TPTP Problem File: SWW628=2.p

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%------------------------------------------------------------------------------
% File     : SWW628=2 : TPTP v7.4.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Mergesort queue-T-WP parameter merge
% 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    : mergesort_queue-T-WP_parameter_merge [Fil14]

% Status   : Theorem
% Rating   : 0.60 v7.4.0, 0.38 v7.3.0, 0.17 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.29 v6.2.0, 0.75 v6.1.0
% Syntax   : Number of formulae    :  114 (  41 unit;  40 type)
%            Number of atoms       :  165 (  64 equality)
%            Maximal formula depth :   27 (   4 average)
%            Number of connectives :  101 (  10   ~;   7   |;  25   &)
%                                         (   7 <=>;  52  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   51 (  25   >;  26   *;   0   +;   0  <<)
%            Number of predicates  :   51 (  43 propositional; 0-3 arity)
%            Number of functors    :   28 (   6 constant; 0-5 arity)
%            Number of variables   :  223 (   0 sgn; 211   !;  12   ?)
%                                         ( 223   :;   0  !>;   0  ?*)
%            Maximal term depth    :    5 (   2 average)
%            Arithmetic symbols    :    9 (   2 prd;   2 fun;   2 num;   3 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(list,type,(
    list: ty > ty )).

tff(nil,type,(
    nil: ty > uni )).

tff(nil_sort1,axiom,(
    ! [A: ty] : sort1(list(A),nil(A)) )).

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

tff(cons_sort1,axiom,(
    ! [A: ty,X: uni,X1: uni] : sort1(list(A),cons(A,X,X1)) )).

tff(match_list,type,(
    match_list1: ( ty * ty * uni * uni * uni ) > uni )).

tff(match_list_sort1,axiom,(
    ! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort1(A1,match_list1(A1,A,X,X1,X2)) )).

tff(match_list_Nil1,axiom,(
    ! [A: ty,A1: ty,Z: uni,Z1: uni] :
      ( sort1(A1,Z)
     => match_list1(A1,A,nil(A),Z,Z1) = Z ) )).

tff(match_list_Cons1,axiom,(
    ! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni,U1: uni] :
      ( sort1(A1,Z1)
     => match_list1(A1,A,cons(A,U,U1),Z,Z1) = Z1 ) )).

tff(nil_Cons1,axiom,(
    ! [A: ty,V: uni,V1: uni] : nil(A) != cons(A,V,V1) )).

tff(cons_proj_1,type,(
    cons_proj_11: ( ty * uni ) > uni )).

tff(cons_proj_1_sort1,axiom,(
    ! [A: ty,X: uni] : sort1(A,cons_proj_11(A,X)) )).

tff(cons_proj_1_def1,axiom,(
    ! [A: ty,U: uni,U1: uni] :
      ( sort1(A,U)
     => cons_proj_11(A,cons(A,U,U1)) = U ) )).

tff(cons_proj_2,type,(
    cons_proj_21: ( ty * uni ) > uni )).

tff(cons_proj_2_sort1,axiom,(
    ! [A: ty,X: uni] : sort1(list(A),cons_proj_21(A,X)) )).

tff(cons_proj_2_def1,axiom,(
    ! [A: ty,U: uni,U1: uni] : cons_proj_21(A,cons(A,U,U1)) = U1 )).

tff(list_inversion1,axiom,(
    ! [A: ty,U: uni] :
      ( U = nil(A)
      | U = cons(A,cons_proj_11(A,U),cons_proj_21(A,U)) ) )).

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

tff(mem_def,axiom,(
    ! [A: ty,X: uni] :
      ( sort1(A,X)
     => ( ~ mem(A,X,nil(A))
        & ! [X1: uni,X2: uni] :
            ( sort1(A,X1)
           => ( mem(A,X,cons(A,X1,X2))
            <=> ( X = X1
                | mem(A,X,X2) ) ) ) ) ) )).

tff(length,type,(
    length2: ( ty * uni ) > $int )).

tff(length_def,axiom,(
    ! [A: ty] :
      ( length2(A,nil(A)) = 0
      & ! [X: uni,X1: uni] : length2(A,cons(A,X,X1)) = $sum(1,length2(A,X1)) ) )).

tff(length_nonnegative,axiom,(
    ! [A: ty,L: uni] : $lesseq(0,length2(A,L)) )).

tff(length_nil,axiom,(
    ! [A: ty,L: uni] :
      ( length2(A,L) = 0
    <=> L = nil(A) ) )).

