%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : HAL003+3 : TPTP v9.0.0. Released v2.6.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:08:44 AM UTC 2025
% Result : Theorem 0.22s 0.41s
% Output : Proof 0.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 35
% Syntax : Number of formulae : 80 ( 21 unt; 0 typ; 0 def)
% Number of atoms : 627 ( 108 equ)
% Maximal formula atoms : 20 ( 7 avg)
% Number of connectives : 925 ( 424 ~; 311 |; 118 &)
% ( 52 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 46 ( 46 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 3 con; 0-3 aty)
% Number of variables : 295 ( 221 !; 45 ?; 295 :)
% Comments :
%------------------------------------------------------------------------------
tff(element_type,type,
element: ( $i * $i ) > $o ).
tff(b_type,type,
b: $i ).
tff(subtract_type,type,
subtract: ( $i * $i * $i ) > $i ).
tff(tptp_fun_B2_11_type,type,
tptp_fun_B2_11: $i > $i ).
tff(tptp_fun_ElCod_3_type,type,
tptp_fun_ElCod_3: ( $i * $i * $i ) > $i ).
tff(g_type,type,
g: $i ).
tff(e_type,type,
e: $i ).
tff(tptp_fun_B1_12_type,type,
tptp_fun_B1_12: $i > $i ).
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(surjection_type,type,
surjection: $i > $o ).
tff(morphism_type,type,
morphism: ( $i * $i * $i ) > $o ).
tff(1,plain,
( ~ surjection(g)
<=> ~ surjection(g) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
~ surjection(g),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',g_surjection) ).
tff(3,plain,
~ surjection(g),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
( morphism(g,b,e)
<=> morphism(g,b,e) ),
inference(rewrite,[status(thm)],[]) ).
tff(5,axiom,
morphism(g,b,e),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',g_morphism) ).
tff(6,plain,
morphism(g,b,e),
inference(modus_ponens,[status(thm)],[5,4]) ).
tff(7,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
refl(( ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(( ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(10,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) ),
inference(quant_intro,[status(thm)],[9]) ).
tff(11,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) ),
inference(transitivity,[status(thm)],[10,8]) ).
tff(12,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(( ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(13,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) ),
inference(quant_intro,[status(thm)],[12]) ).
tff(14,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
trans(monotonicity(monotonicity(rewrite(( ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) )
<=> ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )),
( ( ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) )),
( ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) )),
rewrite(( ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) )),
( ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(15,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ) ),
inference(quant_intro,[status(thm)],[14]) ).
tff(16,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(17,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
trans(monotonicity(rewrite(( ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( element(ElCod,Cod)
=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) )
<=> ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) )),
( ( ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( element(ElCod,Cod)
=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) )
=> surjection(Morphism) )
<=> ( ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) )
=> surjection(Morphism) ) )),
rewrite(( ( ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) )
=> surjection(Morphism) )
<=> ( surjection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) ) )),
( ( ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( element(ElCod,Cod)
=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) )
=> surjection(Morphism) )
<=> ( surjection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(18,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( element(ElCod,Cod)
=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) )
=> surjection(Morphism) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) ) ),
inference(quant_intro,[status(thm)],[17]) ).
tff(19,axiom,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( element(ElCod,Cod)
=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) )
=> surjection(Morphism) ),
file('/export/starexec/sandbox/benchmark/Axioms/HAL001+0.ax',properties_for_surjection) ).
tff(20,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) ),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [ElCod: $i] :
( ~ element(ElCod,Cod)
| ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = ElCod ) ) ) ) ),
inference(modus_ponens,[status(thm)],[20,16]) ).
tff(22,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ),
inference(skolemize,[status(sab)],[21]) ).
tff(23,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ( element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
& ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism,ElDom) = tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ),
inference(modus_ponens,[status(thm)],[22,15]) ).
tff(24,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ),
inference(modus_ponens,[status(thm)],[23,13]) ).
tff(25,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) ),
inference(modus_ponens,[status(thm)],[24,11]) ).
tff(26,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
| surjection(g)
| ~ morphism(g,b,e)
| ~ ( ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
| surjection(g)
| ~ morphism(g,b,e)
| ~ ( ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
| surjection(g)
| ~ morphism(g,b,e)
| ~ ( ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( surjection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ element(tptp_fun_ElCod_3(Cod,Dom,Morphism),Cod)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism,ElDom) != tptp_fun_ElCod_3(Cod,Dom,Morphism) ) ) ) )
| surjection(g)
| ~ morphism(g,b,e)
| ~ ( ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) ) ) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
~ ( ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) ) ),
inference(unit_resolution,[status(thm)],[28,25,6,3]) ).
tff(30,plain,
( ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) )
| element(tptp_fun_ElCod_3(e,b,g),e) ),
inference(tautology,[status(thm)],[]) ).
tff(31,plain,
element(tptp_fun_ElCod_3(e,b,g),e),
inference(unit_resolution,[status(thm)],[30,29]) ).
