nat; n:nat |] ==> m #- n le m";
by (rtac (prems MRS diff_induct) 1);
by (etac leE 3);
by (ALLGOALS
(asm_simp_tac
(nat_ss addsimps (prems @ [le_iff, diff_0, diff_0_eq_0,
diff_succ_succ, naturals_are_ordinals]))));
val diff_le_self = result();
(*** Simplification over add, mult, diff ***)
val arith_typechecks = [add_type, mult_type, diff_type];
val arith_simps = [add_0, add_succ,
mult_0, mult_succ,
diff_0, diff_0_eq_0, diff_succ_succ];
val arith_ss = nat_ss addsimps (arith_simps@arith_typechecks);
(*** Addition ***)
(*Associative law for addition*)
val add_assoc = prove_goal Arith.thy
"m:nat ==> (m #+ n) #+ k = m #+ (n #+ k)"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]);
(*The following two lemmas are used for add_commute and sometimes
elsewhere, since they are safe for rewriting.*)
val add_0_right = prove_goal Arith.thy
"m:nat ==> m #+ 0 = m"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]);
val add_succ_right = prove_goal Arith.thy
"m:nat ==> m #+ succ(n) = succ(m #+ n)"
(fn prems=>
[ (nat_ind_tac "m" prems 1),
(ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]);
(*Commutative law for addition*)
val add_commute = prove_goal Arith.thy
"[| m:nat; n:nat |] ==> m #+ n = n #+ m"
(fn prems=>
[ (nat_ind_tac "n" prems 1),
(ALLGOALS
(asm_simp_tac
(arith_ss addsimps (prems@[add_0_right, add_succ_right])))) ]);
(*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);
val add_left_cancel = result();
(*** 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 addsimps prems))) ]);
(*right successor law for multiplication*)
val mult_succ_right = prove_goal Arith.thy
"!!m n. [| m:nat; n:nat |] ==> m #* succ(n) = m #+ (m #* n)"
(fn _=>
[ (nat_ind_tac "m" [] 1),
(ALLGOALS (asm_simp_tac (arith_ss addsimps [add_assoc RS sym]))),
(*The final goal requires the commutative law for addition*)
(rtac (add_commute RS subst_context) 1),
(REPEAT (assume_tac 1)) ]);
(*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
(arith_ss addsimps (prems@[mult_0_right, mult_succ_right])))) ]);
(*addition distributes over multiplication*)
val add_mult_distrib = prove_goal Arith.thy
"!!m n. [| m:nat; k:nat |] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)"
(fn _=>
[ (etac nat_induct 1),
(ALLGOALS (asm_simp_tac (arith_ss addsimps [add_assoc RS sym]))) ]);
(*Distributive law on the left; requires an extra typing premise*)
val add_mult_distrib_left = prove_goal Arith.thy
"[| 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
val ss = arith_ss addsimps ([mult_commute',add_mult_distrib]@prems)
in [ (simp_tac ss 1) ]
end);
(*Associative law for multiplication*)
val mult_assoc = prove_goal 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 (arith_ss addsimps [add_mult_distrib]))) ]);
(*** 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 addsimps prems))) ]);
(*Addition is the inverse of subtraction*)
goal Arith.thy "!!m n. [| n le m; m:nat |] ==> n #+ (m#-n) = m";
by (forward_tac [lt_nat_in_nat] 1);
by (etac nat_succI 1);
by (etac rev_mp 1);
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
by (ALLGOALS (asm_simp_tac arith_ss));
val add_diff_inverse = result();
(*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 addsimps [mnat])));
val diff_add_inverse = result();
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 addsimps [mnat])));
val diff_add_0 = result();
(*** Remainder ***)
goal Arith.thy "!!m n. [| 0<n; n le m; m:nat |] ==> m #- n < m";
by (forward_tac [lt_nat_in_nat] 1 THEN etac nat_succI 1);
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 (nat_ss addsimps [diff_le_self,diff_succ_succ])));
val div_termination = result();
val div_rls = (*for mod and div*)
nat_typechecks @
[Ord_transrec_type, apply_type, div_termination RS ltD, if_type,
naturals_are_ordinals, not_lt_iff_le RS iffD1];
val div_ss = ZF_ss addsimps [naturals_are_ordinals, div_termination RS ltD,
