TSTP Solution File: HAL001+2 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : HAL001+2 : TPTP v8.1.0. Released v2.6.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n021.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 : Fri Sep 16 22:35:02 EDT 2022
% Result : Theorem 0.19s 0.43s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 135
% Syntax : Number of formulae : 273 ( 74 unt; 22 typ; 0 def)
% Number of atoms : 2566 ( 778 equ)
% Maximal formula atoms : 44 ( 10 avg)
% Number of connectives : 4153 (1991 ~;1587 |; 346 &)
% ( 177 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 153 ( 153 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 21 ( 9 >; 12 *; 0 +; 0 <<)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-4 aty)
% Number of functors : 17 ( 17 usr; 13 con; 0-3 aty)
% Number of variables : 1177 (1060 !; 13 ?;1177 :)
% Comments :
%------------------------------------------------------------------------------
tff(zero_type,type,
zero: $i > $i ).
tff(b_type,type,
b: $i ).
tff(tptp_fun_El_8_type,type,
tptp_fun_El_8: ( $i * $i * $i ) > $i ).
tff(g_type,type,
g: $i ).
tff(e_type,type,
e: $i ).
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(alpha_type,type,
alpha: $i ).
tff(element_type,type,
element: ( $i * $i ) > $o ).
tff(morphism_type,type,
morphism: ( $i * $i * $i ) > $o ).
tff(tptp_fun_ElDom_4_type,type,
tptp_fun_ElDom_4: ( $i * $i * $i ) > $i ).
tff(d_type,type,
d: $i ).
tff(f_type,type,
f: $i ).
tff(gamma_type,type,
gamma: $i ).
tff(injection_type,type,
injection: $i > $o ).
tff(c_type,type,
c: $i ).
tff(beta_type,type,
beta: $i ).
tff(exact_type,type,
exact: ( $i * $i ) > $o ).
tff(r_type,type,
r: $i ).
tff(h_type,type,
h: $i ).
tff(delta_type,type,
delta: $i ).
tff(commute_type,type,
commute: ( $i * $i * $i * $i ) > $o ).
tff(1,plain,
( morphism(alpha,a,b)
<=> morphism(alpha,a,b) ),
inference(rewrite,[status(thm)],]) ).
tff(2,axiom,
morphism(alpha,a,b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alpha_morphism) ).
tff(3,plain,
morphism(alpha,a,b),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(
( ( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(5,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
refl(
( ( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(7,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[6]) ).
tff(8,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(
( ( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(9,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[8]) ).
tff(10,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ) ),
inference(transitivity,[status(thm)],[9,7]) ).
tff(11,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(
( ( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(12,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[11]) ).
tff(13,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(14,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
trans(
monotonicity(
rewrite(
( ( ! [El: $i] :
( element(El,Dom)
=> element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) )
<=> ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )),
( ( morphism(Morphism,Dom,Cod)
=> ( ! [El: $i] :
( element(El,Dom)
=> element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )
<=> ( morphism(Morphism,Dom,Cod)
=> ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ) )),
rewrite(
( ( morphism(Morphism,Dom,Cod)
=> ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ) )),
( ( morphism(Morphism,Dom,Cod)
=> ( ! [El: $i] :
( element(El,Dom)
=> element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(15,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( morphism(Morphism,Dom,Cod)
=> ( ! [El: $i] :
( element(El,Dom)
=> element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[14]) ).
tff(16,axiom,
! [Morphism: $i,Dom: $i,Cod: $i] :
( morphism(Morphism,Dom,Cod)
=> ( ! [El: $i] :
( element(El,Dom)
=> element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/HAL001+0.ax',morphism) ).
tff(17,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[16,15]) ).
tff(18,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[17,13]) ).
tff(19,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ( ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
& ( apply(Morphism,zero(Dom)) = zero(Cod) ) ) ),
inference(skolemize,[status(sab)],[18]) ).
tff(20,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[19,12]) ).
tff(21,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) )
| ( apply(Morphism,zero(Dom)) != zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[20,10]) ).
tff(22,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) ),
inference(modus_ponens,[status(thm)],[21,5]) ).
tff(23,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(alpha,a,b)
| ~ ( ( apply(alpha,zero(a)) != zero(b) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(alpha,El),b) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(alpha,a,b)
| ~ ( ( apply(alpha,zero(a)) != zero(b) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(alpha,El),b) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(24,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(alpha,a,b)
| ~ ( ( apply(alpha,zero(a)) != zero(b) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(alpha,El),b) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(25,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(alpha,a,b)
| ~ ( ( apply(alpha,zero(a)) != zero(b) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(alpha,El),b) ) ) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
~ ( ( apply(alpha,zero(a)) != zero(b) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(alpha,El),b) ) ),
inference(unit_resolution,[status(thm)],[25,22,3]) ).