tff(32,plain,
^ [E: $i] :
refl(( ( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) )
<=> ( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(33,plain,
( ! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) )
<=> ! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) ) ),
inference(quant_intro,[status(thm)],[32]) ).
tff(34,plain,
^ [E: $i] :
rewrite(( ( ~ element(E,e)
| ( element(tptp_fun_B1_12(E),b)
& element(tptp_fun_B2_11(E),b)
& ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) = E ) ) )
<=> ( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(35,plain,
( ! [E: $i] :
( ~ element(E,e)
| ( element(tptp_fun_B1_12(E),b)
& element(tptp_fun_B2_11(E),b)
& ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) = E ) ) )
<=> ! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) ) ),
inference(quant_intro,[status(thm)],[34]) ).
tff(36,plain,
( ! [E: $i] :
( ~ element(E,e)
| ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) )
<=> ! [E: $i] :
( ~ element(E,e)
| ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(37,plain,
^ [E: $i] :
trans(monotonicity(quant_intro(proof_bind(^ [B1: $i,B2: $i] :
rewrite(( ( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) )
<=> ( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ))),
( ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) )
<=> ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) )),
( ( element(E,e)
=> ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) )
<=> ( element(E,e)
=> ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ) )),
rewrite(( ( element(E,e)
=> ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) )
<=> ( ~ element(E,e)
| ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ) )),
( ( element(E,e)
=> ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) )
<=> ( ~ element(E,e)
| ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(38,plain,
( ! [E: $i] :
( element(E,e)
=> ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) )
<=> ! [E: $i] :
( ~ element(E,e)
| ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ) ),
inference(quant_intro,[status(thm)],[37]) ).
tff(39,axiom,
! [E: $i] :
( element(E,e)
=> ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',lemma12) ).
tff(40,plain,
! [E: $i] :
( ~ element(E,e)
| ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ),
inference(modus_ponens,[status(thm)],[39,38]) ).
tff(41,plain,
! [E: $i] :
( ~ element(E,e)
| ? [B1: $i,B2: $i] :
( element(B1,b)
& element(B2,b)
& ( apply(g,subtract(b,B1,B2)) = E ) ) ),
inference(modus_ponens,[status(thm)],[40,36]) ).
tff(42,plain,
! [E: $i] :
( ~ element(E,e)
| ( element(tptp_fun_B1_12(E),b)
& element(tptp_fun_B2_11(E),b)
& ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) = E ) ) ),
inference(skolemize,[status(sab)],[41]) ).
tff(43,plain,
! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) ),
inference(modus_ponens,[status(thm)],[42,35]) ).
tff(44,plain,
! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) ),
inference(modus_ponens,[status(thm)],[43,33]) ).
tff(45,plain,
( ( ~ ! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) )
| ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ) )
<=> ( ~ ! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) )
| ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(46,plain,
( ~ ! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) )
| ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
( ~ ! [E: $i] :
( ~ element(E,e)
| ~ ( ~ element(tptp_fun_B1_12(E),b)
| ~ element(tptp_fun_B2_11(E),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(E),tptp_fun_B2_11(E))) != E ) ) )
| ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ) ),
inference(modus_ponens,[status(thm)],[46,45]) ).
tff(48,plain,
~ ( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ),
inference(unit_resolution,[status(thm)],[47,44,31]) ).
tff(49,plain,
( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) )
| element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b) ),
inference(tautology,[status(thm)],[]) ).
tff(50,plain,
element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b),
inference(unit_resolution,[status(thm)],[49,48]) ).
tff(51,plain,
( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) )
| element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b) ),
inference(tautology,[status(thm)],[]) ).
tff(52,plain,
element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b),
inference(unit_resolution,[status(thm)],[51,48]) ).
tff(53,plain,
^ [Dom: $i,El1: $i,El2: $i] :
refl(( ( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
<=> ( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) ) )),
inference(bind,[status(th)],[]) ).
tff(54,plain,
( ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
<=> ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) ) ),
inference(quant_intro,[status(thm)],[53]) ).
tff(55,plain,
^ [Dom: $i,El1: $i,El2: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( element(El1,Dom)
& element(El2,Dom) )
<=> ~ ( ~ element(El2,Dom)
| ~ element(El1,Dom) ) )),
( ~ ( element(El1,Dom)
& element(El2,Dom) )
<=> ~ ~ ( ~ element(El2,Dom)
| ~ element(El1,Dom) ) )),
rewrite(( ~ ~ ( ~ element(El2,Dom)
| ~ element(El1,Dom) )
<=> ( ~ element(El2,Dom)
| ~ element(El1,Dom) ) )),
( ~ ( element(El1,Dom)
& element(El2,Dom) )
<=> ( ~ element(El2,Dom)
| ~ element(El1,Dom) ) )),
( ( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) )
<=> ( ~ element(El2,Dom)
| ~ element(El1,Dom)
| element(subtract(Dom,El1,El2),Dom) ) )),
rewrite(( ( ~ element(El2,Dom)
| ~ element(El1,Dom)
| element(subtract(Dom,El1,El2),Dom) )
<=> ( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) ) )),
( ( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) )
<=> ( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) ) )),
inference(bind,[status(th)],[]) ).
