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% File     : NUM006^0 : TPTP v8.2.0. Released v3.6.0.
% Domain   : Number Theory
% Axioms   : Church Numerals in Simple Type Theory
% Version  : [Ben08] axioms : Especial.
% English  :

% Refs     : [Ben08] Benzmueller (2008), Email to G. Sutcliffe
% Source   : [Ben08]
% Names    : CHURCH_NUM [Ben08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   28 (  14 unt;  14 typ;  14 def)
%            Number of atoms       :   14 (  14 equ;   0 cnn)
%            Maximal formula atoms :    1 (   0 avg)
%            Number of connectives :   65 (   0   ~;   0   |;   0   &;  65   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    1 (   1 avg;  65 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   91 (  91   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   15 (  14 usr;   0 con; 2-4 aty)
%            Number of variables   :   33 (  33   ^   0   !;   0   ?;  33   :)
% SPC      : 

% Comments :
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thf(zero,type,
    zero: ( $i > $i ) > $i > $i ).

thf(one,type,
    one: ( $i > $i ) > $i > $i ).

thf(two,type,
    two: ( $i > $i ) > $i > $i ).

thf(three,type,
    three: ( $i > $i ) > $i > $i ).

thf(four,type,
    four: ( $i > $i ) > $i > $i ).

thf(five,type,
    five: ( $i > $i ) > $i > $i ).

thf(six,type,
    six: ( $i > $i ) > $i > $i ).

thf(seven,type,
    seven: ( $i > $i ) > $i > $i ).

thf(eight,type,
    eight: ( $i > $i ) > $i > $i ).

thf(nine,type,
    nine: ( $i > $i ) > $i > $i ).

thf(ten,type,
    ten: ( $i > $i ) > $i > $i ).

thf(succ,type,
    succ: ( ( $i > $i ) > $i > $i ) > ( $i > $i ) > $i > $i ).

thf(plus,type,
    plus: ( ( $i > $i ) > $i > $i ) > ( ( $i > $i ) > $i > $i ) > ( $i > $i ) > $i > $i ).

thf(mult,type,
    mult: ( ( $i > $i ) > $i > $i ) > ( ( $i > $i ) > $i > $i ) > ( $i > $i ) > $i > $i ).

thf(zero_ax,definition,
    ( zero
    = ( ^ [X: $i > $i,Y: $i] : Y ) ) ).

thf(one_ax,definition,
    ( one
    = ( ^ [X: $i > $i,Y: $i] : ( X @ Y ) ) ) ).

thf(two_ax,definition,
    ( two
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ Y ) ) ) ) ).

thf(three_ax,definition,
    ( three
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ Y ) ) ) ) ) ).

thf(four_ax,definition,
    ( four
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ ( X @ Y ) ) ) ) ) ) ).

thf(five_ax,definition,
    ( five
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ ( X @ ( X @ Y ) ) ) ) ) ) ) ).

thf(six_ax,definition,
    ( six
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ Y ) ) ) ) ) ) ) ) ).

thf(seven_ax,definition,
    ( seven
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ Y ) ) ) ) ) ) ) ) ) ).

thf(eight_ax,definition,
    ( eight
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ Y ) ) ) ) ) ) ) ) ) ) ).

thf(nine_ax,definition,
    ( nine
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ Y ) ) ) ) ) ) ) ) ) ) ) ).

thf(ten_ax,definition,
    ( ten
    = ( ^ [X: $i > $i,Y: $i] : ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ ( X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(succ_ax,definition,
    ( succ
    = ( ^ [N: ( $i > $i ) > $i > $i,X: $i > $i,Y: $i] : ( X @ ( N @ X @ Y ) ) ) ) ).

thf(plus_ax,definition,
    ( plus
    = ( ^ [M: ( $i > $i ) > $i > $i,N: ( $i > $i ) > $i > $i,X: $i > $i,Y: $i] : ( M @ X @ ( N @ X @ Y ) ) ) ) ).

thf(mult_ax,definition,
    ( mult
    = ( ^ [M: ( $i > $i ) > $i > $i,N: ( $i > $i ) > $i > $i,X: $i > $i,Y: $i] : ( M @ ( N @ X ) @ Y ) ) ) ).

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