tff(27,plain,
( ( apply(alpha,zero(a)) != zero(b) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(alpha,El),b) )
| ( apply(alpha,zero(a)) = zero(b) ) ),
inference(tautology,[status(thm)],]) ).
tff(28,plain,
apply(alpha,zero(a)) = zero(b),
inference(unit_resolution,[status(thm)],[27,26]) ).
tff(29,plain,
( ~ injection(g)
<=> ~ injection(g) ),
inference(rewrite,[status(thm)],]) ).
tff(30,axiom,
~ injection(g),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',g_injection) ).
tff(31,plain,
~ injection(g),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
( morphism(g,b,e)
<=> morphism(g,b,e) ),
inference(rewrite,[status(thm)],]) ).
tff(33,axiom,
morphism(g,b,e),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',g_morphism) ).
tff(34,plain,
morphism(g,b,e),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
refl(
( ( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) )
<=> ( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(36,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[35]) ).
tff(37,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(
( ( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) )
<=> ( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(38,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[37]) ).
tff(39,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(
( ( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) )
<=> ( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(40,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(42,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
trans(
monotonicity(
rewrite(
( ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
<=> ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) )),
( ( ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
=> injection(Morphism) )
<=> ( ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )
=> injection(Morphism) ) )),
rewrite(
( ( ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )
=> injection(Morphism) )
<=> ( injection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ) )),
( ( ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
=> injection(Morphism) )
<=> ( injection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(43,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
=> injection(Morphism) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ) ),
inference(quant_intro,[status(thm)],[42]) ).
tff(44,axiom,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
=> injection(Morphism) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',properties_for_injection_2) ).
tff(45,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ ( morphism(Morphism,Dom,Cod)
& ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ),
inference(modus_ponens,[status(thm)],[45,41]) ).
tff(47,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) ),
inference(skolemize,[status(sab)],[46]) ).
tff(48,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ~ ( element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
& ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) = zero(Cod) ) )
| ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) ) ) ),
inference(modus_ponens,[status(thm)],[47,40]) ).
tff(49,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[48,38]) ).
tff(50,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[49,36]) ).
tff(51,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) )
| injection(g)
| ~ morphism(g,b,e)
| ~ ( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) )
| injection(g)
| ~ morphism(g,b,e)
| ~ ( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(52,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) )
| injection(g)
| ~ morphism(g,b,e)
| ~ ( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(53,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ~ ( ( tptp_fun_El_8(Cod,Dom,Morphism) = zero(Dom) )
| ~ element(tptp_fun_El_8(Cod,Dom,Morphism),Dom)
| ( apply(Morphism,tptp_fun_El_8(Cod,Dom,Morphism)) != zero(Cod) ) ) )
| injection(g)
| ~ morphism(g,b,e)
| ~ ( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) ) ) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
~ ( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) ) ),
inference(unit_resolution,[status(thm)],[53,50,34,31]) ).
tff(55,plain,
( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) )
| ( apply(g,tptp_fun_El_8(e,b,g)) = zero(e) ) ),
inference(tautology,[status(thm)],]) ).
tff(56,plain,
apply(g,tptp_fun_El_8(e,b,g)) = zero(e),
inference(unit_resolution,[status(thm)],[55,54]) ).
tff(57,plain,
( exact(alpha,beta)
<=> exact(alpha,beta) ),
inference(rewrite,[status(thm)],]) ).
tff(58,axiom,
exact(alpha,beta),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alpha_beta_exact) ).
tff(59,plain,
exact(alpha,beta),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
( morphism(beta,b,c)
<=> morphism(beta,b,c) ),
inference(rewrite,[status(thm)],]) ).
tff(61,axiom,
morphism(beta,b,c),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',beta_morphism) ).
tff(62,plain,
morphism(beta,b,c),
inference(modus_ponens,[status(thm)],[61,60]) ).
tff(63,plain,
^ [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
refl(
( ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
<=> ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(64,plain,
( ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
<=> ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) ),
inference(quant_intro,[status(thm)],[63]) ).
tff(65,plain,
^ [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
rewrite(
( ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
<=> ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(66,plain,
( ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
<=> ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) ),
inference(quant_intro,[status(thm)],[65]) ).
tff(67,plain,
( ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
<=> ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) ),
inference(transitivity,[status(thm)],[66,64]) ).
tff(68,plain,
^ [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
<=> ~ ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2) ) )),
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
<=> ~ ~ ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2) ) )),
rewrite(
( ~ ~ ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2) )
<=> ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2) ) )),
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
<=> ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2) ) )),
quant_intro(
proof_bind(
^ [ElCodDom: $i] :
trans(
monotonicity(
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) ) )),
( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ~ ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) ) )),
rewrite(
( ~ ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
<=> ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) ) )),
( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) ) )),
rewrite(
( ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) )
<=> ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )),
( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
<=> ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) ) )),
rewrite(
( ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
<=> ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) ) )),
( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
<=> ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) ) )),
rewrite(
( ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) )
<=> ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) )),
( ( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
& ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ( ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
& ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )),
rewrite(
( ( ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
& ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) )
<=> ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )),
( ( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
& ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ))),
( ! [ElCodDom: $i] :
( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
& ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )),
( ( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
& ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) )
<=> ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) )),
rewrite(
( ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
<=> ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) )),
( ( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
& ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) )
<=> ( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(69,plain,
( ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
& ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) )
<=> ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ) ),
inference(quant_intro,[status(thm)],[68]) ).
