%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : HAL001+2 : TPTP v9.0.0. Released v2.6.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:08:43 AM UTC 2025
% Result : Theorem 0.24s 0.45s
% Output : Proof 0.24s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 118
% Syntax : Number of formulae : 267 ( 75 unt; 0 typ; 0 def)
% Number of atoms : 2883 ( 910 equ)
% Maximal formula atoms : 44 ( 10 avg)
% Number of connectives : 4845 (2383 ~;1871 |; 346 &)
% ( 193 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 154 ( 154 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 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 : 1280 (1162 !; 13 ?;1280 :)
% 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/sandbox/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/sandbox/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/sandbox/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/sandbox/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/sandbox/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/sandbox/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/sandbox/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/sandbox/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)
| ~ exact(alpha,beta)
| ~ morphism(beta,b,c)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) )
<=> ( ~ ! [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)
| ~ exact(alpha,beta)
| ~ morphism(beta,b,c)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(80,plain,
( ( ~ morphism(alpha,a,b)
| ~ morphism(beta,b,c)
| ~ exact(alpha,beta)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) )
<=> ( ~ morphism(alpha,a,b)
| ~ exact(alpha,beta)
| ~ morphism(beta,b,c)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(81,plain,
^ [ElCodDom: $i] :
rewrite(( ~ ( ~ ( ~ 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(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) )),
inference(bind,[status(th)],[]) ).
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 ) ) ) )
<=> ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ),
inference(quant_intro,[status(thm)],[81]) ).
tff(83,plain,
( ( ~ 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 ) ) ) ) )
<=> ( ~ morphism(alpha,a,b)
| ~ morphism(beta,b,c)
| ~ exact(alpha,beta)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ) ),
inference(monotonicity,[status(thm)],[82]) ).
tff(84,plain,
( ( ~ 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 ) ) ) ) )
<=> ( ~ morphism(alpha,a,b)
| ~ exact(alpha,beta)
| ~ morphism(beta,b,c)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ) ),
inference(transitivity,[status(thm)],[83,80]) ).
tff(85,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)
| ~ exact(alpha,beta)
| ~ morphism(beta,b,c)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ) ),
inference(monotonicity,[status(thm)],[84]) ).
tff(86,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)
| ~ exact(alpha,beta)
| ~ morphism(beta,b,c)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ) ),
inference(transitivity,[status(thm)],[85,79]) ).
tff(87,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(88,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)
| ~ exact(alpha,beta)
| ~ morphism(beta,b,c)
| ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ) ),
inference(modus_ponens,[status(thm)],[87,86]) ).
tff(89,plain,
! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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 ) ) ) ),
inference(unit_resolution,[status(thm)],[88,78,3,62,59]) ).
tff(90,plain,
( ~ ! [ElCodDom: $i] :
~ ( ~ ( ~ ( ~ element(ElCodDom,b)
| ( apply(beta,ElCodDom) != zero(c) ) )
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(alpha,ElDom) != ElCodDom ) ) )
| ~ ( ~ 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(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(quant_inst,[status(thm)],[]) ).
tff(91,plain,
~ ( ~ ( ~ ( ~ 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(unit_resolution,[status(thm)],[90,89]) ).
tff(92,plain,
( ~ ( ~ ( ~ 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) ) ) )
| ~ 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(93,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)],[92,91]) ).
tff(94,plain,
( morphism(delta,e,r)
<=> morphism(delta,e,r) ),
inference(rewrite,[status(thm)],[]) ).
tff(95,axiom,
morphism(delta,e,r),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',delta_morphism) ).
tff(96,plain,
morphism(delta,e,r),
inference(modus_ponens,[status(thm)],[95,94]) ).
tff(97,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(98,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(99,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)],[98,97]) ).
tff(100,plain,
~ ( ( apply(delta,zero(e)) != zero(r) )
| ~ ! [El: $i] :
( ~ element(El,e)
| element(apply(delta,El),r) ) ),
inference(unit_resolution,[status(thm)],[99,22,96]) ).
tff(101,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(102,plain,
apply(delta,zero(e)) = zero(r),
inference(unit_resolution,[status(thm)],[101,100]) ).
tff(103,plain,
apply(delta,apply(g,tptp_fun_El_8(e,b,g))) = apply(delta,zero(e)),
inference(monotonicity,[status(thm)],[56]) ).
tff(104,plain,
( commute(beta,h,g,delta)
<=> commute(beta,h,g,delta) ),
inference(rewrite,[status(thm)],[]) ).
tff(105,axiom,
commute(beta,h,g,delta),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',beta_h_g_delta_commute) ).
tff(106,plain,
commute(beta,h,g,delta),
inference(modus_ponens,[status(thm)],[105,104]) ).
