TPTP Problem File: SWW620=2.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : SWW620=2 : TPTP v7.4.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Mergesort array-T-WP parameter find run
% 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_array-T-WP_parameter_find_run [Fil14]

% Status   : Theorem
% Rating   : 0.40 v7.4.0, 0.12 v7.3.0, 0.00 v7.0.0, 0.14 v6.4.0, 0.00 v6.3.0, 0.43 v6.2.0, 0.50 v6.1.0
% Syntax   : Number of formulae    :  157 (  33 unit;  59 type)
%            Number of atoms       :  254 (  76 equality)
%            Maximal formula depth :   20 (   5 average)
%            Number of connectives :  168 (  12   ~;   4   |;  63   &)
%                                         (  11 <=>;  78  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  115 (  43   >;  72   *;   0   +;   0  <<)
%            Number of predicates  :   78 (  62 propositional; 0-7 arity)
%            Number of functors    :   41 (   7 constant; 0-5 arity)
%            Number of variables   :  301 (   0 sgn; 299   !;   2   ?)
%                                         ( 301   :;   0  !>;   0  ?*)
%            Maximal term depth    :    5 (   1 average)
%            Arithmetic symbols    :  143 (   2 prd;   4 fun;   2 num; 135 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_sort4,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(map,type,(
    map: ( ty * ty ) > ty )).

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

tff(get_sort8,axiom,(
    ! [A: ty,B: ty,X: uni,X1: uni] : sort1(B,get(B,A,X,X1)) )).

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

tff(set_sort8,axiom,(
    ! [A: ty,B: ty,X: uni,X1: uni,X2: uni] : sort1(map(A,B),set(B,A,X,X1,X2)) )).

tff(select_eq,axiom,(
    ! [A: ty,B: ty,M: uni,A1: uni,A2: uni,B1: uni] :
      ( sort1(B,B1)
     => ( A1 = A2
       => get(B,A,set(B,A,M,A1,B1),A2) = B1 ) ) )).

tff(select_neq,axiom,(
    ! [A: ty,B: ty,M: uni,A1: uni,A2: uni] :
      ( sort1(A,A1)
     => ( sort1(A,A2)
       => ! [B1: uni] :
            ( A1 != A2
           => get(B,A,set(B,A,M,A1,B1),A2) = get(B,A,M,A2) ) ) ) )).

tff(const1,type,(
    const: ( ty * ty * uni ) > uni )).

tff(const_sort4,axiom,(
    ! [A: ty,B: ty,X: uni] : sort1(map(A,B),const(B,A,X)) )).

tff(const,axiom,(
    ! [A: ty,B: ty,B1: uni,A1: uni] :
      ( sort1(B,B1)
     => get(B,A,const(B,A,B1),A1) = B1 ) )).

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

tff(mk_array,type,(
    mk_array1: ( ty * $int * uni ) > uni )).

tff(mk_array_sort4,axiom,(
    ! [A: ty,X: $int,X1: uni] : sort1(array(A),mk_array1(A,X,X1)) )).

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

tff(length_def1,axiom,(
    ! [A: ty,U: $int,U1: uni] : length1(A,mk_array1(A,U,U1)) = U )).

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

tff(elts_sort4,axiom,(
    ! [A: ty,X: uni] : sort1(map(int,A),elts(A,X)) )).

tff(elts_def1,axiom,(
    ! [A: ty,U: $int,U1: uni] :
      ( sort1(map(int,A),U1)
     => elts(A,mk_array1(A,U,U1)) = U1 ) )).

tff(array_inversion1,axiom,(
    ! [A: ty,U: uni] : U = mk_array1(A,length1(A,U),elts(A,U)) )).

tff(get1,type,(
    get2: ( ty * uni * $int ) > uni )).

tff(get_sort9,axiom,(
    ! [A: ty,X: uni,X1: $int] : sort1(A,get2(A,X,X1)) )).

tff(t2tb,type,(
    t2tb: $int > uni )).

tff(t2tb_sort12,axiom,(
    ! [X: $int] : sort1(int,t2tb(X)) )).

tff(tb2t,type,(
    tb2t: uni > $int )).