tff(70,plain,
( ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(71,plain,
^ [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
trans(
monotonicity(
rewrite(
( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
<=> ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) ) )),
( ( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
=> ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
=> ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) )),
rewrite(
( ( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
=> ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) )),
( ( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
=> ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(72,plain,
( ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
=> ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) )
<=> ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) ),
inference(quant_intro,[status(thm)],[71]) ).
tff(73,axiom,
! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
=> ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/HAL001+0.ax',exact_properties) ).
tff(74,plain,
! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
<=> ? [ElDom: $i] :
( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ),
inference(modus_ponens,[status(thm)],[74,70]) ).
tff(76,plain,
! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ ( exact(Morphism1,Morphism2)
& morphism(Morphism1,Dom,CodDom)
& morphism(Morphism2,CodDom,Cod) )
| ! [ElCodDom: $i] :
( ( ~ ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ( element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
& ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) = ElCodDom ) ) )
& ( ( element(ElCodDom,CodDom)
& ( apply(Morphism2,ElCodDom) = zero(Cod) ) )
| ! [ElDom: $i] :
~ ( element(ElDom,Dom)
& ( apply(Morphism1,ElDom) = ElCodDom ) ) ) ) ),
inference(skolemize,[status(sab)],[75]) ).
tff(77,plain,
! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ),
inference(modus_ponens,[status(thm)],[76,69]) ).
tff(78,plain,
! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) ),
inference(modus_ponens,[status(thm)],[77,67]) ).
tff(79,plain,
( ( ~ ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(beta,b,c)
| ~ exact(alpha,beta)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(ElCodDom,a,alpha)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) ) ) )
<=> ( ~ ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(beta,b,c)
| ~ exact(alpha,beta)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(ElCodDom,a,alpha)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(80,plain,
( ~ ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(beta,b,c)
| ~ exact(alpha,beta)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(ElCodDom,a,alpha)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(81,plain,
( ~ ! [Morphism1: $i,Morphism2: $i,Dom: $i,CodDom: $i,Cod: $i] :
( ~ morphism(Morphism1,Dom,CodDom)
| ~ morphism(Morphism2,CodDom,Cod)
| ~ exact(Morphism1,Morphism2)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1),Dom)
| ( apply(Morphism1,tptp_fun_ElDom_4(ElCodDom,Dom,Morphism1)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,CodDom)
| ( apply(Morphism2,ElCodDom) != zero(Cod) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(Morphism1,ElDom) != ElCodDom ) ) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(beta,b,c)
| ~ exact(alpha,beta)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(ElCodDom,a,alpha)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) ) ) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(ElCodDom,a,alpha)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) ) ),
inference(unit_resolution,[status(thm)],[81,78,3,62,59]) ).
tff(83,plain,
( ~ ! [ElCodDom: $i] :
~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(ElCodDom,a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(ElCodDom,a,alpha)) != ElCodDom ) ) )
| ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) ) )
| ~ ( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) )
| ~ ( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != tptp_fun_El_8(e,b,g) ) ) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(84,plain,
~ ( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) )
| ~ ( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != tptp_fun_El_8(e,b,g) ) ) ) ),
inference(unit_resolution,[status(thm)],[83,82]) ).
tff(85,plain,
( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) )
| ~ ( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != tptp_fun_El_8(e,b,g) ) ) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) ),
inference(tautology,[status(thm)],]) ).
tff(86,plain,
( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) ),
inference(unit_resolution,[status(thm)],[85,84]) ).
tff(87,plain,
( morphism(delta,e,r)
<=> morphism(delta,e,r) ),
inference(rewrite,[status(thm)],]) ).
tff(88,axiom,
morphism(delta,e,r),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',delta_morphism) ).
tff(89,plain,
morphism(delta,e,r),
inference(modus_ponens,[status(thm)],[88,87]) ).
tff(90,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(delta,e,r)
| ~ ( ( apply(delta,zero(e)) != zero(r) )
| ~ ! [El: $i] :
( ~ element(El,e)
| element(apply(delta,El),r) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(delta,e,r)
| ~ ( ( apply(delta,zero(e)) != zero(r) )
| ~ ! [El: $i] :
( ~ element(El,e)
| element(apply(delta,El),r) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(91,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(delta,e,r)
| ~ ( ( apply(delta,zero(e)) != zero(r) )
| ~ ! [El: $i] :
( ~ element(El,e)
| element(apply(delta,El),r) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(92,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(delta,e,r)
| ~ ( ( apply(delta,zero(e)) != zero(r) )
| ~ ! [El: $i] :
( ~ element(El,e)
| element(apply(delta,El),r) ) ) ),
inference(modus_ponens,[status(thm)],[91,90]) ).
