%------------------------------------------------------------------------------
% File     : SWV014_0 : TPTP v9.1.0. Released v9.1.0.
% Domain   : Software Verification
% Axioms   : Anthem standard preamble
% Version  : [Han25] axioms : Especial.
% English  :

% Refs     : [FL+20] Fandinno et al. (2020), Verifying Tight Logic Programs
%          : [FH+23] Fandinno et al. (2023), External Behavior of a Logic P
%          : [Han25] Hansen (2025), Email to Geoff Sutcliffe
% Source   : [Han25]
% Names    : standard_preamble.ax [Han25]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   27 (   3 unt;  12 typ;   0 def)
%            Number of atoms       :   33 (  11 equ)
%            Maximal formula atoms :    4 (   2 avg)
%            Number of connectives :   20 (   2   ~;   4   |;   4   &)
%                                         (   8 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   4 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number arithmetic     :    8 (   1 atm;   0 fun;   0 num;   7 var)
%            Number of types       :    4 (   2 usr;   1 ari)
%            Number of type conns  :   12 (   8   >;   4   *;   0   +;   0  <<)
%            Number of predicates  :    8 (   6 usr;   0 prp; 1-2 aty)
%            Number of functors    :    4 (   4 usr;   2 con; 0-1 aty)
%            Number of variables   :   28 (  26   !;   2   ?;  28   :)
% SPC      : TF0_SAT_EQU_ARI

% Comments : From https://github.com/ZachJHansen/anthem-benchmarks/tree/tptp
%------------------------------------------------------------------------------
tff(general_type,type,
    general: $tType ).

tff(symbol_type,type,
    symbol: $tType ).

tff(f__integer___decl,type,
    f__integer__: $int > general ).

tff(f__symbolic___decl,type,
    f__symbolic__: symbol > general ).

tff(inf_type,type,
    c__infimum__: general ).

tff(sup_type,type,
    c__supremum__: general ).

tff(p__is_integer__decl,type,
    p__is_integer__: general > $o ).

tff(p__is_symbolic__decl,type,
    p__is_symbolic__: general > $o ).

tff(p__less_equal__decl,type,
    p__less_equal__: ( general * general ) > $o ).

tff(p__less__decl,type,
    p__less__: ( general * general ) > $o ).

tff(p__greater_equal__decl,type,
    p__greater_equal__: ( general * general ) > $o ).

tff(p__greater__decl,type,
    p__greater__: ( general * general ) > $o ).

tff(p__is_integer__def_ax,axiom,
    ! [X: general] :
      ( p__is_integer__(X)
    <=> ? [N: $int] : ( X = f__integer__(N) ) ) ).

tff(p__is_symbolic__def_ax,axiom,
    ! [X1: general] :
      ( p__is_symbolic__(X1)
    <=> ? [X2: symbol] : ( X1 = f__symbolic__(X2) ) ) ).

tff(general_universe_ax,axiom,
    ! [X: general] :
      ( ( X = c__infimum__ )
      | p__is_integer__(X)
      | p__is_symbolic__(X)
      | ( X = c__supremum__ ) ) ).

tff(f__integer__def_ax,axiom,
    ! [N1: $int,N2: $int] :
      ( ( f__integer__(N1) = f__integer__(N2) )
    <=> ( N1 = N2 ) ) ).

tff(f__symbolic__def_ax,axiom,
    ! [S1: symbol,S2: symbol] :
      ( ( f__symbolic__(S1) = f__symbolic__(S2) )
    <=> ( S1 = S2 ) ) ).

tff(numeral_ordering_ax,axiom,
    ! [N1: $int,N2: $int] :
      ( p__less_equal__(f__integer__(N1),f__integer__(N2))
    <=> $lesseq(N1,N2) ) ).

tff(antisymmetric_ordering_ax,axiom,
    ! [X1: general,X2: general] :
      ( ( p__less_equal__(X1,X2)
        & p__less_equal__(X2,X1) )
     => ( X1 = X2 ) ) ).

tff(transitive_ordering_ax,axiom,
    ! [X1: general,X2: general,X3: general] :
      ( ( p__less_equal__(X1,X2)
        & p__less_equal__(X2,X3) )
     => p__less_equal__(X1,X3) ) ).

tff(strongly_connected_ordering_ax,axiom,
    ! [X1: general,X2: general] :
      ( p__less_equal__(X1,X2)
      | p__less_equal__(X2,X1) ) ).

tff(p__less__def_ax,axiom,
    ! [X1: general,X2: general] :
      ( p__less__(X1,X2)
    <=> ( p__less_equal__(X1,X2)
        & ( X1 != X2 ) ) ) ).

tff(p__greater_equal__def_ax,axiom,
    ! [X1: general,X2: general] :
      ( p__greater_equal__(X1,X2)
    <=> p__less_equal__(X2,X1) ) ).

tff(p__greater__def_ax,axiom,
    ! [X1: general,X2: general] :
      ( p__greater__(X1,X2)
    <=> ( p__less_equal__(X2,X1)
        & ( X1 != X2 ) ) ) ).

tff(minimal_element_ax,axiom,
    ! [N: $int] : p__less__(c__infimum__,f__integer__(N)) ).

tff(numerals_less_than_symbols_ax,axiom,
    ! [N: $int,S: symbol] : p__less__(f__integer__(N),f__symbolic__(S)) ).

tff(maximal_element_ax,axiom,
    ! [S: symbol] : p__less__(f__symbolic__(S),c__supremum__) ).

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