%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : ANA039-1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 04:59:16 AM UTC 2025
% Result : Unsatisfiable 0.44s 0.60s
% Output : Proof 0.48s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 74
% Syntax : Number of formulae : 170 ( 41 unt; 0 typ; 0 def)
% Number of atoms : 774 ( 0 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 1088 ( 511 ~; 501 |; 0 &)
% ( 76 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 27 ( 27 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 12 ( 11 usr; 1 prp; 0-3 aty)
% Number of functors : 10 ( 10 usr; 4 con; 0-3 aty)
% Number of variables : 447 ( 400 !; 0 ?; 447 :)
% Comments :
%------------------------------------------------------------------------------
tff(c_lessequals_type,type,
c_lessequals: ( $i * $i * $i ) > $o ).
tff(t_a_type,type,
t_a: $i ).
tff(c_times_type,type,
c_times: ( $i * $i * $i ) > $i ).
tff(c_HOL_Oabs_type,type,
c_HOL_Oabs: ( $i * $i ) > $i ).
tff(v_f_type,type,
v_f: $i > $i ).
tff(v_x_type,type,
v_x: $i ).
tff(v_c_type,type,
v_c: $i ).
tff(tc_set_type,type,
tc_set: $i > $i ).
tff(c_SetInterval_OatMost_type,type,
c_SetInterval_OatMost: ( $i * $i ) > $i ).
tff(c_in_type,type,
c_in: ( $i * $i * $i ) > $o ).
tff(v_h_type,type,
v_h: $i > $i ).
tff(class_Orderings_Oord_type,type,
class_Orderings_Oord: $i > $o ).
tff(class_Ring__and__Field_Oordered__idom_type,type,
class_Ring__and__Field_Oordered__idom: $i > $o ).
tff(class_Orderings_Oorder_type,type,
class_Orderings_Oorder: $i > $o ).
tff(c_0_type,type,
c_0: $i ).
tff(class_OrderedGroup_Olordered__ab__group__abs_type,type,
class_OrderedGroup_Olordered__ab__group__abs: $i > $o ).
tff(class_Ring__and__Field_Opordered__semiring_type,type,
class_Ring__and__Field_Opordered__semiring: $i > $o ).
tff(1,plain,
^ [V_U: $i] :
refl(( c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a)
<=> c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a)
<=> ! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a)
<=> ! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conjecture_0) ).
tff(5,plain,
! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [V_U: $i] : c_lessequals(c_HOL_Oabs(v_h(V_U),t_a),c_times(v_c,c_HOL_Oabs(v_f(V_U),t_a),t_a),t_a)
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( class_Ring__and__Field_Oordered__idom(t_a)
<=> class_Ring__and__Field_Oordered__idom(t_a) ),
inference(rewrite,[status(thm)],[]) ).
tff(11,axiom,
class_Ring__and__Field_Oordered__idom(t_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_tcs) ).
tff(12,plain,
class_Ring__and__Field_Oordered__idom(t_a),
inference(modus_ponens,[status(thm)],[11,10]) ).
tff(13,plain,
^ [T: $i] :
refl(( ( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(14,plain,
( ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,plain,
( ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(16,plain,
^ [T: $i] :
rewrite(( ( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Orderings_Oord(T) )
<=> ( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(17,plain,
( ! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Orderings_Oord(T) )
<=> ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[16]) ).
tff(18,axiom,
! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Orderings_Oord(T) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-0.ax',clsrel_Ring__and__Field_Oordered__idom_45) ).
tff(19,plain,
! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[19,15]) ).
tff(21,plain,
! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(skolemize,[status(sab)],[20]) ).
tff(22,plain,
! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[21,14]) ).
tff(23,plain,
( ( ~ ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oord(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) )
<=> ( ~ ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oord(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,plain,
( ~ ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oord(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(25,plain,
( ~ ! [T: $i] :
( class_Orderings_Oord(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oord(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
class_Orderings_Oord(t_a),
inference(unit_resolution,[status(thm)],[25,22,12]) ).
tff(27,plain,
^ [V_i: $i,T_a: $i,V_k: $i] :
refl(( ( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) )
<=> ( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(28,plain,
( ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) )
<=> ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ) ),
inference(quant_intro,[status(thm)],[27]) ).