tff(93,plain,
~ ( ( apply(delta,zero(e)) != zero(r) )
| ~ ! [El: $i] :
( ~ element(El,e)
| element(apply(delta,El),r) ) ),
inference(unit_resolution,[status(thm)],[92,22,89]) ).
tff(94,plain,
( ( apply(delta,zero(e)) != zero(r) )
| ~ ! [El: $i] :
( ~ element(El,e)
| element(apply(delta,El),r) )
| ( apply(delta,zero(e)) = zero(r) ) ),
inference(tautology,[status(thm)],]) ).
tff(95,plain,
apply(delta,zero(e)) = zero(r),
inference(unit_resolution,[status(thm)],[94,93]) ).
tff(96,plain,
apply(delta,apply(g,tptp_fun_El_8(e,b,g))) = apply(delta,zero(e)),
inference(monotonicity,[status(thm)],[56]) ).
tff(97,plain,
( commute(beta,h,g,delta)
<=> commute(beta,h,g,delta) ),
inference(rewrite,[status(thm)],]) ).
tff(98,axiom,
commute(beta,h,g,delta),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',beta_h_g_delta_commute) ).
tff(99,plain,
commute(beta,h,g,delta),
inference(modus_ponens,[status(thm)],[98,97]) ).
tff(100,plain,
( morphism(h,c,r)
<=> morphism(h,c,r) ),
inference(rewrite,[status(thm)],]) ).
tff(101,axiom,
morphism(h,c,r),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',h_morphism) ).
tff(102,plain,
morphism(h,c,r),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
^ [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
rewrite(
( ( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(104,plain,
( ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) ),
inference(quant_intro,[status(thm)],[103]) ).
tff(105,plain,
^ [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
refl(
( ( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
inference(bind,[status(th)],]) ).
tff(106,plain,
( ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) ),
inference(quant_intro,[status(thm)],[105]) ).
tff(107,plain,
^ [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
rewrite(
( ( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
inference(bind,[status(th)],]) ).
tff(108,plain,
( ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) ),
inference(quant_intro,[status(thm)],[107]) ).
tff(109,plain,
( ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) ),
inference(transitivity,[status(thm)],[108,106]) ).
tff(110,plain,
^ [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
<=> ~ ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
<=> ~ ~ ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
rewrite(
( ~ ~ ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) )
<=> ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
<=> ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
( ( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) )),
rewrite(
( ( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
( ( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) )),
inference(bind,[status(th)],]) ).
tff(111,plain,
( ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ) ),
inference(quant_intro,[status(thm)],[110]) ).
tff(112,plain,
( ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(113,plain,
^ [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
trans(
monotonicity(
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod) )
<=> ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod) ) )),
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2) )
<=> ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2) ) )),
rewrite(
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2) )
<=> ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2) ) )),
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2) )
<=> ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2) ) )),
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
<=> ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) ) )),
rewrite(
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
<=> ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) ) )),
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
<=> ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) ) )),
quant_intro(
proof_bind(
^ [ElDom: $i] :
rewrite(
( ( element(ElDom,Dom)
=> ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
<=> ( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ))),
( ! [ElDom: $i] :
( element(ElDom,Dom)
=> ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
<=> ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )),
( ( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
=> ! [ElDom: $i] :
( element(ElDom,Dom)
=> ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
=> ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) )),
rewrite(
( ( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
=> ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) )),
( ( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
=> ! [ElDom: $i] :
( element(ElDom,Dom)
=> ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(114,plain,
( ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
=> ! [ElDom: $i] :
( element(ElDom,Dom)
=> ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
<=> ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ) ),
inference(quant_intro,[status(thm)],[113]) ).
tff(115,axiom,
! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
=> ! [ElDom: $i] :
( element(ElDom,Dom)
=> ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/HAL001+0.ax',commute_properties) ).
tff(116,plain,
! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ),
inference(modus_ponens,[status(thm)],[115,114]) ).
tff(117,plain,
! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ),
inference(modus_ponens,[status(thm)],[116,112]) ).
tff(118,plain,
! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ ( commute(M1,M2,M3,M4)
& morphism(M1,Dom,DomCod1)
& morphism(M2,DomCod1,Cod)
& morphism(M3,Dom,DomCod2)
& morphism(M4,DomCod2,Cod) )
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ),
inference(skolemize,[status(sab)],[117]) ).
tff(119,plain,
! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ),
inference(modus_ponens,[status(thm)],[118,111]) ).
tff(120,plain,
! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) )
| ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4) ),
inference(modus_ponens,[status(thm)],[119,109]) ).
tff(121,plain,
! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) ),
inference(modus_ponens,[status(thm)],[120,104]) ).