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

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

tff(get_def,axiom,(
    ! [A: ty,A1: uni,I: $int] : get2(A,A1,I) = get(A,int,elts(A,A1),t2tb(I)) )).

tff(set1,type,(
    set2: ( ty * uni * $int * uni ) > uni )).

tff(set_sort9,axiom,(
    ! [A: ty,X: uni,X1: $int,X2: uni] : sort1(array(A),set2(A,X,X1,X2)) )).

tff(set_def,axiom,(
    ! [A: ty,A1: uni,I: $int,V: uni] : set2(A,A1,I,V) = mk_array1(A,length1(A,A1),set(A,int,elts(A,A1),t2tb(I),V)) )).

tff(make,type,(
    make1: ( ty * $int * uni ) > uni )).

tff(make_sort4,axiom,(
    ! [A: ty,X: $int,X1: uni] : sort1(array(A),make1(A,X,X1)) )).

tff(make_def,axiom,(
    ! [A: ty,N: $int,V: uni] : make1(A,N,V) = mk_array1(A,N,const(A,int,V)) )).

tff(elt6,type,(
    elt6: $tType )).

tff(elt7,type,(
    elt7: ty )).

tff(le3,type,(
    le4: ( elt6 * elt6 ) > $o )).

tff(refl4,axiom,(
    ! [X: elt6] : le4(X,X) )).

tff(trans4,axiom,(
    ! [X: elt6,Y: elt6,Z: elt6] :
      ( le4(X,Y)
     => ( le4(Y,Z)
       => le4(X,Z) ) ) )).

tff(total4,axiom,(
    ! [X: elt6,Y: elt6] :
      ( le4(X,Y)
      | le4(Y,X) ) )).

tff(array_elt3,type,(
    array_elt3: $tType )).

tff(sorted_sub3,type,(
    sorted_sub4: ( array_elt3 * $int * $int ) > $o )).

tff(t2tb10,type,(
    t2tb10: array_elt3 > uni )).

tff(t2tb_sort13,axiom,(
    ! [X: array_elt3] : sort1(array(elt7),t2tb10(X)) )).

tff(tb2t10,type,(
    tb2t10: uni > array_elt3 )).

tff(bridgeL10,axiom,(
    ! [I: array_elt3] : tb2t10(t2tb10(I)) = I )).

tff(bridgeR10,axiom,(
    ! [J: uni] : t2tb10(tb2t10(J)) = J )).

tff(t2tb11,type,(
    t2tb11: elt6 > uni )).

tff(t2tb_sort14,axiom,(
    ! [X: elt6] : sort1(elt7,t2tb11(X)) )).

tff(tb2t11,type,(
    tb2t11: uni > elt6 )).

tff(bridgeL11,axiom,(
    ! [I: elt6] : tb2t11(t2tb11(I)) = I )).

tff(bridgeR11,axiom,(
    ! [J: uni] :
      ( sort1(elt7,J)
     => t2tb11(tb2t11(J)) = J ) )).

tff(sorted_sub_def3,axiom,(
    ! [A: array_elt3,L: $int,U: $int] :
      ( sorted_sub4(A,L,U)
    <=> ! [I1: $int,I2: $int] :
          ( ( $lesseq(L,I1)
            & $lesseq(I1,I2)
            & $less(I2,U) )
         => le4(tb2t11(get2(elt7,t2tb10(A),I1)),tb2t11(get2(elt7,t2tb10(A),I2))) ) ) )).

tff(sorted3,type,(
    sorted4: array_elt3 > $o )).

tff(sorted_def3,axiom,(
    ! [A: array_elt3] :
      ( sorted4(A)
    <=> ! [I1: $int,I2: $int] :
          ( ( $lesseq(0,I1)
            & $lesseq(I1,I2)
            & $less(I2,length1(elt7,t2tb10(A))) )
         => le4(tb2t11(get2(elt7,t2tb10(A),I1)),tb2t11(get2(elt7,t2tb10(A),I2))) ) ) )).

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

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

tff(mk_ref_sort4,axiom,(
    ! [A: ty,X: uni] : sort1(ref(A),mk_ref(A,X)) )).

