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 : SET $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 [[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)] 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) zero_or_suc = ... : {n:nat}or (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 zero_not_one = ... : not (Eq zero one)