%--------------------------------------------------------------------------
% File     : MGT025+1 : TPTP v8.2.0. Released v2.0.0.
% Domain   : Management (Organisation Theory)
% Problem  : Constant population means opposite growth rates
% Version  : [PB+94] axioms : Reduced & Augmented > Complete.
% English  : If one of the two subpopulations has positive growth rate,
%            then the other subpopulation must have negative growth rate
%            if the total number of organizations is constant.

% Refs     : [PM93]  Peli & Masuch (1993), The Logic of Propogation Strateg
%          : [PM94]  Peli & Masuch (1994), The Logic of Propogation Strateg
%          : [Kam95] Kamps (1995), Email to G. Sutcliffe
% Source   : [Kam95]
% Names    :

% Status   : Theorem
% Rating   : 0.14 v8.1.0, 0.08 v7.5.0, 0.12 v7.4.0, 0.10 v7.2.0, 0.07 v7.1.0, 0.04 v7.0.0, 0.07 v6.4.0, 0.12 v6.3.0, 0.08 v6.1.0, 0.13 v5.5.0, 0.26 v5.4.0, 0.29 v5.3.0, 0.37 v5.2.0, 0.20 v5.1.0, 0.24 v5.0.0, 0.21 v4.1.0, 0.26 v4.0.1, 0.22 v4.0.0, 0.25 v3.7.0, 0.30 v3.5.0, 0.26 v3.4.0, 0.32 v3.3.0, 0.36 v3.1.0, 0.44 v2.7.0, 0.50 v2.6.0, 0.43 v2.5.0, 0.38 v2.4.0, 0.50 v2.3.0, 0.33 v2.2.1, 0.00 v2.1.0
% Syntax   : Number of formulae    :    8 (   0 unt;   0 def)
%            Number of atoms       :   49 (   9 equ)
%            Maximal formula atoms :   10 (   6 avg)
%            Number of connectives :   41 (   0   ~;   6   |;  23   &)
%                                         (   0 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   7 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    9 (   8 usr;   0 prp; 1-4 aty)
%            Number of functors    :    7 (   7 usr;   3 con; 0-2 aty)
%            Number of variables   :   20 (  20   !;   0   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%----Subsitution axioms
%----Problem axioms
%----MP. If there are only first movers and efficient producers in an
%----environment, then the number of organizations is the sum of members
%----in these groups.
fof(mp_only_members,axiom,
    ! [E,X,T] :
      ( ( environment(E)
        & subpopulation(X,E,T)
        & ( greater(cardinality_at_time(X,T),zero)
         => ( X = efficient_producers
            | X = first_movers ) ) )
     => number_of_organizations(E,T) = sum(cardinality_at_time(first_movers,T),cardinality_at_time(efficient_producers,T)) ) ).

%----MP. First movers and efficient producers are subpopulations.
fof(mp_subpopulations,axiom,
    ! [E,T] :
      ( ( environment(E)
        & in_environment(E,T) )
     => ( subpopulation(first_movers,E,T)
        & subpopulation(efficient_producers,E,T) ) ) ).

%----MP. If a constant "a" is the sum of "b" and "c", then either "b" and
%----"c" are also constants, or one of the two additives increases, while
%----the other decreases.
fof(mp_abc_sum_increase,axiom,
    ! [A,B,C] :
      ( ( A = sum(B,C)
        & constant(A) )
     => ( ( constant(B)
          & constant(C) )
        | ( increases(B)
          & decreases(C) )
        | ( decreases(B)
          & increases(C) ) ) ) ).

%----MP. If the number of a non-empty subpopulation is constant or
%----increases or decreases, then its growth rate is, respectively, zero
%----or positive or negative.
fof(mp_growth_rate,axiom,
    ! [X,E,T] :
      ( ( environment(E)
        & in_environment(E,T)
        & subpopulation(X,E,T)
        & greater(cardinality_at_time(X,T),zero) )
     => ( ( constant(cardinality_at_time(X,T))
         => growth_rate(X,T) = zero )
        & ( increases(cardinality_at_time(X,T))
         => greater(growth_rate(X,T),zero) )
        & ( decreases(cardinality_at_time(X,T))
         => greater(zero,growth_rate(X,T)) ) ) ) ).

%----MP. If the population is composed of first movers and of efficient
%----producers, then their number is not zero.
fof(mp_non_zero_producers,axiom,
    ! [E,T] :
      ( ( environment(E)
        & subpopulations(first_movers,efficient_producers,E,T) )
     => ( greater(cardinality_at_time(first_movers,T),zero)
        & greater(cardinality_at_time(efficient_producers,T),zero) ) ) ).

%----MP. The time points when FM and EP are present in the environment
%----occur during the environment sustains.
fof(mp_time_point_occur,axiom,
    ! [E,T] :
      ( ( environment(E)
        & subpopulations(first_movers,efficient_producers,E,T) )
     => in_environment(E,T) ) ).

%----A9. The population contains only first movers and efficient producers.
fof(a9,hypothesis,
    ! [E,X,T] :
      ( ( environment(E)
        & subpopulation(X,E,T)
        & greater(cardinality_at_time(X,T),zero) )
     => ( X = efficient_producers
        | X = first_movers ) ) ).

%----GOAL: L7. If one of the two subpopulations has positive growth rate,
%----then the other subpopulation must have negative growth rate if the
%----total number of organizations is constant.
fof(prove_l7,conjecture,
    ! [E,T] :
      ( ( environment(E)
        & subpopulations(first_movers,efficient_producers,E,T)
        & constant(number_of_organizations(E,T)) )
     => ( ( growth_rate(first_movers,T) = zero
          & growth_rate(efficient_producers,T) = zero )
        | ( greater(growth_rate(first_movers,T),zero)
          & greater(zero,growth_rate(efficient_producers,T)) )
        | ( greater(growth_rate(efficient_producers,T),zero)
          & greater(zero,growth_rate(first_movers,T)) ) ) ) ).

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