## TPTP Problem File: ALG017^7.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : ALG017^7 : TPTP v7.0.0. Released v5.5.0.
% Domain   : General Algebra
% Problem  : Groups 4: CPROPS-PAIRWISE-EXCLUSIVE-PROBLEM-1
% Version  : [Ben12] axioms.
% English  :

% Refs     : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
%          : [CM+04] Colton et al. (2004), Automatic Generation of Classifi
%          : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source   : [Ben12]
% Names    : s4-cumul-GAL017+1 [Ben12]

% Status   : Theorem
% Rating   : 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0
% Syntax   : Number of formulae    :  104 (   0 unit;  43 type;  32 defn)
%            Number of atoms       : 3000 (  36 equality; 181 variable)
%            Maximal formula depth :   73 (   8 average)
%            Number of connectives : 2872 (   5   ~;   5   |;   9   &;2843   @)
%                                         (   0 <=>;  10  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  183 ( 183   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   47 (  43   :;   0   =)
%            Number of variables   :  114 (   2 sgn;  44   !;   7   ?;  63   ^)
%                                         ( 114   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : Goedel translation of ALG017+1
%------------------------------------------------------------------------------
%----Include axioms for Modal logic S4 under cumulative domains
include('Axioms/LCL015^0.ax').
include('Axioms/LCL013^5.ax').
include('Axioms/LCL015^1.ax').
%------------------------------------------------------------------------------
thf(unit_type,type,(
unit: mu )).

thf(existence_of_unit_ax,axiom,(
! [V: \$i] :
( exists_in_world @ unit @ V ) )).

thf(inv_type,type,(
inv: mu > mu )).

thf(existence_of_inv_ax,axiom,(
! [V: \$i,V1: mu] :
( exists_in_world @ ( inv @ V1 ) @ V ) )).

thf(e3_type,type,(
e3: mu )).

thf(existence_of_e3_ax,axiom,(
! [V: \$i] :
( exists_in_world @ e3 @ V ) )).

thf(e2_type,type,(
e2: mu )).

thf(existence_of_e2_ax,axiom,(
! [V: \$i] :
( exists_in_world @ e2 @ V ) )).

thf(e1_type,type,(
e1: mu )).

thf(existence_of_e1_ax,axiom,(
! [V: \$i] :
( exists_in_world @ e1 @ V ) )).

thf(e0_type,type,(
e0: mu )).

thf(existence_of_e0_ax,axiom,(
! [V: \$i] :
( exists_in_world @ e0 @ V ) )).

thf(op_type,type,(
op: mu > mu > mu )).

thf(existence_of_op_ax,axiom,(
! [V: \$i,V2: mu,V1: mu] :
( exists_in_world @ ( op @ V2 @ V1 ) @ V ) )).

thf(reflexivity,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] :
( mbox_s4 @ ( qmltpeq @ X @ X ) ) ) ) )).

thf(symmetry,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Y: mu] :
( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ X ) ) ) ) ) ) ) ) )).

thf(transitivity,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Y: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Z: mu] :
( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ Z ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ) ) ) ) )).

thf(inv_substitution_1,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ A ) @ ( inv @ B ) ) ) ) ) ) ) ) ) )).

thf(op_substitution_1,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] :
( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ A @ C ) @ ( op @ B @ C ) ) ) ) ) ) ) ) ) ) ) )).

thf(op_substitution_2,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] :
( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ C @ A ) @ ( op @ C @ B ) ) ) ) ) ) ) ) ) ) ) )).

thf(ax1,axiom,
( mvalid @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e3 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e3 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e3 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e3 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e2 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e2 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e2 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e3 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e3 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e3 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e3 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e1 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e1 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e1 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e1 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e3 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e3 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e3 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e3 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e0 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e0 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e0 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e0 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e1 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e1 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e1 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e1 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e2 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e2 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e2 ) @ e3 ) ) ) ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