tff(29,plain,
( ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) )
<=> ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,plain,
^ [V_i: $i,T_a: $i,V_k: $i] :
rewrite(( ( ~ class_Orderings_Oord(T_a)
| ~ c_lessequals(V_i,V_k,T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
<=> ( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| ~ c_lessequals(V_i,V_k,T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
<=> ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,axiom,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| ~ c_lessequals(V_i,V_k,T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-2.ax',cls_SetInterval_OatMost__iff__iff2_0) ).
tff(33,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ),
inference(modus_ponens,[status(thm)],[33,29]) ).
tff(35,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) ),
inference(modus_ponens,[status(thm)],[35,28]) ).
tff(37,plain,
( ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) )
| ~ class_Orderings_Oord(t_a)
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a) )
<=> ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) )
| ~ class_Orderings_Oord(t_a)
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(38,plain,
( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) )
| ~ class_Orderings_Oord(t_a)
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| ~ c_lessequals(V_i,V_k,T_a) )
| ~ class_Orderings_Oord(t_a)
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a) ),
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a),
inference(unit_resolution,[status(thm)],[39,36,26,9]) ).
tff(41,plain,
( ~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
<=> ~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conjecture_2) ).
tff(43,plain,
~ c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
^ [V_i: $i,T_a: $i,V_k: $i] :
refl(( ( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
<=> ( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(45,plain,
( ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
<=> ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ) ),
inference(quant_intro,[status(thm)],[44]) ).
tff(46,plain,
( ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
<=> ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(47,plain,
^ [V_i: $i,T_a: $i,V_k: $i] :
rewrite(( ( ~ class_Orderings_Oord(T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| c_lessequals(V_i,V_k,T_a) )
<=> ( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(48,plain,
( ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| c_lessequals(V_i,V_k,T_a) )
<=> ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ) ),
inference(quant_intro,[status(thm)],[47]) ).
tff(49,axiom,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a)
| c_lessequals(V_i,V_k,T_a) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-2.ax',cls_SetInterval_OatMost__iff__iff1_0) ).
tff(50,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ),
inference(modus_ponens,[status(thm)],[50,46]) ).
tff(52,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ),
inference(skolemize,[status(sab)],[51]) ).
tff(53,plain,
! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) ),
inference(modus_ponens,[status(thm)],[52,45]) ).
tff(54,plain,
( ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Orderings_Oord(t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) )
<=> ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Orderings_Oord(t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(55,plain,
( ( ~ class_Orderings_Oord(t_a)
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) )
<=> ( c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Orderings_Oord(t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(56,plain,
( ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| ~ class_Orderings_Oord(t_a)
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) )
<=> ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Orderings_Oord(t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) ) ),
inference(monotonicity,[status(thm)],[55]) ).
tff(57,plain,
( ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| ~ class_Orderings_Oord(t_a)
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) )
<=> ( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Orderings_Oord(t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) ) ),
inference(transitivity,[status(thm)],[56,54]) ).
tff(58,plain,
( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| ~ class_Orderings_Oord(t_a)
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(59,plain,
( ~ ! [V_i: $i,T_a: $i,V_k: $i] :
( ~ class_Orderings_Oord(T_a)
| c_lessequals(V_i,V_k,T_a)
| ~ c_in(V_i,c_SetInterval_OatMost(V_k,T_a),T_a) )
| c_lessequals(c_HOL_Oabs(v_h(v_x),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Orderings_Oord(t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) ),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
( ~ class_Orderings_Oord(t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a) ),
inference(unit_resolution,[status(thm)],[59,53,43]) ).
tff(61,plain,
~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a),
inference(unit_resolution,[status(thm)],[60,26]) ).
tff(62,plain,
^ [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
refl(( ( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) )
<=> ( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ) )),
inference(bind,[status(th)],[]) ).
tff(63,plain,
( ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) )
<=> ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,plain,
( ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) )
<=> ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(65,plain,
^ [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
rewrite(( ( ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a))
| c_in(V_c,V_B,T_a) )
<=> ( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ) )),
inference(bind,[status(th)],[]) ).
tff(66,plain,
( ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a))
| c_in(V_c,V_B,T_a) )
<=> ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ) ),
inference(quant_intro,[status(thm)],[65]) ).
tff(67,axiom,
! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a))
| c_in(V_c,V_B,T_a) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-0.ax',cls_Set_OsubsetD_0) ).
tff(68,plain,
! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ),
inference(modus_ponens,[status(thm)],[67,66]) ).