tff(122,plain,
( ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(g,b,e)
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) )
<=> ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(g,b,e)
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(123,plain,
( ( ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(g,b,e)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) )
<=> ( ~ morphism(g,b,e)
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(124,plain,
( ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(g,b,e)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) )
<=> ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(g,b,e)
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) ) ),
inference(monotonicity,[status(thm)],[123]) ).
tff(125,plain,
( ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(g,b,e)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) )
<=> ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(g,b,e)
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) ) ),
inference(transitivity,[status(thm)],[124,122]) ).
tff(126,plain,
( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(g,b,e)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(127,plain,
( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(g,b,e)
| ~ morphism(beta,b,c)
| ~ morphism(h,c,r)
| ~ morphism(delta,e,r)
| ~ commute(beta,h,g,delta)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ) ),
inference(modus_ponens,[status(thm)],[126,125]) ).
tff(128,plain,
! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ),
inference(unit_resolution,[status(thm)],[127,121,62,89,34,102,99]) ).
tff(129,plain,
( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) )
| element(tptp_fun_El_8(e,b,g),b) ),
inference(tautology,[status(thm)],]) ).
tff(130,plain,
element(tptp_fun_El_8(e,b,g),b),
inference(unit_resolution,[status(thm)],[129,54]) ).
tff(131,plain,
( ( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) = apply(delta,apply(g,tptp_fun_El_8(e,b,g))) ) )
<=> ( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) = apply(delta,apply(g,tptp_fun_El_8(e,b,g))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(132,plain,
( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) = apply(delta,apply(g,tptp_fun_El_8(e,b,g))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(133,plain,
( ~ ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) = apply(delta,apply(g,tptp_fun_El_8(e,b,g))) ) ),
inference(modus_ponens,[status(thm)],[132,131]) ).
tff(134,plain,
apply(h,apply(beta,tptp_fun_El_8(e,b,g))) = apply(delta,apply(g,tptp_fun_El_8(e,b,g))),
inference(unit_resolution,[status(thm)],[133,130,128]) ).
tff(135,plain,
apply(h,apply(beta,tptp_fun_El_8(e,b,g))) = zero(r),
inference(transitivity,[status(thm)],[134,96,95]) ).
tff(136,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(beta,b,c)
| ~ ( ( apply(beta,zero(b)) != zero(c) )
| ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(beta,b,c)
| ~ ( ( apply(beta,zero(b)) != zero(c) )
| ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(137,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(beta,b,c)
| ~ ( ( apply(beta,zero(b)) != zero(c) )
| ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(138,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(beta,b,c)
| ~ ( ( apply(beta,zero(b)) != zero(c) )
| ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ) ) ),
inference(modus_ponens,[status(thm)],[137,136]) ).
tff(139,plain,
~ ( ( apply(beta,zero(b)) != zero(c) )
| ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ) ),
inference(unit_resolution,[status(thm)],[138,22,62]) ).
tff(140,plain,
( ( apply(beta,zero(b)) != zero(c) )
| ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) )
| ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ) ),
inference(tautology,[status(thm)],]) ).
tff(141,plain,
! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ),
inference(unit_resolution,[status(thm)],[140,139]) ).
tff(142,plain,
( ( ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| element(apply(beta,tptp_fun_El_8(e,b,g)),c) )
<=> ( ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| element(apply(beta,tptp_fun_El_8(e,b,g)),c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(143,plain,
( ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| element(apply(beta,tptp_fun_El_8(e,b,g)),c) ),
inference(quant_inst,[status(thm)],]) ).
tff(144,plain,
( ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| element(apply(beta,tptp_fun_El_8(e,b,g)),c) ),
inference(modus_ponens,[status(thm)],[143,142]) ).
tff(145,plain,
element(apply(beta,tptp_fun_El_8(e,b,g)),c),
inference(unit_resolution,[status(thm)],[144,141,130]) ).
tff(146,plain,
( injection(h)
<=> injection(h) ),
inference(rewrite,[status(thm)],]) ).
tff(147,axiom,
injection(h),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',h_injection) ).
tff(148,plain,
injection(h),
inference(modus_ponens,[status(thm)],[147,146]) ).
tff(149,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
refl(
( ( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(150,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[149]) ).
tff(151,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
rewrite(
( ( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(152,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[151]) ).
tff(153,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) ),
inference(transitivity,[status(thm)],[152,150]) ).