thf(ax2,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e0 ) @ e0 ) @ ( op @ e0 @ ( op @ e0 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e0 ) @ e1 ) @ ( op @ e0 @ ( op @ e0 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e0 ) @ e2 ) @ ( op @ e0 @ ( op @ e0 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e0 ) @ e3 ) @ ( op @ e0 @ ( op @ e0 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e1 ) @ e0 ) @ ( op @ e0 @ ( op @ e1 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e1 ) @ e1 ) @ ( op @ e0 @ ( op @ e1 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e1 ) @ e2 ) @ ( op @ e0 @ ( op @ e1 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e1 ) @ e3 ) @ ( op @ e0 @ ( op @ e1 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e2 ) @ e0 ) @ ( op @ e0 @ ( op @ e2 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e2 ) @ e1 ) @ ( op @ e0 @ ( op @ e2 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e2 ) @ e2 ) @ ( op @ e0 @ ( op @ e2 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e2 ) @ e3 ) @ ( op @ e0 @ ( op @ e2 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e3 ) @ e0 ) @ ( op @ e0 @ ( op @ e3 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e3 ) @ e1 ) @ ( op @ e0 @ ( op @ e3 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e3 ) @ e2 ) @ ( op @ e0 @ ( op @ e3 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e0 @ e3 ) @ e3 ) @ ( op @ e0 @ ( op @ e3 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e0 ) @ e0 ) @ ( op @ e1 @ ( op @ e0 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e0 ) @ e1 ) @ ( op @ e1 @ ( op @ e0 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e0 ) @ e2 ) @ ( op @ e1 @ ( op @ e0 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e0 ) @ e3 ) @ ( op @ e1 @ ( op @ e0 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e1 ) @ e0 ) @ ( op @ e1 @ ( op @ e1 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e1 ) @ e1 ) @ ( op @ e1 @ ( op @ e1 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e1 ) @ e2 ) @ ( op @ e1 @ ( op @ e1 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e1 ) @ e3 ) @ ( op @ e1 @ ( op @ e1 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e2 ) @ e0 ) @ ( op @ e1 @ ( op @ e2 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e2 ) @ e1 ) @ ( op @ e1 @ ( op @ e2 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e2 ) @ e2 ) @ ( op @ e1 @ ( op @ e2 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e2 ) @ e3 ) @ ( op @ e1 @ ( op @ e2 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e3 ) @ e0 ) @ ( op @ e1 @ ( op @ e3 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e3 ) @ e1 ) @ ( op @ e1 @ ( op @ e3 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e3 ) @ e2 ) @ ( op @ e1 @ ( op @ e3 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e1 @ e3 ) @ e3 ) @ ( op @ e1 @ ( op @ e3 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e0 ) @ e0 ) @ ( op @ e2 @ ( op @ e0 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e0 ) @ e1 ) @ ( op @ e2 @ ( op @ e0 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e0 ) @ e2 ) @ ( op @ e2 @ ( op @ e0 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e0 ) @ e3 ) @ ( op @ e2 @ ( op @ e0 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e1 ) @ e0 ) @ ( op @ e2 @ ( op @ e1 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e1 ) @ e1 ) @ ( op @ e2 @ ( op @ e1 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e1 ) @ e2 ) @ ( op @ e2 @ ( op @ e1 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e1 ) @ e3 ) @ ( op @ e2 @ ( op @ e1 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e2 ) @ e0 ) @ ( op @ e2 @ ( op @ e2 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e2 ) @ e1 ) @ ( op @ e2 @ ( op @ e2 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e2 ) @ e2 ) @ ( op @ e2 @ ( op @ e2 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e2 ) @ e3 ) @ ( op @ e2 @ ( op @ e2 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e3 ) @ e0 ) @ ( op @ e2 @ ( op @ e3 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e3 ) @ e1 ) @ ( op @ e2 @ ( op @ e3 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e3 ) @ e2 ) @ ( op @ e2 @ ( op @ e3 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e2 @ e3 ) @ e3 ) @ ( op @ e2 @ ( op @ e3 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e0 ) @ e0 ) @ ( op @ e3 @ ( op @ e0 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e0 ) @ e1 ) @ ( op @ e3 @ ( op @ e0 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e0 ) @ e2 ) @ ( op @ e3 @ ( op @ e0 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e0 ) @ e3 ) @ ( op @ e3 @ ( op @ e0 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e1 ) @ e0 ) @ ( op @ e3 @ ( op @ e1 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e1 ) @ e1 ) @ ( op @ e3 @ ( op @ e1 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e1 ) @ e2 ) @ ( op @ e3 @ ( op @ e1 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e1 ) @ e3 ) @ ( op @ e3 @ ( op @ e1 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e2 ) @ e0 ) @ ( op @ e3 @ ( op @ e2 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e2 ) @ e1 ) @ ( op @ e3 @ ( op @ e2 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e2 ) @ e2 ) @ ( op @ e3 @ ( op @ e2 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e2 ) @ e3 ) @ ( op @ e3 @ ( op @ e2 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e3 ) @ e0 ) @ ( op @ e3 @ ( op @ e3 @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e3 ) @ e1 ) @ ( op @ e3 @ ( op @ e3 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e3 ) @ e2 ) @ ( op @ e3 @ ( op @ e3 @ e2 ) ) ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( op @ e3 @ e3 ) @ e3 ) @ ( op @ e3 @ ( op @ e3 @ e3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

thf(ax3,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ unit @ e0 ) @ e0 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ unit ) @ e0 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ unit @ e1 ) @ e1 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ unit ) @ e1 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ unit @ e2 ) @ e2 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ unit ) @ e2 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ unit @ e3 ) @ e3 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ unit ) @ e3 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ unit @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ unit @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ unit @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ unit @ e3 ) ) ) ) ) ) ) ) ) ) ) ) ) )).