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

tff(contents_sort4,axiom,(
    ! [A: ty,X: uni] : sort1(A,contents(A,X)) )).

tff(contents_def4,axiom,(
    ! [A: ty,U: uni] :
      ( sort1(A,U)
     => contents(A,mk_ref(A,U)) = U ) )).

tff(ref_inversion4,axiom,(
    ! [A: ty,U: uni] :
      ( sort1(ref(A),U)
     => U = mk_ref(A,contents(A,U)) ) )).

tff(occ,type,(
    occ1: ( ty * uni * uni * $int * $int ) > $int )).

tff(occ_empty,axiom,(
    ! [A: ty,V: uni,M: uni,L: $int,U: $int] :
      ( $lesseq(U,L)
     => occ1(A,V,M,L,U) = 0 ) )).

tff(occ_right_no_add,axiom,(
    ! [A: ty,V: uni,M: uni,L: $int,U: $int] :
      ( sort1(A,V)
     => ( $less(L,U)
       => ( get(A,int,M,t2tb($difference(U,1))) != V
         => occ1(A,V,M,L,U) = occ1(A,V,M,L,$difference(U,1)) ) ) ) )).

tff(occ_right_add,axiom,(
    ! [A: ty,V: uni,M: uni,L: $int,U: $int] :
      ( $less(L,U)
     => ( get(A,int,M,t2tb($difference(U,1))) = V
       => occ1(A,V,M,L,U) = $sum(1,occ1(A,V,M,L,$difference(U,1))) ) ) )).

tff(occ_bounds,axiom,(
    ! [A: ty,V: uni,M: uni,L: $int,U: $int] :
      ( $lesseq(L,U)
     => ( $lesseq(0,occ1(A,V,M,L,U))
        & $lesseq(occ1(A,V,M,L,U),$difference(U,L)) ) ) )).

tff(occ_append,axiom,(
    ! [A: ty,V: uni,M: uni,L: $int,Mid: $int,U: $int] :
      ( ( $lesseq(L,Mid)
        & $lesseq(Mid,U) )
     => occ1(A,V,M,L,U) = $sum(occ1(A,V,M,L,Mid),occ1(A,V,M,Mid,U)) ) )).

tff(occ_neq,axiom,(
    ! [A: ty,V: uni,M: uni,L: $int,U: $int] :
      ( sort1(A,V)
     => ( ! [I: $int] :
            ( ( $lesseq(L,I)
              & $less(I,U) )
           => get(A,int,M,t2tb(I)) != V )
       => occ1(A,V,M,L,U) = 0 ) ) )).

tff(occ_exists,axiom,(
    ! [A: ty,V: uni,M: uni,L: $int,U: $int] :
      ( sort1(A,V)
     => ( $less(0,occ1(A,V,M,L,U))
       => ? [I: $int] :
            ( $lesseq(L,I)
            & $less(I,U)
            & get(A,int,M,t2tb(I)) = V ) ) ) )).

tff(occ_pos,axiom,(
    ! [A: ty,M: uni,L: $int,U: $int,I: $int] :
      ( ( $lesseq(L,I)
        & $less(I,U) )
     => $less(0,occ1(A,get(A,int,M,t2tb(I)),M,L,U)) ) )).

tff(occ_eq,axiom,(
    ! [A: ty,V: uni,M1: uni,M2: uni,L: $int,U: $int] :
      ( ! [I: $int] :
          ( ( $lesseq(L,I)
            & $less(I,U) )
         => get(A,int,M1,t2tb(I)) = get(A,int,M2,t2tb(I)) )
     => occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) ) )).

tff(permut,type,(
    permut2: ( ty * uni * uni * $int * $int ) > $o )).

tff(permut_def,axiom,(
    ! [A: ty,M1: uni,M2: uni,L: $int,U: $int] :
      ( ( permut2(A,M1,M2,L,U)
       => ! [V: uni] : occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) )
      & ( ! [V: uni] :
            ( sort1(A,V)
           => occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) )
       => permut2(A,M1,M2,L,U) ) ) )).