tff(154,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
<=> ~ ( ~ injection(Morphism)
| ~ morphism(Morphism,Dom,Cod) ) )),
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
<=> ~ ~ ( ~ injection(Morphism)
| ~ morphism(Morphism,Dom,Cod) ) )),
rewrite(
( ~ ~ ( ~ injection(Morphism)
| ~ morphism(Morphism,Dom,Cod) )
<=> ( ~ injection(Morphism)
| ~ morphism(Morphism,Dom,Cod) ) )),
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
<=> ( ~ injection(Morphism)
| ~ morphism(Morphism,Dom,Cod) ) )),
quant_intro(
proof_bind(
^ [El: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
<=> ~ ( ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )),
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
<=> ~ ~ ( ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )),
rewrite(
( ~ ~ ( ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) )
<=> ( ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )),
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
<=> ( ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )),
( ( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) )
<=> ( ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) )
| ( El = zero(Dom) ) ) )),
rewrite(
( ( ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) )
| ( El = zero(Dom) ) )
<=> ( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )),
( ( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) )
<=> ( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ))),
( ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) )
<=> ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )),
( ( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )
<=> ( ~ injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) )),
rewrite(
( ( ~ injection(Morphism)
| ~ morphism(Morphism,Dom,Cod)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) )),
( ( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )
<=> ( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(155,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ) ),
inference(quant_intro,[status(thm)],[154]) ).
tff(156,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(157,plain,
^ [Morphism: $i,Dom: $i,Cod: $i] :
trans(
monotonicity(
quant_intro(
proof_bind(
^ [El: $i] :
rewrite(
( ( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) )
<=> ( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ))),
( ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) )
<=> ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )),
( ( ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
=> ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
<=> ( ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
=> ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) )),
rewrite(
( ( ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
=> ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) )
<=> ( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) )),
( ( ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
=> ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
<=> ( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(158,plain,
( ! [Morphism: $i,Dom: $i,Cod: $i] :
( ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
=> ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) )
<=> ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ) ),
inference(quant_intro,[status(thm)],[157]) ).
tff(159,axiom,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
=> ! [El: $i] :
( ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
=> ( El = zero(Dom) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',injection_properties_2) ).
tff(160,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ),
inference(modus_ponens,[status(thm)],[159,158]) ).
tff(161,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ),
inference(modus_ponens,[status(thm)],[160,156]) ).
tff(162,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ ( injection(Morphism)
& morphism(Morphism,Dom,Cod) )
| ! [El: $i] :
( ~ ( element(El,Dom)
& ( apply(Morphism,El) = zero(Cod) ) )
| ( El = zero(Dom) ) ) ),
inference(skolemize,[status(sab)],[161]) ).
tff(163,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[162,155]) ).
tff(164,plain,
! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) ),
inference(modus_ponens,[status(thm)],[163,153]) ).
tff(165,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(166,plain,
( ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) )
<=> ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(167,plain,
^ [El: $i] :
rewrite(
( ( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) )
<=> ( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) )),
inference(bind,[status(th)],]) ).
tff(168,plain,
( ! [El: $i] :
( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) )
<=> ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ),
inference(quant_intro,[status(thm)],[167]) ).
tff(169,plain,
( ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) ) )
<=> ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ) ),
inference(monotonicity,[status(thm)],[168]) ).
tff(170,plain,
( ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) ) )
<=> ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ) ),
inference(transitivity,[status(thm)],[169,166]) ).
tff(171,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ) ),
inference(monotonicity,[status(thm)],[170]) ).
tff(172,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ) ),
inference(transitivity,[status(thm)],[171,165]) ).
tff(173,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(174,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ) ),
inference(modus_ponens,[status(thm)],[173,172]) ).
tff(175,plain,
! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) ),
inference(unit_resolution,[status(thm)],[174,164,102,148]) ).
tff(176,plain,
( ( ~ ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) )
<=> ( ~ ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(177,plain,
( ~ ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(178,plain,
( ~ ! [El: $i] :
( ~ element(El,c)
| ( apply(h,El) != zero(r) )
| ( El = zero(c) ) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) ),
inference(modus_ponens,[status(thm)],[177,176]) ).
tff(179,plain,
apply(beta,tptp_fun_El_8(e,b,g)) = zero(c),
inference(unit_resolution,[status(thm)],[178,175,145,135]) ).
tff(180,plain,
( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) ),
inference(tautology,[status(thm)],]) ).
tff(181,plain,
( ~ ( ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) != zero(c) )
| ~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ) ),
inference(unit_resolution,[status(thm)],[180,130]) ).
tff(182,plain,
~ ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) ) ),
inference(unit_resolution,[status(thm)],[181,179,86]) ).
tff(183,plain,
( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) )
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = tptp_fun_El_8(e,b,g) ) ),
inference(tautology,[status(thm)],]) ).
tff(184,plain,
apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = tptp_fun_El_8(e,b,g),
inference(unit_resolution,[status(thm)],[183,182]) ).
tff(185,plain,
apply(g,apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = apply(g,tptp_fun_El_8(e,b,g)),
inference(monotonicity,[status(thm)],[184]) ).
tff(186,plain,
( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != tptp_fun_El_8(e,b,g) )
| element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a) ),
inference(tautology,[status(thm)],]) ).
tff(187,plain,
element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a),
inference(unit_resolution,[status(thm)],[186,182]) ).
