%--------------------------------------------------------------------------
% File     : NUM-ROZEK : TPTP v8.1.2.
% Domain   : Data Structures
% Axioms   : Lists - Head, Tail, and Length
%--------------------------------------------------------------------------

%-- List Basics

tff(list_t, type, list: $tType).

tff(list_nil, type, nil : list).

tff(list_cons, type,
    cons: (
        ($int * list) > list
    )
).

%-- Head

tff(head_t, type,
    head: list > $int
).

tff(head_cons, axiom,
    ![H: $int, T: list]: (
        head(cons(H, T)) = H
    )
).

%-- Tail

tff(tail_t, type,
    tail: list > list
).

tff(tail_cons, axiom,
    ![H: $int, T: list]: (
        tail(cons(H, T)) = T
    )
).

% Note: The existence of head and tail definition shows that
% cons is injective.

%-- Length

tff(length_t, type,
    length: list > $int
).

tff(length_nil, axiom, length(nil) = 0).

tff(length_cons, axiom,
    ![H: $int, T: list]: (
        length(cons(H, T)) = $sum(1, length(T))
    )
).