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## Natural Numbers: Basic Definitions

The basic definitions are held in the file lib_nat. We find it useful to define the double iterator and double induction, and the constants one to ten. This file also defines the algebraic functions addition, multiplication and exponentiation. We also define pred (predecessor) in one step from this we define truncated subtraction (minus).

``` ** Module lib_nat Imports lib_bool
\$nat : Type(nat)
\$zero : nat
\$suc : nat->nat
\$nat_elim :
{C_nat:nat->Type}(C_nat zero)->
({x1:nat}(C_nat x1)->C_nat (suc x1))->{z:nat}C_nat z
** Label (!nat!) nat
** Label (!nat elim!) nat_elim
** Label (!nat zero!) zero
** Label (!nat suc!) suc
[[C_nat:nat->Type][f_zero:C_nat zero]
[f_suc:{x1:nat}(C_nat x1)->C_nat (suc x1)][x1:nat]
nat_elim C_nat f_zero f_suc zero  ==>  f_zero
|| nat_elim C_nat f_zero f_suc (suc x1)  ==>
f_suc x1 (nat_elim C_nat f_zero f_suc x1)]

Gen (!nat is zero!) as nat_is_zero = ... : nat->Type
Gen (!nat is suc!) as nat_is_suc = ... : nat->Type
Gen (!nat zero injective!) as nat_zero_injective = ... :
(Eq zero zero)->{P|Type}P->P
Gen (!nat suc injective!) as nat_suc_injective = ... :
{ix0,iy0|nat}(Eq (suc ix0) (suc iy0))->{P|Type}((Eq ix0 iy0)->P)->P
nat_rec = ... : {t|Type}t->(nat->t->t)->nat->t
nat_iter = ... : {t|Type}t->(t->t)->nat->t
nat_ind = ... :
{P:nat->Prop}(P zero)->({n:nat}(P n)->P (suc n))->{n:nat}P n
nat_double_elim = ... :
{T:nat->nat->Type}(T zero zero)->
({n:nat}(T zero n)->T zero (suc n))->
({m:nat}({n:nat}T m n)->T (suc m) zero)->
({m:nat}({n:nat}T m n)->{n:nat}(T (suc m) n)->T (suc m) (suc n))->
{m,n:nat}T m n
nat_diagonal_iter = ... :
{C|Type}(nat->C)->(nat->C)->(C->C)->nat->nat->C
nat_diagonal_ind = ... :
{phi:nat->nat->Prop}({m:nat}phi zero m)->({n:nat}phi (suc n) zero)->
({n,m:nat}(phi n m)->phi (suc n) (suc m))->{n,m:nat}phi n m
nat_d_rec_for_int = ... :
{C:nat->nat->Type}(C zero zero)->({m,n:nat}(C m n)->C (suc m) n)->
({m,n:nat}(C m n)->C m (suc n))->{m,n:nat}C m n
is_suc = ... : nat->Prop
is_zero = ... : nat->Prop
zero_not_suc = ... : {n:nat}not (Eq zero (suc n))
suc_not_zero = ... : not (is_suc zero)
suc_n_not_n = ... : {n:nat}not (Eq (suc n) n)
** Config Qnify nat suc_n_not_n
zero_or_suc = ... : {n:nat}(Eq n zero \/ is_suc n)
one = ... : nat
two = ... : nat
three = ... : nat
four = ... : nat
five = ... : nat
six = ... : nat
seven = ... : nat
eight = ... : nat
nine = ... : nat
ten = ... : nat
plus = ... : nat->nat->nat
times = ... : nat->nat->nat
exp = ... : nat->nat->nat
pred = ... : nat->nat
minus = ... : nat->nat->nat
ackerman = ... : nat->nat->nat
if_zero = ... : {t|Type}nat->t->t->t
fact = ... : nat->nat
zero_not_one = ... : not (Eq zero one)
```

Conor McBride
11/13/1998