tff(permut_trans,axiom,(
    ! [A: ty,A1: uni,A2: uni,A3: uni,L: $int,U: $int] :
      ( permut2(A,A1,A2,L,U)
     => ( permut2(A,A2,A3,L,U)
       => permut2(A,A1,A3,L,U) ) ) )).

tff(permut_exists,axiom,(
    ! [A: ty,A1: uni,A2: uni,L: $int,U: $int,I: $int] :
      ( permut2(A,A1,A2,L,U)
     => ( ( $lesseq(L,I)
          & $less(I,U) )
       => ? [J: $int] :
            ( $lesseq(L,J)
            & $less(J,U)
            & get(A,int,A1,t2tb(J)) = get(A,int,A2,t2tb(I)) ) ) ) )).

tff(map_eq_sub,type,(
    map_eq_sub1: ( ty * uni * uni * $int * $int ) > $o )).

tff(map_eq_sub_def,axiom,(
    ! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
      ( map_eq_sub1(A,A1,A2,L,U)
    <=> ! [I: $int] :
          ( ( $lesseq(L,I)
            & $less(I,U) )
         => get(A,int,A1,t2tb(I)) = get(A,int,A2,t2tb(I)) ) ) )).

tff(array_eq_sub,type,(
    array_eq_sub1: ( ty * uni * uni * $int * $int ) > $o )).

tff(array_eq_sub_def,axiom,(
    ! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
      ( array_eq_sub1(A,A1,A2,L,U)
    <=> ( length1(A,A1) = length1(A,A2)
        & $lesseq(0,L)
        & $lesseq(L,length1(A,A1))
        & $lesseq(0,U)
        & $lesseq(U,length1(A,A1))
        & map_eq_sub1(A,elts(A,A1),elts(A,A2),L,U) ) ) )).

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

tff(array_eq_def,axiom,(
    ! [A: ty,A1: uni,A2: uni] :
      ( array_eq(A,A1,A2)
    <=> ( length1(A,A1) = length1(A,A2)
        & map_eq_sub1(A,elts(A,A1),elts(A,A2),0,length1(A,A1)) ) ) )).

tff(exchange,type,(
    exchange2: ( ty * uni * uni * $int * $int * $int * $int ) > $o )).

tff(exchange_def,axiom,(
    ! [A: ty,A1: uni,A2: uni,L: $int,U: $int,I: $int,J: $int] :
      ( exchange2(A,A1,A2,L,U,I,J)
    <=> ( $lesseq(L,I)
        & $less(I,U)
        & $lesseq(L,J)
        & $less(J,U)
        & get(A,int,A1,t2tb(I)) = get(A,int,A2,t2tb(J))
        & get(A,int,A1,t2tb(J)) = get(A,int,A2,t2tb(I))
        & ! [K: $int] :
            ( ( $lesseq(L,K)
              & $less(K,U) )
           => ( K != I
             => ( K != J
               => get(A,int,A1,t2tb(K)) = get(A,int,A2,t2tb(K)) ) ) ) ) ) )).

tff(exchange_set,axiom,(
    ! [A: ty,A1: uni,L: $int,U: $int,I: $int,J: $int] :
      ( ( $lesseq(L,I)
        & $less(I,U) )
     => ( ( $lesseq(L,J)
          & $less(J,U) )
       => exchange2(A,A1,set(A,int,set(A,int,A1,t2tb(I),get(A,int,A1,t2tb(J))),t2tb(J),get(A,int,A1,t2tb(I))),L,U,I,J) ) ) )).

tff(exchange1,type,(
    exchange3: ( ty * uni * uni * $int * $int ) > $o )).

tff(exchange_def1,axiom,(
    ! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
      ( exchange3(A,A1,A2,I,J)
    <=> ( length1(A,A1) = length1(A,A2)
        & exchange2(A,elts(A,A1),elts(A,A2),0,length1(A,A1),I,J) ) ) )).

tff(permut1,type,(
    permut3: ( ty * uni * uni * $int * $int ) > $o )).