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

tff(infix_plpl_sort1,axiom,(
    ! [A: ty,X: uni,X1: uni] : sort1(list(A),infix_plpl(A,X,X1)) )).

tff(infix_plpl_def,axiom,(
    ! [A: ty,L2: uni] :
      ( infix_plpl(A,nil(A),L2) = L2
      & ! [X: uni,X1: uni] : infix_plpl(A,cons(A,X,X1),L2) = cons(A,X,infix_plpl(A,X1,L2)) ) )).

tff(append_assoc,axiom,(
    ! [A: ty,L1: uni,L2: uni,L3: uni] : infix_plpl(A,L1,infix_plpl(A,L2,L3)) = infix_plpl(A,infix_plpl(A,L1,L2),L3) )).

tff(append_l_nil,axiom,(
    ! [A: ty,L: uni] : infix_plpl(A,L,nil(A)) = L )).

tff(append_length,axiom,(
    ! [A: ty,L1: uni,L2: uni] : length2(A,infix_plpl(A,L1,L2)) = $sum(length2(A,L1),length2(A,L2)) )).

tff(mem_append,axiom,(
    ! [A: ty,X: uni,L1: uni,L2: uni] :
      ( mem(A,X,infix_plpl(A,L1,L2))
    <=> ( mem(A,X,L1)
        | mem(A,X,L2) ) ) )).

tff(mem_decomp,axiom,(
    ! [A: ty,X: uni,L: uni] :
      ( mem(A,X,L)
     => ? [L1: uni,L2: uni] :
          ( sort1(list(A),L1)
          & sort1(list(A),L2)
          & L = infix_plpl(A,L1,cons(A,X,L2)) ) ) )).

tff(num_occ,type,(
    num_occ1: ( ty * uni * uni ) > $int )).

tff(num_occ_def,axiom,(
    ! [A: ty,X: uni] :
      ( sort1(A,X)
     => ( num_occ1(A,X,nil(A)) = 0
        & ! [X1: uni,X2: uni] :
            ( sort1(A,X1)
           => ( ( X = X1
               => num_occ1(A,X,cons(A,X1,X2)) = $sum(1,num_occ1(A,X,X2)) )
              & ( X != X1
               => num_occ1(A,X,cons(A,X1,X2)) = $sum(0,num_occ1(A,X,X2)) ) ) ) ) ) )).

tff(mem_Num_Occ,axiom,(
    ! [A: ty,X: uni,L: uni] :
      ( mem(A,X,L)
    <=> $less(0,num_occ1(A,X,L)) ) )).

tff(append_Num_Occ,axiom,(
    ! [A: ty,X: uni,L1: uni,L2: uni] : num_occ1(A,X,infix_plpl(A,L1,L2)) = $sum(num_occ1(A,X,L1),num_occ1(A,X,L2)) )).

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

tff(reverse_sort1,axiom,(
    ! [A: ty,X: uni] : sort1(list(A),reverse(A,X)) )).

tff(reverse_def,axiom,(
    ! [A: ty] :
      ( reverse(A,nil(A)) = nil(A)
      & ! [X: uni,X1: uni] : reverse(A,cons(A,X,X1)) = infix_plpl(A,reverse(A,X1),cons(A,X,nil(A))) ) )).

tff(reverse_append,axiom,(
    ! [A: ty,L1: uni,L2: uni,X: uni] : infix_plpl(A,reverse(A,cons(A,X,L1)),L2) = infix_plpl(A,reverse(A,L1),cons(A,X,L2)) )).

tff(reverse_cons,axiom,(
    ! [A: ty,L: uni,X: uni] : reverse(A,cons(A,X,L)) = infix_plpl(A,reverse(A,L),cons(A,X,nil(A))) )).

tff(reverse_reverse,axiom,(
    ! [A: ty,L: uni] : reverse(A,reverse(A,L)) = L )).

tff(reverse_mem,axiom,(
    ! [A: ty,L: uni,X: uni] :
      ( mem(A,X,L)
    <=> mem(A,X,reverse(A,L)) ) )).

tff(reverse_length,axiom,(
    ! [A: ty,L: uni] : length2(A,reverse(A,L)) = length2(A,L) )).

tff(reverse_num_occ,axiom,(
    ! [A: ty,X: uni,L: uni] : num_occ1(A,X,L) = num_occ1(A,X,reverse(A,L)) )).

