%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : HEN005-4 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:08:47 AM UTC 2025
% Result : Unsatisfiable 0.13s 0.38s
% Output : Proof 0.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 32
% Syntax : Number of formulae : 77 ( 41 unt; 0 typ; 0 def)
% Number of atoms : 175 ( 92 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 169 ( 76 ~; 71 |; 0 &)
% ( 22 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of FOOLs : 5 ( 5 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 4 con; 0-2 aty)
% Number of variables : 118 ( 109 !; 0 ?; 118 :)
% Comments :
%------------------------------------------------------------------------------
tff(zero_type,type,
zero: $i ).
tff(divide_type,type,
divide: ( $i * $i ) > $i ).
tff(c_type,type,
c: $i ).
tff(a_type,type,
a: $i ).
tff(b_type,type,
b: $i ).
tff(less_equal_type,type,
less_equal: ( $i * $i ) > $o ).
tff(1,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z))
<=> less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z))
<=> ! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z))
<=> ! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)),
file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax',quotient_property) ).
tff(5,plain,
! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z))
| less_equal(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
less_equal(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
<=> ( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
<=> ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
<=> ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) ),
file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax',quotient_less_equal1) ).
tff(14,plain,
! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c))
| ( divide(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)) = zero ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c))
| ( divide(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)) = zero ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,plain,
( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c))
| ( divide(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)) = zero ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c))
| ( divide(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)) = zero ) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
divide(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)) = zero,
inference(unit_resolution,[status(thm)],[19,16,9]) ).
tff(21,plain,
( less_equal(a,b)
<=> less_equal(a,b) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
less_equal(a,b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_LE_b) ).
tff(23,plain,
less_equal(a,b),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(a,b)
| ( divide(a,b) = zero ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(a,b)
| ( divide(a,b) = zero ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(25,plain,
( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(a,b)
| ( divide(a,b) = zero ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(26,plain,
( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(a,b)
| ( divide(a,b) = zero ) ),
inference(modus_ponens,[status(thm)],[25,24]) ).
tff(27,plain,
divide(a,b) = zero,
inference(unit_resolution,[status(thm)],[26,16,23]) ).
tff(28,plain,
divide(divide(a,b),c) = divide(zero,c),
inference(monotonicity,[status(thm)],[27]) ).
tff(29,plain,
divide(zero,c) = divide(divide(a,b),c),
inference(symmetry,[status(thm)],[28]) ).
tff(30,plain,
^ [X: $i] :
refl(( ( divide(zero,X) = zero )
<=> ( divide(zero,X) = zero ) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [X: $i] : ( divide(zero,X) = zero )
<=> ! [X: $i] : ( divide(zero,X) = zero ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [X: $i] : ( divide(zero,X) = zero )
<=> ! [X: $i] : ( divide(zero,X) = zero ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,axiom,
! [X: $i] : ( divide(zero,X) = zero ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',zero_divide_anything_is_zero) ).
tff(34,plain,
! [X: $i] : ( divide(zero,X) = zero ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [X: $i] : ( divide(zero,X) = zero ),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [X: $i] : ( divide(zero,X) = zero ),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
( ~ ! [X: $i] : ( divide(zero,X) = zero )
| ( divide(zero,c) = zero ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(38,plain,
divide(zero,c) = zero,
inference(unit_resolution,[status(thm)],[37,36]) ).
tff(39,plain,
zero = divide(zero,c),
inference(symmetry,[status(thm)],[38]) ).
tff(40,plain,
( less_equal(b,c)
<=> less_equal(b,c) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,axiom,
less_equal(b,c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_LE_c) ).
tff(42,plain,
less_equal(b,c),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(b,c)
| ( divide(b,c) = zero ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(b,c)
| ( divide(b,c) = zero ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(44,plain,
( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(b,c)
| ( divide(b,c) = zero ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(45,plain,
( ~ ! [Y: $i,X: $i] :
( ~ less_equal(X,Y)
| ( divide(X,Y) = zero ) )
| ~ less_equal(b,c)
| ( divide(b,c) = zero ) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
divide(b,c) = zero,
inference(unit_resolution,[status(thm)],[45,16,42]) ).
tff(47,plain,
divide(b,c) = divide(divide(a,b),c),
inference(transitivity,[status(thm)],[46,39,29]) ).