tff(69,plain,
! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ),
inference(modus_ponens,[status(thm)],[68,64]) ).
tff(70,plain,
! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ),
inference(skolemize,[status(sab)],[69]) ).
tff(71,plain,
! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) ),
inference(modus_ponens,[status(thm)],[70,63]) ).
tff(72,plain,
( ( ~ ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) )
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) )
<=> ( ~ ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) )
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(73,plain,
( ~ ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) )
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(74,plain,
( ~ ! [V_c: $i,T_a: $i,V_B: $i,V_A: $i] :
( c_in(V_c,V_B,T_a)
| ~ c_in(V_c,V_A,T_a)
| ~ c_lessequals(V_A,V_B,tc_set(T_a)) )
| c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_in(c_HOL_Oabs(v_h(v_x),t_a),c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),t_a)
| ~ c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) ),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
~ c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)),
inference(unit_resolution,[status(thm)],[74,71,61,40]) ).
tff(76,plain,
^ [T: $i] :
refl(( ( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(77,plain,
( ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[76]) ).
tff(78,plain,
( ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(79,plain,
^ [T: $i] :
rewrite(( ( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Orderings_Oorder(T) )
<=> ( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(80,plain,
( ! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Orderings_Oorder(T) )
<=> ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[79]) ).
tff(81,axiom,
! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Orderings_Oorder(T) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-0.ax',clsrel_Ring__and__Field_Oordered__idom_44) ).
tff(82,plain,
! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[82,78]) ).
tff(84,plain,
! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(skolemize,[status(sab)],[83]) ).
tff(85,plain,
! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[84,77]) ).
tff(86,plain,
( ( ~ ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oorder(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) )
<=> ( ~ ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oorder(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,plain,
( ~ ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oorder(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(88,plain,
( ~ ! [T: $i] :
( class_Orderings_Oorder(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Orderings_Oorder(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(modus_ponens,[status(thm)],[87,86]) ).
tff(89,plain,
class_Orderings_Oorder(t_a),
inference(unit_resolution,[status(thm)],[88,85,12]) ).
tff(90,plain,
^ [V_x: $i,T_a: $i,V_y: $i] :
refl(( ( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
<=> ( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ) )),
inference(bind,[status(th)],[]) ).
tff(91,plain,
( ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
<=> ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ) ),
inference(quant_intro,[status(thm)],[90]) ).
tff(92,plain,
( ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
<=> ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(93,plain,
^ [V_x: $i,T_a: $i,V_y: $i] :
rewrite(( ( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
<=> ( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ) )),
inference(bind,[status(th)],[]) ).
tff(94,plain,
( ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
<=> ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ) ),
inference(quant_intro,[status(thm)],[93]) ).
tff(95,axiom,
! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-2.ax',cls_SetInterval_OatMost__subset__iff__iff2_0) ).
tff(96,plain,
! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ),
inference(modus_ponens,[status(thm)],[95,94]) ).
tff(97,plain,
! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ),
inference(modus_ponens,[status(thm)],[96,92]) ).
tff(98,plain,
! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ),
inference(skolemize,[status(sab)],[97]) ).
tff(99,plain,
! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) ),
inference(modus_ponens,[status(thm)],[98,91]) ).
tff(100,plain,
( ( ~ ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
| ~ class_Orderings_Oorder(t_a)
| ~ c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) )
<=> ( ~ ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
| ~ class_Orderings_Oorder(t_a)
| ~ c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(101,plain,
( ~ ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
| ~ class_Orderings_Oorder(t_a)
| ~ c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(102,plain,
( ~ ! [V_x: $i,T_a: $i,V_y: $i] :
( ~ class_Orderings_Oorder(T_a)
| ~ c_lessequals(V_x,V_y,T_a)
| c_lessequals(c_SetInterval_OatMost(V_x,T_a),c_SetInterval_OatMost(V_y,T_a),tc_set(T_a)) )
| ~ class_Orderings_Oorder(t_a)
| ~ c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| c_lessequals(c_SetInterval_OatMost(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),c_SetInterval_OatMost(c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),tc_set(t_a)) ),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
~ c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a),
inference(unit_resolution,[status(thm)],[102,99,89,75]) ).
tff(104,plain,
^ [T: $i] :
refl(( ( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(105,plain,
( ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[104]) ).