thf(ax4,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ ( inv @ e0 ) ) @ unit ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( inv @ e0 ) @ e0 ) @ unit ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ ( inv @ e1 ) ) @ unit ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( inv @ e1 ) @ e1 ) @ unit ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ ( inv @ e2 ) ) @ unit ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( inv @ e2 ) @ e2 ) @ unit ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ ( inv @ e3 ) ) @ unit ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ ( inv @ e3 ) @ e3 ) @ unit ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e3 ) ) ) ) ) @ ( mand @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e3 ) ) ) ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e0 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e1 ) ) @ ( mor @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

thf(ax5,axiom,
( mvalid @ ( mbox_s4 @ ( qmltpeq @ ( inv @ unit ) @ unit ) ) )).

thf(ax6,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( inv @ ( inv @ e0 ) ) @ e0 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( inv @ ( inv @ e1 ) ) @ e1 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( inv @ ( inv @ e2 ) ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ ( inv @ e3 ) ) @ e3 ) ) ) ) ) )).

thf(ax7,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e0 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e1 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e3 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e0 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e0 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e1 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e3 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e0 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e1 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e3 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e0 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e1 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ e3 ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e3 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e3 ) @ e3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

thf(ax8,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ ( inv @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ ( inv @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e0 ) @ ( inv @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ ( inv @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e1 ) @ ( inv @ e3 ) ) ) ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( inv @ e2 ) @ ( inv @ e3 ) ) ) ) ) ) ) ) ) ) )).

thf(ax9,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ ( op @ e1 @ e0 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ ( op @ e2 @ e0 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ ( op @ e3 @ e0 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ ( op @ e2 @ e0 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ ( op @ e3 @ e0 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ ( op @ e3 @ e0 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ ( op @ e1 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ ( op @ e2 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ ( op @ e3 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ ( op @ e2 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ ( op @ e3 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e1 ) @ ( op @ e3 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ ( op @ e1 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ ( op @ e2 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ ( op @ e3 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e2 ) @ ( op @ e2 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e2 ) @ ( op @ e3 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ ( op @ e3 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e3 ) @ ( op @ e1 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e3 ) @ ( op @ e2 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e3 ) @ ( op @ e3 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e3 ) @ ( op @ e2 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e3 ) @ ( op @ e3 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e3 ) @ ( op @ e3 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ ( op @ e0 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ ( op @ e0 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ ( op @ e0 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ ( op @ e0 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e1 ) @ ( op @ e0 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e2 ) @ ( op @ e0 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ ( op @ e1 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ ( op @ e1 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e0 ) @ ( op @ e1 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ ( op @ e1 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ ( op @ e1 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e2 ) @ ( op @ e1 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ ( op @ e2 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ ( op @ e2 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e0 ) @ ( op @ e2 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e1 ) @ ( op @ e2 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e1 ) @ ( op @ e2 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ ( op @ e2 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e0 ) @ ( op @ e3 @ e1 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e0 ) @ ( op @ e3 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e0 ) @ ( op @ e3 @ e3 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e1 ) @ ( op @ e3 @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e1 ) @ ( op @ e3 @ e3 ) ) ) ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e2 ) @ ( op @ e3 @ e3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

thf(ax10,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ e0 @ e1 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ e0 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ e0 @ e3 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ e1 @ e2 ) ) ) ) @ ( mand @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ e1 @ e3 ) ) ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ e2 @ e3 ) ) ) ) ) ) ) ) ) )).

thf(ax11,axiom,
( mvalid @ ( mand @ ( mbox_s4 @ ( qmltpeq @ e0 @ ( op @ e3 @ e3 ) ) ) @ ( mbox_s4 @ ( qmltpeq @ e1 @ ( op @ e2 @ e3 ) ) ) ) )).

thf(co1,conjecture,
( mvalid @ ( mbox_s4 @ ( mnot @ ( mand @ ( mor @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e0 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e0 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e0 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e0 ) ) ) ) ) @ ( mor @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e1 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e1 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e1 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e1 ) ) ) ) ) @ ( mor @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e2 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e2 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e3 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e3 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e3 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e3 ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mnot @ ( mor @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e0 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e0 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e0 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e0 ) ) ) ) ) @ ( mor @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e1 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e1 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e1 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e1 ) ) ) ) ) @ ( mor @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e2 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e2 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e2 ) ) ) ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e0 @ e0 ) @ e3 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e1 @ e1 ) @ e3 ) ) @ ( mand @ ( mbox_s4 @ ( qmltpeq @ ( op @ e2 @ e2 ) @ e3 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op @ e3 @ e3 ) @ e3 ) ) ) ) ) ) ) ) ) ) ) ) ) )).

%------------------------------------------------------------------------------
```