tff(permut_def1,axiom,(
    ! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
      ( permut3(A,A1,A2,L,U)
    <=> ( length1(A,A1) = length1(A,A2)
        & $lesseq(0,L)
        & $lesseq(L,length1(A,A1))
        & $lesseq(0,U)
        & $lesseq(U,length1(A,A1))
        & permut2(A,elts(A,A1),elts(A,A2),L,U) ) ) )).

tff(permut_sub,type,(
    permut_sub1: ( ty * uni * uni * $int * $int ) > $o )).

tff(permut_sub_def,axiom,(
    ! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
      ( permut_sub1(A,A1,A2,L,U)
    <=> ( map_eq_sub1(A,elts(A,A1),elts(A,A2),0,L)
        & permut3(A,A1,A2,L,U)
        & map_eq_sub1(A,elts(A,A1),elts(A,A2),U,length1(A,A1)) ) ) )).

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

tff(permut_all_def,axiom,(
    ! [A: ty,A1: uni,A2: uni] :
      ( permut_all(A,A1,A2)
    <=> ( length1(A,A1) = length1(A,A2)
        & permut2(A,elts(A,A1),elts(A,A2),0,length1(A,A1)) ) ) )).

tff(exchange_permut_sub,axiom,(
    ! [A: ty,A1: uni,A2: uni,I: $int,J: $int,L: $int,U: $int] :
      ( exchange3(A,A1,A2,I,J)
     => ( ( $lesseq(L,I)
          & $less(I,U) )
       => ( ( $lesseq(L,J)
            & $less(J,U) )
         => ( $lesseq(0,L)
           => ( $lesseq(U,length1(A,A1))
             => permut_sub1(A,A1,A2,L,U) ) ) ) ) ) )).

tff(permut_sub_weakening,axiom,(
    ! [A: ty,A1: uni,A2: uni,L1: $int,U1: $int,L2: $int,U2: $int] :
      ( permut_sub1(A,A1,A2,L1,U1)
     => ( ( $lesseq(0,L2)
          & $lesseq(L2,L1) )
       => ( ( $lesseq(U1,U2)
            & $lesseq(U2,length1(A,A1)) )
         => permut_sub1(A,A1,A2,L2,U2) ) ) ) )).

tff(exchange_permut_all,axiom,(
    ! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
      ( exchange3(A,A1,A2,I,J)
     => permut_all(A,A1,A2) ) )).

tff(abs,type,(
    abs: $int > $int )).

tff(abs_def,axiom,(
    ! [X: $int] :
      ( ( $lesseq(0,X)
       => abs(X) = X )
      & ( ~ $lesseq(0,X)
       => abs(X) = $uminus(X) ) ) )).

tff(abs_le,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(abs(X),Y)
    <=> ( $lesseq($uminus(Y),X)
        & $lesseq(X,Y) ) ) )).

tff(abs_pos,axiom,(
    ! [X: $int] : $lesseq(0,abs(X)) )).

tff(div,type,(
    div: ( $int * $int ) > $int )).

tff(mod,type,(
    mod: ( $int * $int ) > $int )).

tff(div_mod,axiom,(
    ! [X: $int,Y: $int] :
      ( Y != 0
     => X = $sum($product(Y,div(X,Y)),mod(X,Y)) ) )).

tff(div_bound,axiom,(
    ! [X: $int,Y: $int] :
      ( ( $lesseq(0,X)
        & $less(0,Y) )
     => ( $lesseq(0,div(X,Y))
        & $lesseq(div(X,Y),X) ) ) )).

tff(mod_bound,axiom,(
    ! [X: $int,Y: $int] :
      ( Y != 0
     => ( $less($uminus(abs(Y)),mod(X,Y))
        & $less(mod(X,Y),abs(Y)) ) ) )).

tff(div_sign_pos,axiom,(
    ! [X: $int,Y: $int] :
      ( ( $lesseq(0,X)
        & $less(0,Y) )
     => $lesseq(0,div(X,Y)) ) )).

tff(div_sign_neg,axiom,(
    ! [X: $int,Y: $int] :
      ( ( $lesseq(X,0)
        & $less(0,Y) )
     => $lesseq(div(X,Y),0) ) )).