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

tff(permut_def,axiom,(
    ! [A: ty,L1: uni,L2: uni] :
      ( ( permut(A,L1,L2)
       => ! [X: uni] : num_occ1(A,X,L1) = num_occ1(A,X,L2) )
      & ( ! [X: uni] :
            ( sort1(A,X)
           => num_occ1(A,X,L1) = num_occ1(A,X,L2) )
       => permut(A,L1,L2) ) ) )).

tff(permut_refl,axiom,(
    ! [A: ty,L: uni] : permut(A,L,L) )).

tff(permut_sym,axiom,(
    ! [A: ty,L1: uni,L2: uni] :
      ( permut(A,L1,L2)
     => permut(A,L2,L1) ) )).

tff(permut_trans,axiom,(
    ! [A: ty,L1: uni,L2: uni,L3: uni] :
      ( permut(A,L1,L2)
     => ( permut(A,L2,L3)
       => permut(A,L1,L3) ) ) )).

tff(permut_cons,axiom,(
    ! [A: ty,X: uni,L1: uni,L2: uni] :
      ( permut(A,L1,L2)
     => permut(A,cons(A,X,L1),cons(A,X,L2)) ) )).

tff(permut_swap,axiom,(
    ! [A: ty,X: uni,Y: uni,L: uni] : permut(A,cons(A,X,cons(A,Y,L)),cons(A,Y,cons(A,X,L))) )).

tff(permut_cons_append,axiom,(
    ! [A: ty,X: uni,L1: uni,L2: uni] : permut(A,infix_plpl(A,cons(A,X,L1),L2),infix_plpl(A,L1,cons(A,X,L2))) )).

tff(permut_assoc,axiom,(
    ! [A: ty,L1: uni,L2: uni,L3: uni] : permut(A,infix_plpl(A,infix_plpl(A,L1,L2),L3),infix_plpl(A,L1,infix_plpl(A,L2,L3))) )).

tff(permut_append,axiom,(
    ! [A: ty,L1: uni,L2: uni,K1: uni,K2: uni] :
      ( permut(A,L1,K1)
     => ( permut(A,L2,K2)
       => permut(A,infix_plpl(A,L1,L2),infix_plpl(A,K1,K2)) ) ) )).

tff(permut_append_swap,axiom,(
    ! [A: ty,L1: uni,L2: uni] : permut(A,infix_plpl(A,L1,L2),infix_plpl(A,L2,L1)) )).

tff(permut_mem,axiom,(
    ! [A: ty,X: uni,L1: uni,L2: uni] :
      ( permut(A,L1,L2)
     => ( mem(A,X,L1)
       => mem(A,X,L2) ) ) )).

tff(permut_length,axiom,(
    ! [A: ty,L1: uni,L2: uni] :
      ( permut(A,L1,L2)
     => length2(A,L1) = length2(A,L2) ) )).

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

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

tff(mk_t_sort1,axiom,(
    ! [A: ty,X: uni] : sort1(t(A),mk_t(A,X)) )).

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

tff(elts_sort1,axiom,(
    ! [A: ty,X: uni] : sort1(list(A),elts(A,X)) )).

tff(elts_def1,axiom,(
    ! [A: ty,U: uni] : elts(A,mk_t(A,U)) = U )).

tff(t_inversion1,axiom,(
    ! [A: ty,U: uni] : U = mk_t(A,elts(A,U)) )).

tff(length1,type,(
    length3: ( ty * uni ) > $int )).

tff(length_def1,axiom,(
    ! [A: ty,Q: uni] : length3(A,Q) = length2(A,elts(A,Q)) )).

tff(elt,type,(
    elt1: $tType )).

tff(elt1,type,(
    elt: ty )).

tff(le,type,(
    le1: ( elt1 * elt1 ) > $o )).

tff(refl1,axiom,(
    ! [X: elt1] : le1(X,X) )).

tff(trans1,axiom,(
    ! [X: elt1,Y: elt1,Z: elt1] :
      ( le1(X,Y)
     => ( le1(Y,Z)
       => le1(X,Z) ) ) )).

tff(total1,axiom,(
    ! [X: elt1,Y: elt1] :
      ( le1(X,Y)
      | le1(Y,X) ) )).

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

tff(sorted,type,(
    sorted1: list_elt > $o )).

tff(t2tb,type,(
    t2tb: list_elt > uni )).

tff(t2tb_sort,axiom,(
    ! [X: list_elt] : sort1(list(elt),t2tb(X)) )).

tff(tb2t,type,(
    tb2t: uni > list_elt )).

tff(bridgeL,axiom,(
    ! [I: list_elt] : tb2t(t2tb(I)) = I )).

tff(bridgeR,axiom,(
    ! [J: uni] : t2tb(tb2t(J)) = J )).

tff(sorted_Nil,axiom,(
    sorted1(tb2t(nil(elt))) )).

tff(t2tb1,type,(
    t2tb1: elt1 > uni )).

tff(t2tb_sort1,axiom,(
    ! [X: elt1] : sort1(elt,t2tb1(X)) )).

tff(tb2t1,type,(
    tb2t1: uni > elt1 )).

tff(bridgeL1,axiom,(
    ! [I: elt1] : tb2t1(t2tb1(I)) = I )).