tff(188,plain,
( commute(alpha,g,f,gamma)
<=> commute(alpha,g,f,gamma) ),
inference(rewrite,[status(thm)],]) ).
tff(189,axiom,
commute(alpha,g,f,gamma),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alpha_g_f_gamma_commute) ).
tff(190,plain,
commute(alpha,g,f,gamma),
inference(modus_ponens,[status(thm)],[189,188]) ).
tff(191,plain,
( morphism(f,a,d)
<=> morphism(f,a,d) ),
inference(rewrite,[status(thm)],]) ).
tff(192,axiom,
morphism(f,a,d),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',f_morphism) ).
tff(193,plain,
morphism(f,a,d),
inference(modus_ponens,[status(thm)],[192,191]) ).
tff(194,plain,
( morphism(gamma,d,e)
<=> morphism(gamma,d,e) ),
inference(rewrite,[status(thm)],]) ).
tff(195,axiom,
morphism(gamma,d,e),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gamma_morphism) ).
tff(196,plain,
morphism(gamma,d,e),
inference(modus_ponens,[status(thm)],[195,194]) ).
tff(197,plain,
( ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(gamma,d,e)
| ~ morphism(f,a,d)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) )
<=> ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(gamma,d,e)
| ~ morphism(f,a,d)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(198,plain,
( ( ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(f,a,d)
| ~ morphism(gamma,d,e)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) )
<=> ( ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(gamma,d,e)
| ~ morphism(f,a,d)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(199,plain,
( ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(f,a,d)
| ~ morphism(gamma,d,e)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) )
<=> ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(gamma,d,e)
| ~ morphism(f,a,d)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) ) ),
inference(monotonicity,[status(thm)],[198]) ).
tff(200,plain,
( ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(f,a,d)
| ~ morphism(gamma,d,e)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) )
<=> ( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(gamma,d,e)
| ~ morphism(f,a,d)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) ) ),
inference(transitivity,[status(thm)],[199,197]) ).
tff(201,plain,
( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(f,a,d)
| ~ morphism(gamma,d,e)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(202,plain,
( ~ ! [M1: $i,M2: $i,M3: $i,M4: $i,Dom: $i,DomCod1: $i,DomCod2: $i,Cod: $i] :
( ~ morphism(M1,Dom,DomCod1)
| ~ morphism(M2,DomCod1,Cod)
| ~ morphism(M3,Dom,DomCod2)
| ~ morphism(M4,DomCod2,Cod)
| ~ commute(M1,M2,M3,M4)
| ! [ElDom: $i] :
( ~ element(ElDom,Dom)
| ( apply(M2,apply(M1,ElDom)) = apply(M4,apply(M3,ElDom)) ) ) )
| ~ morphism(alpha,a,b)
| ~ morphism(g,b,e)
| ~ morphism(gamma,d,e)
| ~ morphism(f,a,d)
| ~ commute(alpha,g,f,gamma)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ) ),
inference(modus_ponens,[status(thm)],[201,200]) ).
tff(203,plain,
! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ),
inference(unit_resolution,[status(thm)],[202,121,3,196,193,34,190]) ).
tff(204,plain,
( ( ~ ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(g,apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) ) )
<=> ( ~ ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(g,apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(205,plain,
( ~ ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(g,apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(206,plain,
( ~ ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(g,apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) ) ),
inference(modus_ponens,[status(thm)],[205,204]) ).
tff(207,plain,
apply(g,apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))),
inference(unit_resolution,[status(thm)],[206,203,187]) ).
tff(208,plain,
apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = apply(g,apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))),
inference(symmetry,[status(thm)],[207]) ).
tff(209,plain,
apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = zero(e),
inference(transitivity,[status(thm)],[208,185,56]) ).
tff(210,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(f,a,d)
| ~ ( ( apply(f,zero(a)) != zero(d) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(f,a,d)
| ~ ( ( apply(f,zero(a)) != zero(d) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(211,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(f,a,d)
| ~ ( ( apply(f,zero(a)) != zero(d) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(212,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ ( ( apply(Morphism,zero(Dom)) != zero(Cod) )
| ~ ! [El: $i] :
( ~ element(El,Dom)
| element(apply(Morphism,El),Cod) ) ) )
| ~ morphism(f,a,d)
| ~ ( ( apply(f,zero(a)) != zero(d) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ) ) ),
inference(modus_ponens,[status(thm)],[211,210]) ).
tff(213,plain,
~ ( ( apply(f,zero(a)) != zero(d) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ) ),
inference(unit_resolution,[status(thm)],[212,22,193]) ).
tff(214,plain,
( ( apply(f,zero(a)) != zero(d) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) )
| ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ) ),
inference(tautology,[status(thm)],]) ).
tff(215,plain,
! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ),
inference(unit_resolution,[status(thm)],[214,213]) ).