tff(107,plain,
( morphism(h,c,r)
<=> morphism(h,c,r) ),
inference(rewrite,[status(thm)],[]) ).
tff(108,axiom,
morphism(h,c,r),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',h_morphism) ).
tff(109,plain,
morphism(h,c,r),
inference(modus_ponens,[status(thm)],[108,107]) ).
tff(110,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(111,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)],[110]) ).
tff(112,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(113,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)],[112]) ).
tff(114,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(115,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)],[114]) ).
tff(116,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)],[115,113]) ).
tff(117,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(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)) ) ) )
<=> ! [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)],[117]) ).
tff(119,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(120,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(121,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)],[120]) ).
tff(122,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/sandbox/benchmark/Axioms/HAL001+0.ax',commute_properties) ).
tff(123,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)],[122,121]) ).
tff(124,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)],[123,119]) ).
tff(125,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)],[124]) ).
tff(126,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)],[125,118]) ).
tff(127,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)],[126,116]) ).
tff(128,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)],[127,111]) ).
tff(129,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)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ commute(beta,h,g,delta) )
<=> ( ~ ! [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)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ commute(beta,h,g,delta) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(130,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)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ commute(beta,h,g,delta) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(131,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)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ commute(beta,h,g,delta) ) ),
inference(monotonicity,[status(thm)],[130]) ).
tff(132,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)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ commute(beta,h,g,delta) ) ),
inference(transitivity,[status(thm)],[131,129]) ).
tff(133,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(134,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)
| ! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) )
| ~ commute(beta,h,g,delta) ),
inference(modus_ponens,[status(thm)],[133,132]) ).
tff(135,plain,
! [ElDom: $i] :
( ~ element(ElDom,b)
| ( apply(h,apply(beta,ElDom)) = apply(delta,apply(g,ElDom)) ) ),
inference(unit_resolution,[status(thm)],[134,128,62,96,34,109,106]) ).
tff(136,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(137,plain,
element(tptp_fun_El_8(e,b,g),b),
inference(unit_resolution,[status(thm)],[136,54]) ).
tff(138,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(139,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(140,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)],[139,138]) ).
tff(141,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)],[140,137,135]) ).
tff(142,plain,
apply(h,apply(beta,tptp_fun_El_8(e,b,g))) = zero(r),
inference(transitivity,[status(thm)],[141,103,102]) ).
tff(143,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(144,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(145,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)],[144,143]) ).
tff(146,plain,
~ ( ( apply(beta,zero(b)) != zero(c) )
| ~ ! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ) ),
inference(unit_resolution,[status(thm)],[145,22,62]) ).
tff(147,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(148,plain,
! [El: $i] :
( ~ element(El,b)
| element(apply(beta,El),c) ),
inference(unit_resolution,[status(thm)],[147,146]) ).
tff(149,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(150,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(151,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)],[150,149]) ).
tff(152,plain,
element(apply(beta,tptp_fun_El_8(e,b,g)),c),
inference(unit_resolution,[status(thm)],[151,148,137]) ).
tff(153,plain,
( injection(h)
<=> injection(h) ),
inference(rewrite,[status(thm)],[]) ).
tff(154,axiom,
injection(h),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',h_injection) ).
tff(155,plain,
injection(h),
inference(modus_ponens,[status(thm)],[154,153]) ).
tff(156,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(157,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)],[156]) ).
tff(158,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(159,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)],[158]) ).
tff(160,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)],[159,157]) ).
tff(161,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(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) ) ) )
<=> ! [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)],[161]) ).
tff(163,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(164,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(165,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)],[164]) ).
tff(166,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/sandbox/benchmark/theBenchmark.p',injection_properties_2) ).
tff(167,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)],[166,165]) ).
tff(168,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)],[167,163]) ).
tff(169,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)],[168]) ).
tff(170,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)],[169,162]) ).
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) ) ) ),
inference(modus_ponens,[status(thm)],[170,160]) ).
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] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( 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] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(173,plain,
( ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) )
<=> ( ~ morphism(h,c,r)
| ~ injection(h)
| ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(174,plain,
^ [El: $i] :
rewrite(( ( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) )
<=> ( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) )),
inference(bind,[status(th)],[]) ).
tff(175,plain,
( ! [El: $i] :
( ( El = zero(c) )
| ~ element(El,c)
| ( apply(h,El) != zero(r) ) )
<=> ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ),
inference(quant_intro,[status(thm)],[174]) ).
tff(176,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] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ) ),
inference(monotonicity,[status(thm)],[175]) ).
tff(177,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] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ) ),
inference(transitivity,[status(thm)],[176,173]) ).