tff(106,plain,
( ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(107,plain,
^ [T: $i] :
rewrite(( ( ~ class_Ring__and__Field_Oordered__idom(T)
| class_OrderedGroup_Olordered__ab__group__abs(T) )
<=> ( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(108,plain,
( ! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_OrderedGroup_Olordered__ab__group__abs(T) )
<=> ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[107]) ).
tff(109,axiom,
! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_OrderedGroup_Olordered__ab__group__abs(T) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-0.ax',clsrel_Ring__and__Field_Oordered__idom_50) ).
tff(110,plain,
! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[109,108]) ).
tff(111,plain,
! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[110,106]) ).
tff(112,plain,
! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(skolemize,[status(sab)],[111]) ).
tff(113,plain,
! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[112,105]) ).
tff(114,plain,
( ( ~ ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_OrderedGroup_Olordered__ab__group__abs(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) )
<=> ( ~ ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_OrderedGroup_Olordered__ab__group__abs(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(115,plain,
( ~ ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_OrderedGroup_Olordered__ab__group__abs(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(116,plain,
( ~ ! [T: $i] :
( class_OrderedGroup_Olordered__ab__group__abs(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_OrderedGroup_Olordered__ab__group__abs(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(modus_ponens,[status(thm)],[115,114]) ).
tff(117,plain,
class_OrderedGroup_Olordered__ab__group__abs(t_a),
inference(unit_resolution,[status(thm)],[116,113,12]) ).
tff(118,plain,
^ [V_a: $i,T_a: $i] :
refl(( ( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) )
<=> ( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(119,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ) ),
inference(quant_intro,[status(thm)],[118]) ).
tff(120,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(121,axiom,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_OrderedGroup_Oabs__ge__zero_0) ).
tff(122,plain,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ),
inference(modus_ponens,[status(thm)],[121,120]) ).
tff(123,plain,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ),
inference(skolemize,[status(sab)],[122]) ).
tff(124,plain,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ),
inference(modus_ponens,[status(thm)],[123,119]) ).
tff(125,plain,
( ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a) )
<=> ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(126,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(127,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a) ),
inference(modus_ponens,[status(thm)],[126,125]) ).
tff(128,plain,
c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a),
inference(unit_resolution,[status(thm)],[127,124,117]) ).
tff(129,plain,
^ [V_a: $i,T_a: $i] :
refl(( ( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) )
<=> ( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(130,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) ) ),
inference(quant_intro,[status(thm)],[129]) ).
tff(131,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(132,axiom,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_OrderedGroup_Oabs__ge__self_0) ).
tff(133,plain,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) ),
inference(modus_ponens,[status(thm)],[132,131]) ).
tff(134,plain,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) ),
inference(skolemize,[status(sab)],[133]) ).
tff(135,plain,
! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) ),
inference(modus_ponens,[status(thm)],[134,130]) ).
tff(136,plain,
( ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) )
<=> ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(137,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(138,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
| c_lessequals(V_a,c_HOL_Oabs(V_a,T_a),T_a) )
| ~ class_OrderedGroup_Olordered__ab__group__abs(t_a)
| c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) ),
inference(modus_ponens,[status(thm)],[137,136]) ).
tff(139,plain,
c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a),
inference(unit_resolution,[status(thm)],[138,135,117]) ).
tff(140,plain,
^ [T: $i] :
refl(( ( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(141,plain,
( ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[140]) ).
tff(142,plain,
( ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
<=> ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(143,plain,
^ [T: $i] :
rewrite(( ( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Ring__and__Field_Opordered__semiring(T) )
<=> ( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) )),
inference(bind,[status(th)],[]) ).
tff(144,plain,
( ! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Ring__and__Field_Opordered__semiring(T) )
<=> ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ) ),
inference(quant_intro,[status(thm)],[143]) ).
tff(145,axiom,
! [T: $i] :
( ~ class_Ring__and__Field_Oordered__idom(T)
| class_Ring__and__Field_Opordered__semiring(T) ),
file('/export/starexec/sandbox2/benchmark/Axioms/MSC001-0.ax',clsrel_Ring__and__Field_Oordered__idom_42) ).
tff(146,plain,
! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[145,144]) ).
tff(147,plain,
! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[146,142]) ).
tff(148,plain,
! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(skolemize,[status(sab)],[147]) ).
tff(149,plain,
! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) ),
inference(modus_ponens,[status(thm)],[148,141]) ).