tff(48,plain,
zero = divide(b,c),
inference(symmetry,[status(thm)],[46]) ).
tff(49,plain,
divide(divide(a,c),zero) = divide(divide(a,c),divide(b,c)),
inference(monotonicity,[status(thm)],[48]) ).
tff(50,plain,
^ [X: $i] :
refl(( ( divide(X,zero) = X )
<=> ( divide(X,zero) = X ) )),
inference(bind,[status(th)],[]) ).
tff(51,plain,
( ! [X: $i] : ( divide(X,zero) = X )
<=> ! [X: $i] : ( divide(X,zero) = X ) ),
inference(quant_intro,[status(thm)],[50]) ).
tff(52,plain,
( ! [X: $i] : ( divide(X,zero) = X )
<=> ! [X: $i] : ( divide(X,zero) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(53,axiom,
! [X: $i] : ( divide(X,zero) = X ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',x_divide_zero_is_x) ).
tff(54,plain,
! [X: $i] : ( divide(X,zero) = X ),
inference(modus_ponens,[status(thm)],[53,52]) ).
tff(55,plain,
! [X: $i] : ( divide(X,zero) = X ),
inference(skolemize,[status(sab)],[54]) ).
tff(56,plain,
! [X: $i] : ( divide(X,zero) = X ),
inference(modus_ponens,[status(thm)],[55,51]) ).
tff(57,plain,
( ~ ! [X: $i] : ( divide(X,zero) = X )
| ( divide(divide(a,c),zero) = divide(a,c) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(58,plain,
divide(divide(a,c),zero) = divide(a,c),
inference(unit_resolution,[status(thm)],[57,56]) ).
tff(59,plain,
divide(a,c) = divide(divide(a,c),zero),
inference(symmetry,[status(thm)],[58]) ).
tff(60,plain,
divide(a,c) = divide(divide(a,c),divide(b,c)),
inference(transitivity,[status(thm)],[59,49]) ).
tff(61,plain,
divide(divide(a,c),divide(b,c)) = divide(divide(divide(a,c),divide(b,c)),divide(divide(a,b),c)),
inference(monotonicity,[status(thm)],[60,47]) ).
tff(62,plain,
divide(a,c) = zero,
inference(transitivity,[status(thm)],[59,49,61,20]) ).
tff(63,plain,
( ~ less_equal(a,c)
<=> ~ less_equal(a,c) ),
inference(rewrite,[status(thm)],[]) ).
tff(64,axiom,
~ less_equal(a,c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_a_LE_c) ).
tff(65,plain,
~ less_equal(a,c),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
^ [Y: $i,X: $i] :
refl(( ( ( divide(X,Y) != zero )
| less_equal(X,Y) )
<=> ( ( divide(X,Y) != zero )
| less_equal(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(67,plain,
( ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) )
<=> ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) ) ),
inference(quant_intro,[status(thm)],[66]) ).
tff(68,plain,
( ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) )
<=> ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(69,axiom,
! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax',quotient_less_equal2) ).
tff(70,plain,
! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) ),
inference(skolemize,[status(sab)],[70]) ).
tff(72,plain,
! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) ),
inference(modus_ponens,[status(thm)],[71,67]) ).
tff(73,plain,
( ( ~ ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) )
| ( divide(a,c) != zero )
| less_equal(a,c) )
<=> ( ~ ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) )
| ( divide(a,c) != zero )
| less_equal(a,c) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(74,plain,
( ~ ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) )
| ( divide(a,c) != zero )
| less_equal(a,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(75,plain,
( ~ ! [Y: $i,X: $i] :
( ( divide(X,Y) != zero )
| less_equal(X,Y) )
| ( divide(a,c) != zero )
| less_equal(a,c) ),
inference(modus_ponens,[status(thm)],[74,73]) ).
tff(76,plain,
divide(a,c) != zero,
inference(unit_resolution,[status(thm)],[75,72,65]) ).
tff(77,plain,
$false,
inference(unit_resolution,[status(thm)],[76,62]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : HEN005-4 : TPTP v9.0.0. Released v1.0.0.
% 0.12/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n006.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 17:12:20 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.13/0.38 % SZS status Unsatisfiable
% 0.13/0.38 % SZS output start Proof
% See solution above
% 0.13/0.39 % E exiting
%------------------------------------------------------------------------------