TPTP Problem File: SWW588=2.p

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%------------------------------------------------------------------------------
% File     : SWW588=2 : TPTP v7.4.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Division-T-WP parameter division
% Version  : Especial : Let and conditional terms encoded away.
% English  :

% Refs     : [Fil14] Filliatre (2014), Email to Geoff Sutcliffe
%          : [BF+]   Bobot et al. (URL), Toccata: Certified Programs and Cert
% Source   : [Fil14]
% Names    : division-T-WP_parameter_division [Fil14]

% Status   : Theorem
% Rating   : 0.30 v7.4.0, 0.25 v7.3.0, 0.17 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.57 v6.2.0, 0.62 v6.1.0
% Syntax   : Number of formulae    :   31 (   6 unit;  18 type)
%            Number of atoms       :   36 (  14 equality)
%            Maximal formula depth :   18 (   4 average)
%            Number of connectives :   25 (   2   ~;   1   |;  10   &)
%                                         (   0 <=>;  12  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   12 (   6   >;   6   *;   0   +;   0  <<)
%            Number of predicates  :   25 (  21 propositional; 0-2 arity)
%            Number of functors    :   13 (   5 constant; 0-4 arity)
%            Number of variables   :   31 (   0 sgn;  30   !;   1   ?)
%                                         (  31   :;   0  !>;   0  ?*)
%            Maximal term depth    :    3 (   1 average)
%            Arithmetic symbols    :   17 (   2 prd;   3 fun;   2 num;  10 var)
% SPC      : TF0_THM_EQU_ARI

% Comments :
%------------------------------------------------------------------------------
tff(uni,type,(
    uni: $tType )).

tff(ty,type,(
    ty: $tType )).

tff(sort,type,(
    sort1: ( ty * uni ) > $o )).

tff(witness,type,(
    witness1: ty > uni )).

tff(witness_sort1,axiom,(
    ! [A: ty] : sort1(A,witness1(A)) )).

tff(int,type,(
    int: ty )).

tff(real,type,(
    real: ty )).

tff(bool,type,(
    bool1: $tType )).

tff(bool1,type,(
    bool: ty )).

tff(true,type,(
    true1: bool1 )).

tff(false,type,(
    false1: bool1 )).

tff(match_bool,type,(
    match_bool1: ( ty * bool1 * uni * uni ) > uni )).

tff(match_bool_sort1,axiom,(
    ! [A: ty,X: bool1,X1: uni,X2: uni] : sort1(A,match_bool1(A,X,X1,X2)) )).

tff(match_bool_True,axiom,(
    ! [A: ty,Z: uni,Z1: uni] :
      ( sort1(A,Z)
     => match_bool1(A,true1,Z,Z1) = Z ) )).

tff(match_bool_False,axiom,(
    ! [A: ty,Z: uni,Z1: uni] :
      ( sort1(A,Z1)
     => match_bool1(A,false1,Z,Z1) = Z1 ) )).

tff(true_False,axiom,(
    true1 != false1 )).

tff(bool_inversion,axiom,(
    ! [U: bool1] :
      ( U = true1
      | U = false1 ) )).

tff(tuple0,type,(
    tuple02: $tType )).

tff(tuple01,type,(
    tuple0: ty )).

tff(tuple02,type,(
    tuple03: tuple02 )).

tff(tuple0_inversion,axiom,(
    ! [U: tuple02] : U = tuple03 )).

tff(qtmark,type,(
    qtmark: ty )).

tff(compatOrderMult,axiom,(
    ! [X: $int,Y: $int,Z: $int] :
      ( $lesseq(X,Y)
     => ( $lesseq(0,Z)
       => $lesseq($product(X,Z),$product(Y,Z)) ) ) )).

tff(ref,type,(
    ref: ty > ty )).

tff(mk_ref,type,(
    mk_ref: ( ty * uni ) > uni )).

tff(mk_ref_sort1,axiom,(
    ! [A: ty,X: uni] : sort1(ref(A),mk_ref(A,X)) )).

tff(contents,type,(
    contents: ( ty * uni ) > uni )).

tff(contents_sort1,axiom,(
    ! [A: ty,X: uni] : sort1(A,contents(A,X)) )).

tff(contents_def1,axiom,(
    ! [A: ty,U: uni] :
      ( sort1(A,U)
     => contents(A,mk_ref(A,U)) = U ) )).

tff(ref_inversion1,axiom,(
    ! [A: ty,U: uni] :
      ( sort1(ref(A),U)
     => U = mk_ref(A,contents(A,U)) ) )).

tff(wP_parameter_division,conjecture,(
    ! [A: $int,B: $int] :
      ( ( $lesseq(0,A)
        & $less(0,B) )
     => ( $sum($product(0,B),A) = A
        & $lesseq(0,A)
        & ! [R: $int,Q: $int] :
            ( ( $sum($product(Q,B),R) = A
              & $lesseq(0,R) )
           => ( ( $lesseq(B,R)
               => ! [Q1: $int] :
                    ( Q1 = $sum(Q,1)
                   => ! [R1: $int] :
                        ( R1 = $difference(R,B)
                       => ( $sum($product(Q1,B),R1) = A
                          & $lesseq(0,R1)
                          & $lesseq(0,R)
                          & $less(R1,R) ) ) ) )
              & ( ~ $lesseq(B,R)
               => ? [R1: $int] :
                    ( $sum($product(Q,B),R1) = A
                    & $lesseq(0,R1)
                    & $less(R1,B) ) ) ) ) ) ) )).

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