TPTP Problem File: ALG014^7.p
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%------------------------------------------------------------------------------
% File : ALG014^7 : TPTP v8.2.0. Released v5.5.0.
% Domain : General Algebra
% Problem : Groups 4: CPROPS-COVERING-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-GAL014+1 [Ben12]
% Status : Theorem
% Rating : 1.00 v5.5.0
% Syntax : Number of formulae : 104 ( 40 unt; 43 typ; 32 def)
% Number of atoms : 1202 ( 36 equ; 0 cnn)
% Maximal formula atoms : 240 ( 19 avg)
% Number of connectives : 2861 ( 5 ~; 5 |; 9 &;2832 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 73 ( 8 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 183 ( 183 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 51 usr; 15 con; 0-3 aty)
% Number of variables : 114 ( 63 ^; 44 !; 7 ?; 114 :)
% SPC : TH0_THM_EQU_NAR
% Comments : Goedel translation of ALG014+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 @ unit @ e0 ) ) ).
thf(ax6,axiom,
mvalid @ ( mbox_s4 @ ( qmltpeq @ ( inv @ unit ) @ unit ) ) ).
thf(ax7,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(ax8,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(ax9,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(ax10,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(ax11,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(co1,conjecture,
mvalid @ ( 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 ) ) ) ) ) @ ( mor @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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