tff(178,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] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ) ),
inference(monotonicity,[status(thm)],[177]) ).
tff(179,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] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ) ),
inference(transitivity,[status(thm)],[178,172]) ).
tff(180,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(181,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] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ) ),
inference(modus_ponens,[status(thm)],[180,179]) ).
tff(182,plain,
! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) ),
inference(unit_resolution,[status(thm)],[181,171,109,155]) ).
tff(183,plain,
( ( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = 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) ) )
<=> ( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = 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) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(184,plain,
( ( ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) )
<=> ( ( apply(beta,tptp_fun_El_8(e,b,g)) = 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) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(185,plain,
( ( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) )
<=> ( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = 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) ) ) ),
inference(monotonicity,[status(thm)],[184]) ).
tff(186,plain,
( ( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) )
<=> ( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = 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) ) ) ),
inference(transitivity,[status(thm)],[185,183]) ).
tff(187,plain,
( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(h,apply(beta,tptp_fun_El_8(e,b,g))) != zero(r) )
| ~ element(apply(beta,tptp_fun_El_8(e,b,g)),c)
| ( apply(beta,tptp_fun_El_8(e,b,g)) = zero(c) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(188,plain,
( ~ ! [El: $i] :
( ( apply(h,El) != zero(r) )
| ~ element(El,c)
| ( El = zero(c) ) )
| ( apply(beta,tptp_fun_El_8(e,b,g)) = 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) ) ),
inference(modus_ponens,[status(thm)],[187,186]) ).
tff(189,plain,
apply(beta,tptp_fun_El_8(e,b,g)) = zero(c),
inference(unit_resolution,[status(thm)],[188,182,152,142]) ).
tff(190,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(191,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)],[190,137]) ).
tff(192,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)],[191,189,93]) ).
tff(193,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(194,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)],[193,192]) ).
tff(195,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)],[194]) ).
tff(196,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(197,plain,
element(tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha),a),
inference(unit_resolution,[status(thm)],[196,192]) ).
tff(198,plain,
( commute(alpha,g,f,gamma)
<=> commute(alpha,g,f,gamma) ),
inference(rewrite,[status(thm)],[]) ).
tff(199,axiom,
commute(alpha,g,f,gamma),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',alpha_g_f_gamma_commute) ).
tff(200,plain,
commute(alpha,g,f,gamma),
inference(modus_ponens,[status(thm)],[199,198]) ).
tff(201,plain,
( morphism(f,a,d)
<=> morphism(f,a,d) ),
inference(rewrite,[status(thm)],[]) ).
tff(202,axiom,
morphism(f,a,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',f_morphism) ).
tff(203,plain,
morphism(f,a,d),
inference(modus_ponens,[status(thm)],[202,201]) ).
tff(204,plain,
( morphism(gamma,d,e)
<=> morphism(gamma,d,e) ),
inference(rewrite,[status(thm)],[]) ).
tff(205,axiom,
morphism(gamma,d,e),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',gamma_morphism) ).
tff(206,plain,
morphism(gamma,d,e),
inference(modus_ponens,[status(thm)],[205,204]) ).
tff(207,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)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ commute(alpha,g,f,gamma) )
<=> ( ~ ! [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)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ commute(alpha,g,f,gamma) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(208,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)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ commute(alpha,g,f,gamma) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(209,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)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ commute(alpha,g,f,gamma) ) ),
inference(monotonicity,[status(thm)],[208]) ).
tff(210,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)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ commute(alpha,g,f,gamma) ) ),
inference(transitivity,[status(thm)],[209,207]) ).
tff(211,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(212,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)
| ! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) )
| ~ commute(alpha,g,f,gamma) ),
inference(modus_ponens,[status(thm)],[211,210]) ).
tff(213,plain,
! [ElDom: $i] :
( ~ element(ElDom,a)
| ( apply(g,apply(alpha,ElDom)) = apply(gamma,apply(f,ElDom)) ) ),
inference(unit_resolution,[status(thm)],[212,128,3,206,203,34,200]) ).
tff(214,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(215,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(216,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)],[215,214]) ).
tff(217,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)],[216,213,197]) ).
tff(218,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)],[217]) ).
tff(219,plain,
apply(gamma,apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha))) = zero(e),
inference(transitivity,[status(thm)],[218,195,56]) ).
tff(220,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(221,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(222,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)],[221,220]) ).
tff(223,plain,
~ ( ( apply(f,zero(a)) != zero(d) )
| ~ ! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ) ),
inference(unit_resolution,[status(thm)],[222,22,203]) ).
tff(224,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(225,plain,
! [El: $i] :
( ~ element(El,a)
| element(apply(f,El),d) ),
inference(unit_resolution,[status(thm)],[224,223]) ).