tff(150,plain,
( ( ~ ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Ring__and__Field_Opordered__semiring(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) )
<=> ( ~ ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Ring__and__Field_Opordered__semiring(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(151,plain,
( ~ ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Ring__and__Field_Opordered__semiring(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(152,plain,
( ~ ! [T: $i] :
( class_Ring__and__Field_Opordered__semiring(T)
| ~ class_Ring__and__Field_Oordered__idom(T) )
| class_Ring__and__Field_Opordered__semiring(t_a)
| ~ class_Ring__and__Field_Oordered__idom(t_a) ),
inference(modus_ponens,[status(thm)],[151,150]) ).
tff(153,plain,
class_Ring__and__Field_Opordered__semiring(t_a),
inference(unit_resolution,[status(thm)],[152,149,12]) ).
tff(154,plain,
^ [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
refl(( ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
<=> ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(155,plain,
( ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
<=> ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ) ),
inference(quant_intro,[status(thm)],[154]) ).
tff(156,plain,
( ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
<=> ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(157,plain,
^ [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
trans(monotonicity(rewrite(( ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(V_a,V_b,T_a)
| ~ c_lessequals(c_0,V_c,T_a) )
<=> ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ) )),
( ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(V_a,V_b,T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a) )
<=> ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a) ) )),
rewrite(( ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a) )
<=> ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ) )),
( ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(V_a,V_b,T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a) )
<=> ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(158,plain,
( ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(V_a,V_b,T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a) )
<=> ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ) ),
inference(quant_intro,[status(thm)],[157]) ).
tff(159,axiom,
! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| ~ c_lessequals(V_a,V_b,T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Ring__and__Field_Opordered__semiring__class_Omult__right__mono_0) ).
tff(160,plain,
! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ),
inference(modus_ponens,[status(thm)],[159,158]) ).
tff(161,plain,
! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ),
inference(modus_ponens,[status(thm)],[160,156]) ).
tff(162,plain,
! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ),
inference(skolemize,[status(sab)],[161]) ).
tff(163,plain,
! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) ),
inference(modus_ponens,[status(thm)],[162,155]) ).
tff(164,plain,
( ( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Ring__and__Field_Opordered__semiring(t_a) )
<=> ( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Ring__and__Field_Opordered__semiring(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(165,plain,
( ( ~ class_Ring__and__Field_Opordered__semiring(t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) )
<=> ( ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Ring__and__Field_Opordered__semiring(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(166,plain,
( ( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ class_Ring__and__Field_Opordered__semiring(t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) )
<=> ( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Ring__and__Field_Opordered__semiring(t_a) ) ),
inference(monotonicity,[status(thm)],[165]) ).
tff(167,plain,
( ( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ class_Ring__and__Field_Opordered__semiring(t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) )
<=> ( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Ring__and__Field_Opordered__semiring(t_a) ) ),
inference(transitivity,[status(thm)],[166,164]) ).
tff(168,plain,
( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ class_Ring__and__Field_Opordered__semiring(t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(169,plain,
( ~ ! [V_c: $i,V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_c,T_a),T_a)
| ~ c_lessequals(c_0,V_c,T_a)
| ~ c_lessequals(V_a,V_b,T_a) )
| ~ c_lessequals(v_c,c_HOL_Oabs(v_c,t_a),t_a)
| ~ c_lessequals(c_0,c_HOL_Oabs(v_f(v_x),t_a),t_a)
| c_lessequals(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_a),t_a),c_times(c_HOL_Oabs(v_c,t_a),c_HOL_Oabs(v_f(v_x),t_a),t_a),t_a)
| ~ class_Ring__and__Field_Opordered__semiring(t_a) ),
inference(modus_ponens,[status(thm)],[168,167]) ).
tff(170,plain,
$false,
inference(unit_resolution,[status(thm)],[169,163,153,139,128,103]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : ANA039-1 : TPTP v9.0.0. Released v3.2.0.
% 0.12/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.19/0.34 % WCLimit : 300
% 0.19/0.34 % DateTime : Thu Jun 19 11:23:41 EDT 2025
% 0.19/0.34 % CPUTime :
% 0.44/0.60 % SZS status Unsatisfiable
% 0.44/0.60 % SZS output start Proof
% See solution above
% 0.48/0.65 % E exiting
%------------------------------------------------------------------------------