tff(56,plain,
( ! [Dom: $i,El1: $i,El2: $i] :
( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) )
<=> ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) ) ),
inference(quant_intro,[status(thm)],[55]) ).
tff(57,plain,
( ! [Dom: $i,El1: $i,El2: $i] :
( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) )
<=> ! [Dom: $i,El1: $i,El2: $i] :
( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,plain,
^ [Dom: $i,El1: $i,El2: $i] :
rewrite(( ( ( element(El1,Dom)
& element(El2,Dom) )
=> element(subtract(Dom,El1,El2),Dom) )
<=> ( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) ) )),
inference(bind,[status(th)],[]) ).
tff(59,plain,
( ! [Dom: $i,El1: $i,El2: $i] :
( ( element(El1,Dom)
& element(El2,Dom) )
=> element(subtract(Dom,El1,El2),Dom) )
<=> ! [Dom: $i,El1: $i,El2: $i] :
( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) ) ),
inference(quant_intro,[status(thm)],[58]) ).
tff(60,axiom,
! [Dom: $i,El1: $i,El2: $i] :
( ( element(El1,Dom)
& element(El2,Dom) )
=> element(subtract(Dom,El1,El2),Dom) ),
file('/export/starexec/sandbox/benchmark/Axioms/HAL001+0.ax',subtract_in_domain) ).
tff(61,plain,
! [Dom: $i,El1: $i,El2: $i] :
( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) ),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
! [Dom: $i,El1: $i,El2: $i] :
( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) ),
inference(modus_ponens,[status(thm)],[61,57]) ).
tff(63,plain,
! [Dom: $i,El1: $i,El2: $i] :
( ~ ( element(El1,Dom)
& element(El2,Dom) )
| element(subtract(Dom,El1,El2),Dom) ),
inference(skolemize,[status(sab)],[62]) ).
tff(64,plain,
! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) ),
inference(modus_ponens,[status(thm)],[63,56]) ).
tff(65,plain,
! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) ),
inference(modus_ponens,[status(thm)],[64,54]) ).
tff(66,plain,
( ( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b) )
<=> ( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(67,plain,
( ( element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b) )
<=> ( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(68,plain,
( ( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b) )
<=> ( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b) ) ),
inference(monotonicity,[status(thm)],[67]) ).
tff(69,plain,
( ( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b) )
<=> ( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b) ) ),
inference(transitivity,[status(thm)],[68,66]) ).
tff(70,plain,
( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(71,plain,
( ~ ! [Dom: $i,El1: $i,El2: $i] :
( element(subtract(Dom,El1,El2),Dom)
| ~ element(El2,Dom)
| ~ element(El1,Dom) )
| ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b) ),
inference(modus_ponens,[status(thm)],[70,69]) ).
tff(72,plain,
element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b),
inference(unit_resolution,[status(thm)],[71,65,52,50]) ).
tff(73,plain,
( ~ element(tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),b)
| ~ element(tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) )
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) = tptp_fun_ElCod_3(e,b,g) ) ),
inference(tautology,[status(thm)],[]) ).
tff(74,plain,
apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) = tptp_fun_ElCod_3(e,b,g),
inference(unit_resolution,[status(thm)],[73,48]) ).
tff(75,plain,
( ~ element(tptp_fun_ElCod_3(e,b,g),e)
| ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) ) ),
inference(tautology,[status(thm)],[]) ).
tff(76,plain,
! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) ),
inference(unit_resolution,[status(thm)],[75,29]) ).
tff(77,plain,
( ( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) )
| ~ element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) )
<=> ( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) )
| ~ element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(78,plain,
( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) )
| ~ element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(79,plain,
( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(g,ElDom) != tptp_fun_ElCod_3(e,b,g) ) )
| ~ element(subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g))),b)
| ( apply(g,subtract(b,tptp_fun_B1_12(tptp_fun_ElCod_3(e,b,g)),tptp_fun_B2_11(tptp_fun_ElCod_3(e,b,g)))) != tptp_fun_ElCod_3(e,b,g) ) ),
inference(modus_ponens,[status(thm)],[78,77]) ).
tff(80,plain,
$false,
inference(unit_resolution,[status(thm)],[79,76,74,72]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : HAL003+3 : TPTP v9.0.0. Released v2.6.0.
% 0.03/0.13 % Command : run_E %s %d THM
% 0.15/0.35 % Computer : n001.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Thu Jun 19 17:14:33 EDT 2025
% 0.22/0.35 % CPUTime :
% 0.22/0.41 % SZS status Theorem
% 0.22/0.41 % SZS output start Proof
% See solution above
% 0.22/0.44 % E exiting
%------------------------------------------------------------------------------