tff(216,plain,
( ( ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d) )
<=> ( ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(217,plain,
( ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d) ),
inference(quant_inst,[status(thm)],]) ).
tff(218,plain,
( ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d) ),
inference(modus_ponens,[status(thm)],[217,216]) ).
tff(219,plain,
element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d),
inference(unit_resolution,[status(thm)],[218,215,187]) ).
tff(220,plain,
( injection(gamma)
<=> injection(gamma) ),
inference(rewrite,[status(thm)],]) ).
tff(221,axiom,
injection(gamma),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',gamma_injection) ).
tff(222,plain,
injection(gamma),
inference(modus_ponens,[status(thm)],[221,220]) ).
tff(223,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(gamma,d,e)
| ~ injection(gamma)
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(gamma,d,e)
| ~ injection(gamma)
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(224,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(gamma,d,e)
| ~ injection(gamma)
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(225,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(gamma,d,e)
| ~ injection(gamma)
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) ) ),
inference(modus_ponens,[status(thm)],[224,223]) ).
tff(226,plain,
! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) ),
inference(unit_resolution,[status(thm)],[225,164,196,222]) ).
tff(227,plain,
( ( ~ ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = zero(d) )
| ~ element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d)
| ( apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) != zero(e) ) )
<=> ( ~ ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = zero(d) )
| ~ element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d)
| ( apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) != zero(e) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(228,plain,
( ~ ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = zero(d) )
| ~ element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d)
| ( apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) != zero(e) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(229,plain,
( ~ ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = zero(d) )
| ~ element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d)
| ( apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) != zero(e) ) ),
inference(modus_ponens,[status(thm)],[228,227]) ).
tff(230,plain,
apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = zero(d),
inference(unit_resolution,[status(thm)],[229,226,219,209]) ).
tff(231,plain,
( injection(f)
<=> injection(f) ),
inference(rewrite,[status(thm)],]) ).
tff(232,axiom,
injection(f),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',f_injection) ).
tff(233,plain,
injection(f),
inference(modus_ponens,[status(thm)],[232,231]) ).
tff(234,plain,
( ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(f,a,d)
| ~ injection(f)
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) ) )
<=> ( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(f,a,d)
| ~ injection(f)
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(235,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(f,a,d)
| ~ injection(f)
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(236,plain,
( ~ ! [Morphism: $i,Dom: $i,Cod: $i] :
( ~ morphism(Morphism,Dom,Cod)
| ~ injection(Morphism)
| ! [El: $i] :
( ( El = zero(Dom) )
| ~ element(El,Dom)
| ( apply(Morphism,El) != zero(Cod) ) ) )
| ~ morphism(f,a,d)
| ~ injection(f)
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) ) ),
inference(modus_ponens,[status(thm)],[235,234]) ).
tff(237,plain,
! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) ),
inference(unit_resolution,[status(thm)],[236,164,193,233]) ).
tff(238,plain,
( ( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) )
<=> ( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(239,plain,
( ( ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) )
<=> ( ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(240,plain,
( ( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) )
<=> ( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) ) ),
inference(monotonicity,[status(thm)],[239]) ).
tff(241,plain,
( ( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) )
<=> ( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) ) ),
inference(transitivity,[status(thm)],[240,238]) ).
tff(242,plain,
( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(243,plain,
( ~ ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a)
| ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) ),
inference(modus_ponens,[status(thm)],[242,241]) ).
tff(244,plain,
( ( tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a) )
| ( apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) != zero(d) ) ),
inference(unit_resolution,[status(thm)],[243,237,187]) ).
tff(245,plain,
tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a),
inference(unit_resolution,[status(thm)],[244,230]) ).
tff(246,plain,
apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = apply(alpha,zero(a)),
inference(monotonicity,[status(thm)],[245]) ).
tff(247,plain,
tptp_fun_El_8(e,b,g) = apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),
inference(symmetry,[status(thm)],[184]) ).
tff(248,plain,
tptp_fun_El_8(e,b,g) = zero(b),
inference(transitivity,[status(thm)],[247,246,28]) ).
tff(249,plain,
( ( tptp_fun_El_8(e,b,g) = zero(b) )
| ~ element(tptp_fun_El_8(e,b,g),b)
| ( apply(g,tptp_fun_El_8(e,b,g)) != zero(e) )
| ( tptp_fun_El_8(e,b,g) != zero(b) ) ),
inference(tautology,[status(thm)],]) ).
tff(250,plain,
tptp_fun_El_8(e,b,g) != zero(b),
inference(unit_resolution,[status(thm)],[249,54]) ).
tff(251,plain,
$false,
inference(unit_resolution,[status(thm)],[250,248]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : HAL001+2 : TPTP v8.1.0. Released v2.6.0.
% 0.11/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n021.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 31 21:31:12 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.19/0.43 % SZS status Theorem
% 0.19/0.43 % SZS output start Proof
% See solution above
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