tff(mod_sign_pos,axiom,(
    ! [X: $int,Y: $int] :
      ( ( $lesseq(0,X)
        & Y != 0 )
     => $lesseq(0,mod(X,Y)) ) )).

tff(mod_sign_neg,axiom,(
    ! [X: $int,Y: $int] :
      ( ( $lesseq(X,0)
        & Y != 0 )
     => $lesseq(mod(X,Y),0) ) )).

tff(rounds_toward_zero,axiom,(
    ! [X: $int,Y: $int] :
      ( Y != 0
     => $lesseq(abs($product(div(X,Y),Y)),abs(X)) ) )).

tff(div_1,axiom,(
    ! [X: $int] : div(X,1) = X )).

tff(mod_1,axiom,(
    ! [X: $int] : mod(X,1) = 0 )).

tff(div_inf,axiom,(
    ! [X: $int,Y: $int] :
      ( ( $lesseq(0,X)
        & $less(X,Y) )
     => div(X,Y) = 0 ) )).

tff(mod_inf,axiom,(
    ! [X: $int,Y: $int] :
      ( ( $lesseq(0,X)
        & $less(X,Y) )
     => mod(X,Y) = X ) )).

tff(div_mult,axiom,(
    ! [X: $int,Y: $int,Z: $int] :
      ( ( $less(0,X)
        & $lesseq(0,Y)
        & $lesseq(0,Z) )
     => div($sum($product(X,Y),Z),X) = $sum(Y,div(Z,X)) ) )).

tff(mod_mult,axiom,(
    ! [X: $int,Y: $int,Z: $int] :
      ( ( $less(0,X)
        & $lesseq(0,Y)
        & $lesseq(0,Z) )
     => mod($sum($product(X,Y),Z),X) = mod(Z,X) ) )).

tff(min,type,(
    min: ( $int * $int ) > $int )).

tff(max,type,(
    max: ( $int * $int ) > $int )).

tff(max_is_ge,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(X,max(X,Y))
      & $lesseq(Y,max(X,Y)) ) )).

tff(max_is_some,axiom,(
    ! [X: $int,Y: $int] :
      ( max(X,Y) = X
      | max(X,Y) = Y ) )).

tff(min_is_le,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(min(X,Y),X)
      & $lesseq(min(X,Y),Y) ) )).

tff(min_is_some,axiom,(
    ! [X: $int,Y: $int] :
      ( min(X,Y) = X
      | min(X,Y) = Y ) )).

tff(max_x,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(Y,X)
     => max(X,Y) = X ) )).

tff(max_y,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(X,Y)
     => max(X,Y) = Y ) )).

tff(min_x,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(X,Y)
     => min(X,Y) = X ) )).

tff(min_y,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(Y,X)
     => min(X,Y) = Y ) )).

tff(max_sym,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(Y,X)
     => max(X,Y) = max(Y,X) ) )).

tff(min_sym,axiom,(
    ! [X: $int,Y: $int] :
      ( $lesseq(Y,X)
     => min(X,Y) = min(Y,X) ) )).

tff(map_int_elt3,type,(
    map_int_elt3: $tType )).

tff(t2tb12,type,(
    t2tb12: map_int_elt3 > uni )).

tff(t2tb_sort15,axiom,(
    ! [X: map_int_elt3] : sort1(map(int,elt7),t2tb12(X)) )).

tff(tb2t12,type,(
    tb2t12: uni > map_int_elt3 )).

tff(bridgeL12,axiom,(
    ! [I: map_int_elt3] : tb2t12(t2tb12(I)) = I )).

tff(bridgeR12,axiom,(
    ! [J: uni] :
      ( sort1(map(int,elt7),J)
     => t2tb12(tb2t12(J)) = J ) )).

tff(wP_parameter_find_run,conjecture,(
    ! [A: $int,A1: map_int_elt3,Lo: $int] :
      ( ( $lesseq(0,A)
        & $lesseq(0,Lo)
        & $less(Lo,A) )
     => sorted_sub4(tb2t10(mk_array1(elt7,A,t2tb12(A1))),Lo,$sum(Lo,1)) ) )).

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