tff(226,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(227,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(228,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)],[227,226]) ).
tff(229,plain,
element(apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)),d),
inference(unit_resolution,[status(thm)],[228,225,197]) ).
tff(230,plain,
( injection(gamma)
<=> injection(gamma) ),
inference(rewrite,[status(thm)],[]) ).
tff(231,axiom,
injection(gamma),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',gamma_injection) ).
tff(232,plain,
injection(gamma),
inference(modus_ponens,[status(thm)],[231,230]) ).
tff(233,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) ) ) )
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ~ injection(gamma)
| ~ morphism(gamma,d,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) ) ) )
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ~ injection(gamma)
| ~ morphism(gamma,d,e) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(234,plain,
( ( ~ morphism(gamma,d,e)
| ~ injection(gamma)
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) ) )
<=> ( ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ~ injection(gamma)
| ~ morphism(gamma,d,e) ) ),
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(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) ) ) )
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ~ injection(gamma)
| ~ morphism(gamma,d,e) ) ),
inference(monotonicity,[status(thm)],[234]) ).
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(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) ) ) )
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ~ injection(gamma)
| ~ morphism(gamma,d,e) ) ),
inference(transitivity,[status(thm)],[235,233]) ).
tff(237,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(238,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) ) ) )
| ! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) )
| ~ injection(gamma)
| ~ morphism(gamma,d,e) ),
inference(modus_ponens,[status(thm)],[237,236]) ).
tff(239,plain,
! [El: $i] :
( ( El = zero(d) )
| ~ element(El,d)
| ( apply(gamma,El) != zero(e) ) ),
inference(unit_resolution,[status(thm)],[238,171,206,232]) ).
tff(240,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(241,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(242,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)],[241,240]) ).
tff(243,plain,
apply(f,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = zero(d),
inference(unit_resolution,[status(thm)],[242,239,229,219]) ).
tff(244,plain,
( injection(f)
<=> injection(f) ),
inference(rewrite,[status(thm)],[]) ).
tff(245,axiom,
injection(f),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',f_injection) ).
tff(246,plain,
injection(f),
inference(modus_ponens,[status(thm)],[245,244]) ).
tff(247,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) ) ) )
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ injection(f)
| ~ morphism(f,a,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) ) ) )
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ injection(f)
| ~ morphism(f,a,d) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(248,plain,
( ( ~ morphism(f,a,d)
| ~ injection(f)
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) ) )
<=> ( ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ injection(f)
| ~ morphism(f,a,d) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(249,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) ) ) )
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ injection(f)
| ~ morphism(f,a,d) ) ),
inference(monotonicity,[status(thm)],[248]) ).
tff(250,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) ) ) )
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ injection(f)
| ~ morphism(f,a,d) ) ),
inference(transitivity,[status(thm)],[249,247]) ).
tff(251,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(252,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) ) ) )
| ! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) )
| ~ injection(f)
| ~ morphism(f,a,d) ),
inference(modus_ponens,[status(thm)],[251,250]) ).
tff(253,plain,
! [El: $i] :
( ( El = zero(a) )
| ~ element(El,a)
| ( apply(f,El) != zero(d) ) ),
inference(unit_resolution,[status(thm)],[252,171,203,246]) ).
tff(254,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(255,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(256,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)],[255]) ).
tff(257,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)],[256,254]) ).
tff(258,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(259,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)],[258,257]) ).
tff(260,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)],[259,253,197]) ).
tff(261,plain,
tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha) = zero(a),
inference(unit_resolution,[status(thm)],[260,243]) ).
tff(262,plain,
apply(alpha,tptp_fun_ElDom_4(tptp_fun_El_8(e,b,g),a,alpha)) = apply(alpha,zero(a)),
inference(monotonicity,[status(thm)],[261]) ).
tff(263,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)],[194]) ).
tff(264,plain,
tptp_fun_El_8(e,b,g) = zero(b),
inference(transitivity,[status(thm)],[263,262,28]) ).
tff(265,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(266,plain,
tptp_fun_El_8(e,b,g) != zero(b),
inference(unit_resolution,[status(thm)],[265,54]) ).
tff(267,plain,
$false,
inference(unit_resolution,[status(thm)],[266,264]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.15 % Problem : HAL001+2 : TPTP v9.0.0. Released v2.6.0.
% 0.08/0.15 % Command : run_E %s %d THM
% 0.15/0.36 % Computer : n013.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Thu Jun 19 17:10:23 EDT 2025
% 0.15/0.37 % CPUTime :
% 0.24/0.45 % SZS status Theorem
% 0.24/0.45 % SZS output start Proof
% See solution above
% 0.24/0.53 % E exiting
%------------------------------------------------------------------------------