TPTP Problem File: SWW652=2.p

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%------------------------------------------------------------------------------
% File     : SWW652=2 : TPTP v7.4.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Vacid 0 build maze-T-Ineq1
% 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    : vacid_0_build_maze-T-Ineq1 [Fil14]

% Status   : Theorem
% Rating   : 0.80 v7.4.0, 0.62 v7.3.0, 0.67 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.86 v6.2.0, 0.88 v6.1.0
% Syntax   : Number of formulae    :   56 (   9 unit;  32 type)
%            Number of atoms       :   68 (  21 equality)
%            Maximal formula depth :   15 (   4 average)
%            Number of connectives :   45 (   1   ~;   3   |;  17   &)
%                                         (   4 <=>;  20  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   27 (  14   >;  13   *;   0   +;   0  <<)
%            Number of predicates  :   43 (  35 propositional; 0-3 arity)
%            Number of functors    :   16 (   5 constant; 0-4 arity)
%            Number of variables   :   68 (   0 sgn;  58   !;  10   ?)
%                                         (  68   :;   0  !>;   0  ?*)
%            Maximal term depth    :    3 (   1 average)
%            Arithmetic symbols    :   38 (   2 prd;   2 fun;   2 num;  32 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(uf_pure,type,(
    uf_pure1: $tType )).

tff(uf_pure1,type,(
    uf_pure: ty )).

tff(repr,type,(
    repr1: ( uf_pure1 * $int * $int ) > $o )).

tff(size,type,(
    size1: uf_pure1 > $int )).

tff(num,type,(
    num1: uf_pure1 > $int )).

tff(repr_function_1,axiom,(
    ! [U: uf_pure1,X: $int] :
      ( ( $lesseq(0,X)
        & $less(X,size1(U)) )
     => ? [Y: $int] :
          ( $lesseq(0,Y)
          & $less(Y,size1(U))
          & repr1(U,X,Y) ) ) )).

tff(repr_function_2,axiom,(
    ! [U: uf_pure1,X: $int,Y: $int,Z: $int] :
      ( ( $lesseq(0,X)
        & $less(X,size1(U)) )
     => ( repr1(U,X,Y)
       => ( repr1(U,X,Z)
         => Y = Z ) ) ) )).

tff(same,type,(
    same1: ( uf_pure1 * $int * $int ) > $o )).

tff(same_def,axiom,(
    ! [U: uf_pure1,X: $int,Y: $int] :
      ( same1(U,X,Y)
    <=> ! [R: $int] :
          ( repr1(U,X,R)
        <=> repr1(U,Y,R) ) ) )).

tff(same_reprs,type,(
    same_reprs1: ( uf_pure1 * uf_pure1 ) > $o )).

tff(same_reprs_def,axiom,(
    ! [U1: uf_pure1,U2: uf_pure1] :
      ( same_reprs1(U1,U2)
    <=> ! [X: $int,R: $int] :
          ( repr1(U1,X,R)
        <=> repr1(U2,X,R) ) ) )).

tff(oneClass,axiom,(
    ! [U: uf_pure1] :
      ( num1(U) = 1
     => ! [X: $int,Y: $int] :
          ( ( $lesseq(0,X)
            & $less(X,size1(U)) )
         => ( ( $lesseq(0,Y)
              & $less(Y,size1(U)) )
           => same1(U,X,Y) ) ) ) )).

tff(uf,type,(
    uf1: $tType )).

tff(uf1,type,(
    uf: ty )).

tff(mk_uf,type,(
    mk_uf1: uf_pure1 > uf1 )).

tff(state,type,(
    state1: uf1 > uf_pure1 )).

tff(state_def1,axiom,(
    ! [U: uf_pure1] : state1(mk_uf1(U)) = U )).

tff(uf_inversion1,axiom,(
    ! [U: uf1] : U = mk_uf1(state1(U)) )).

tff(graph,type,(
    graph1: $tType )).

tff(graph1,type,(
    graph: ty )).

tff(path,type,(
    path1: ( graph1 * $int * $int ) > $o )).

tff(path_refl,axiom,(
    ! [G: graph1,X: $int] : path1(G,X,X) )).

tff(path_sym,axiom,(
    ! [G: graph1,X: $int,Y: $int] :
      ( path1(G,X,Y)
     => path1(G,Y,X) ) )).

tff(path_trans,axiom,(
    ! [G: graph1,X: $int,Y: $int,Z: $int] :
      ( path1(G,X,Y)
     => ( path1(G,Y,Z)
       => path1(G,X,Z) ) ) )).

tff(path_inversion,axiom,(
    ! [Z: graph1,Z1: $int,Z2: $int] :
      ( path1(Z,Z1,Z2)
     => ( ? [G: graph1,X: $int] :
            ( Z = G
            & Z1 = X
            & Z2 = X )
        | ? [G: graph1,X: $int,Y: $int] :
            ( path1(G,X,Y)
            & Z = G
            & Z1 = Y
            & Z2 = X )
        | ? [G: graph1,X: $int,Y: $int,Z3: $int] :
            ( path1(G,X,Y)
            & path1(G,Y,Z3)
            & Z = G
            & Z1 = X
            & Z2 = Z3 ) ) ) )).

tff(ineq1,conjecture,(
    ! [N: $int,X: $int,Y: $int] :
      ( $lesseq(0,N)
     => ( ( $lesseq(0,X)
          & $less(X,N) )
       => ( ( $lesseq(0,Y)
            & $less(Y,N) )
         => $less($sum($product(X,N),Y),$product(N,N)) ) ) ) )).

%------------------------------------------------------------------------------