not_lt_iff_le RS iffD2];
(*Type checking depends upon termination!*)
goalw Arith.thy [mod_def] "!!m n. [| 0<n; m:nat; n:nat |] ==> m mod n : nat";
by (REPEAT (ares_tac div_rls 1 ORELSE etac lt_trans 1));
val mod_type = result();
goal Arith.thy "!!m n. [| 0<n; m<n |] ==> m mod n = m";
by (rtac (mod_def RS def_transrec RS trans) 1);
by (asm_simp_tac div_ss 1);
val mod_less = result();
goal Arith.thy "!!m n. [| 0<n; n le m; m:nat |] ==> m mod n = (m#-n) mod n";
by (forward_tac [lt_nat_in_nat] 1 THEN etac nat_succI 1);
by (rtac (mod_def RS def_transrec RS trans) 1);
by (asm_simp_tac div_ss 1);
val mod_geq = result();
(*** Quotient ***)
(*Type checking depends upon termination!*)
goalw Arith.thy [div_def]
"!!m n. [| 0<n; m:nat; n:nat |] ==> m div n : nat";
by (REPEAT (ares_tac div_rls 1 ORELSE etac lt_trans 1));
val div_type = result();
goal Arith.thy "!!m n. [| 0<n; m<n |] ==> m div n = 0";
by (rtac (div_def RS def_transrec RS trans) 1);
by (asm_simp_tac div_ss 1);
val div_less = result();
goal Arith.thy
"!!m n. [| 0<n; n le m; m:nat |] ==> m div n = succ((m#-n) div n)";
by (forward_tac [lt_nat_in_nat] 1 THEN etac nat_succI 1);
by (rtac (div_def RS def_transrec RS trans) 1);
by (asm_simp_tac div_ss 1);
val div_geq = result();
(*Main Result.*)
goal Arith.thy
"!!m n. [| 0<n; m:nat; n:nat |] ==> (m div n)#*n #+ m mod n = m";
by (etac complete_induct 1);
by (res_inst_tac [("Q","x<n")] (excluded_middle RS disjE) 1);
(*case x<n*)
by (asm_simp_tac (arith_ss addsimps [mod_less, div_less]) 2);
(*case n le x*)
by (asm_full_simp_tac
(arith_ss addsimps [not_lt_iff_le, naturals_are_ordinals,
mod_geq, div_geq, add_assoc,
div_termination RS ltD, add_diff_inverse]) 1);
val mod_div_equality = result();
(**** Additional theorems about "le" ****)
goal Arith.thy "!!m n. [| m:nat; n:nat |] ==> m le m #+ n";
by (etac nat_induct 1);
by (ALLGOALS (asm_simp_tac arith_ss));
val add_le_self = result();
goal Arith.thy "!!m n. [| m:nat; n:nat |] ==> m le n #+ m";
by (rtac (add_commute RS ssubst) 1);
by (REPEAT (ares_tac [add_le_self] 1));
val add_le_self2 = result();
(** Monotonicity of addition **)
(*strict, in 1st argument*)
goal Arith.thy "!!i j k. [| i<j; j:nat; k:nat |] ==> i#+k < j#+k";
by (forward_tac [lt_nat_in_nat] 1);
by (assume_tac 1);
by (etac succ_lt_induct 1);
by (ALLGOALS (asm_simp_tac (arith_ss addsimps [leI])));
val add_lt_mono1 = result();
(*strict, in both arguments*)
goal Arith.thy "!!i j k l. [| i<j; k<l; j:nat; l:nat |] ==> i#+k < j#+l";
by (rtac (add_lt_mono1 RS lt_trans) 1);
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat] 1));
by (EVERY [rtac (add_commute RS ssubst) 1,
rtac (add_commute RS ssubst) 3,
rtac add_lt_mono1 5]);
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat] 1));
val add_lt_mono = result();
(*A [clumsy] way of lifting < monotonicity to le monotonicity *)
val lt_mono::ford::prems = goal Ord.thy
"[| !!i j. [| i<j; j:k |] ==> f(i) < f(j); \
\ !!i. i:k ==> Ord(f(i)); \
\ i le j; j:k \
\ |] ==> f(i) le f(j)";
by (cut_facts_tac prems 1);
by (fast_tac (lt_cs addSIs [lt_mono,ford] addSEs [leE]) 1);
val Ord_lt_mono_imp_le_mono = result();
(*le monotonicity, 1st argument*)
goal Arith.thy
"!!i j k. [| i le j; j:nat; k:nat |] ==> i#+k le j#+k";
by (res_inst_tac [("f", "%j.j#+k")] Ord_lt_mono_imp_le_mono 1);
by (REPEAT (ares_tac [add_lt_mono1, add_type RS naturals_are_ordinals] 1));
val add_le_mono1 = result();
(* le monotonicity, BOTH arguments*)
goal Arith.thy
"!!i j k. [| i le j; k le l; j:nat; l:nat |] ==> i#+k le j#+l";
by (rtac (add_le_mono1 RS le_trans) 1);
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat, nat_succI] 1));
by (EVERY [rtac (add_commute RS ssubst) 1,
rtac (add_commute RS ssubst) 3,
rtac add_le_mono1 5]);
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat, nat_succI] 1));
val add_le_mono = result();