tff(bridgeR1,axiom,(
    ! [J: uni] :
      ( sort1(elt,J)
     => t2tb1(tb2t1(J)) = J ) )).

tff(sorted_One,axiom,(
    ! [X: elt1] : sorted1(tb2t(cons(elt,t2tb1(X),nil(elt)))) )).

tff(sorted_Two,axiom,(
    ! [X: elt1,Y: elt1,L: list_elt] :
      ( le1(X,Y)
     => ( sorted1(tb2t(cons(elt,t2tb1(Y),t2tb(L))))
       => sorted1(tb2t(cons(elt,t2tb1(X),cons(elt,t2tb1(Y),t2tb(L))))) ) ) )).

tff(sorted_inversion,axiom,(
    ! [Z: list_elt] :
      ( sorted1(Z)
     => ( Z = tb2t(nil(elt))
        | ? [X: elt1] : Z = tb2t(cons(elt,t2tb1(X),nil(elt)))
        | ? [X: elt1,Y: elt1,L: list_elt] :
            ( le1(X,Y)
            & sorted1(tb2t(cons(elt,t2tb1(Y),t2tb(L))))
            & Z = tb2t(cons(elt,t2tb1(X),cons(elt,t2tb1(Y),t2tb(L)))) ) ) ) )).

tff(sorted_mem,axiom,(
    ! [X: elt1,L: list_elt] :
      ( ( ! [Y: elt1] :
            ( mem(elt,t2tb1(Y),t2tb(L))
           => le1(X,Y) )
        & sorted1(L) )
    <=> sorted1(tb2t(cons(elt,t2tb1(X),t2tb(L)))) ) )).

tff(sorted_append,axiom,(
    ! [L1: list_elt,L2: list_elt] :
      ( ( sorted1(L1)
        & sorted1(L2)
        & ! [X: elt1,Y: elt1] :
            ( mem(elt,t2tb1(X),t2tb(L1))
           => ( mem(elt,t2tb1(Y),t2tb(L2))
             => le1(X,Y) ) ) )
    <=> sorted1(tb2t(infix_plpl(elt,t2tb(L1),t2tb(L2)))) ) )).

tff(wP_parameter_merge,conjecture,(
    ! [Q1: list_elt,Q2: list_elt,Q: list_elt] :
      ( ( Q = tb2t(nil(elt))
        & sorted1(Q1)
        & sorted1(Q2) )
     => ! [Q3: list_elt,Q21: list_elt,Q11: list_elt] :
          ( ( sorted1(Q11)
            & sorted1(Q21)
            & sorted1(Q3)
            & ! [X: elt1,Y: elt1] :
                ( mem(elt,t2tb1(X),t2tb(Q3))
               => ( mem(elt,t2tb1(Y),t2tb(Q11))
                 => le1(X,Y) ) )
            & ! [X: elt1,Y: elt1] :
                ( mem(elt,t2tb1(X),t2tb(Q3))
               => ( mem(elt,t2tb1(Y),t2tb(Q21))
                 => le1(X,Y) ) )
            & permut(elt,infix_plpl(elt,infix_plpl(elt,t2tb(Q3),t2tb(Q11)),t2tb(Q21)),infix_plpl(elt,t2tb(Q1),t2tb(Q2))) )
         => ( $less(0,length2(elt,t2tb(Q11)))
           => ( length2(elt,t2tb(Q11)) != 0
             => ( length2(elt,t2tb(Q21)) != 0
               => ( Q11 != tb2t(nil(elt))
                 => ! [X1: elt1] :
                      ( ? [X: elt1,X2: list_elt] :
                          ( Q11 = tb2t(cons(elt,t2tb1(X),t2tb(X2)))
                          & X1 = X )
                     => ( Q21 != tb2t(nil(elt))
                       => ! [X2: elt1] :
                            ( ? [X: elt1,X3: list_elt] :
                                ( Q21 = tb2t(cons(elt,t2tb1(X),t2tb(X3)))
                                & X2 = X )
                           => ( ~ le1(X1,X2)
                             => ( Q21 != tb2t(nil(elt))
                               => ! [Q22: list_elt,O: elt1] :
                                    ( ? [X: elt1,X3: list_elt] :
                                        ( Q21 = tb2t(cons(elt,t2tb1(X),t2tb(X3)))
                                        & O = X
                                        & Q22 = X3 )
                                   => ! [Q4: list_elt] :
                                        ( Q4 = tb2t(infix_plpl(elt,t2tb(Q3),cons(elt,t2tb1(O),nil(elt))))
                                       => permut(elt,infix_plpl(elt,infix_plpl(elt,t2tb(Q4),t2tb(Q11)),t2tb(Q22)),infix_plpl(elt,t2tb(Q1),t2tb(Q2))) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

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