%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SCT161+1 : TPTP v9.0.0. Released v5.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.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 05:24:33 AM UTC 2025
% Result : Theorem 126.14s 126.35s
% Output : Proof 126.40s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 43
% Syntax : Number of formulae : 101 ( 28 unt; 0 typ; 0 def)
% Number of atoms : 997 ( 17 equ)
% Maximal formula atoms : 32 ( 9 avg)
% Number of connectives : 1927 (1103 ~; 651 |; 41 &)
% ( 117 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of FOOLs : 72 ( 72 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 19 ( 19 usr; 7 con; 0-4 aty)
% Number of variables : 428 ( 368 !; 24 ?; 428 :)
% Comments :
%------------------------------------------------------------------------------
tff(hBOOL_type,type,
hBOOL: $i > $o ).
tff(hAPP_type,type,
hAPP: ( $i * $i ) > $i ).
tff(tptp_fun_B_m_4_type,type,
tptp_fun_B_m_4: ( $i * $i ) > $i ).
tff(c_Orderings_Oord__class_Oless__eq_type,type,
c_Orderings_Oord__class_Oless__eq: ( $i * $i ) > $i ).
tff(c_Finite__Set_Ocard_type,type,
c_Finite__Set_Ocard: ( $i * $i ) > $i ).
tff(c_Orderings_Otop__class_Otop_type,type,
c_Orderings_Otop__class_Otop: $i > $i ).
tff(tc_fun_type,type,
tc_fun: ( $i * $i ) > $i ).
tff(tc_HOL_Obool_type,type,
tc_HOL_Obool: $i ).
tff(tc_Arrow__Order__Mirabelle_Oindi_type,type,
tc_Arrow__Order__Mirabelle_Oindi: $i ).
tff(tc_Nat_Onat_type,type,
tc_Nat_Onat: $i ).
tff(c_Orderings_Oord__class_Oless_type,type,
c_Orderings_Oord__class_Oless: $i > $i ).
tff(v_Sledgehammer_058p__2_type,type,
v_Sledgehammer_058p__2: $i ).
tff(v_h_____type,type,
v_h____: $i ).
tff(c_SetInterval_Oord__class_OatLeastLessThan_type,type,
c_SetInterval_Oord__class_OatLeastLessThan: ( $i * $i * $i ) > $i ).
tff(c_Groups_Ozero__class_Ozero_type,type,
c_Groups_Ozero__class_Ozero: $i > $i ).
tff(c_member_type,type,
c_member: $i > $i ).
tff(tptp_fun_B_x_5_type,type,
tptp_fun_B_x_5: ( $i * $i ) > $i ).
tff(c_Set_Oimage_type,type,
c_Set_Oimage: ( $i * $i * $i * $i ) > $i ).
tff(v_Lab_____type,type,
v_Lab____: $i ).
tff(v_Lba_____type,type,
v_Lba____: $i ).
tff(class_Orderings_Olinorder_type,type,
class_Orderings_Olinorder: $i > $o ).
tff(1,plain,
^ [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
rewrite(( ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
^ [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
refl(( ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(4,plain,
( ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(quant_intro,[status(thm)],[3]) ).
tff(5,plain,
( ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(pull_quant,[status(thm)],[]) ).
tff(6,plain,
^ [V_P_2: $i,V_n_2: $i] :
trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(( ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
<=> ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) ) )),
( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
<=> ~ ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) ) )),
pull_quant(( ~ ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
<=> ? [B_m: $i] :
~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) ) )),
( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
<=> ? [B_m: $i] :
~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) ) )),
trans(monotonicity(pull_quant(( ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ! [B_x: $i] :
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )),
( ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ~ ! [B_x: $i] :
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )),
pull_quant(( ~ ! [B_x: $i] :
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) )
<=> ? [B_x: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )),
( ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ? [B_x: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )),
( ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ( ? [B_m: $i] :
~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ? [B_x: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
pull_quant(( ( ? [B_m: $i] :
~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ? [B_x: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ? [B_x: $i,B_m: $i] :
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
( ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ? [B_x: $i,B_m: $i] :
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
( ~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ~ ? [B_x: $i,B_m: $i] :
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
pull_quant(( ~ ? [B_x: $i,B_m: $i] :
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ! [B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
( ~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ! [B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(7,plain,
( ! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(quant_intro,[status(thm)],[6]) ).
tff(8,plain,
( ! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(transitivity,[status(thm)],[7,5]) ).
tff(9,plain,
( ! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(transitivity,[status(thm)],[8,4]) ).
tff(10,plain,
^ [V_P_2: $i,V_n_2: $i] :
rewrite(( ~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(transitivity,[status(thm)],[11,9]) ).
tff(13,plain,
^ [V_P_2: $i,V_n_2: $i] :
trans(monotonicity(trans(monotonicity(quant_intro(proof_bind(^ [B_m: $i] :
trans(monotonicity(rewrite(( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) ) )),
( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ~ ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) ) )),
rewrite(( ~ ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
<=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) ) )),
( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) ) ))),
( ! [B_m: $i] :
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) ) )),
rewrite(( ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) )
<=> ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )),
( ( ! [B_m: $i] :
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
| ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
<=> ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) ) )),
rewrite(( ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
<=> ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) ) )),
( ( ! [B_m: $i] :
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
| ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
<=> ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) ) )),
rewrite(( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
& hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
~ ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )),
( ( ( ! [B_m: $i] :
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
| ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
& ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
& hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
~ ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ( ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
& ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ) )),
rewrite(( ( ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
& ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ) )),
( ( ( ! [B_m: $i] :
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
| ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
& ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
& hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
~ ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(14,plain,
( ! [V_P_2: $i,V_n_2: $i] :
( ( ! [B_m: $i] :
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
| ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
& ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
& hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
~ ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) ) )
<=> ! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,plain,
( ! [V_P_2: $i,V_n_2: $i] :
( ? [B_m: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ? [B_x: $i] :
( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) )
<=> ! [V_P_2: $i,V_n_2: $i] :
( ? [B_m: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ? [B_x: $i] :
( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(16,axiom,
! [V_P_2: $i,V_n_2: $i] :
( ? [B_m: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ? [B_x: $i] :
( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ex__nat__less__eq) ).
tff(17,plain,
! [V_P_2: $i,V_n_2: $i] :
( ? [B_m: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ? [B_x: $i] :
( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) ),
inference(modus_ponens,[status(thm)],[16,15]) ).
tff(18,plain,
! [V_P_2: $i,V_n_2: $i] :
( ( ! [B_m: $i] :
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
& hBOOL(hAPP(V_P_2,B_m)) )
| ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
& ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
& hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
~ ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
& hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(skolemize,[status(sab)],[17]) ).
tff(19,plain,
! [V_P_2: $i,V_n_2: $i] :
~ ( ~ ( ! [B_m: $i] :
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m)) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ! [B_x: $i] :
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ) ),
inference(modus_ponens,[status(thm)],[18,14]) ).
tff(20,plain,
! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ),
inference(modus_ponens,[status(thm)],[19,12]) ).
tff(21,plain,
! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) ),
inference(modus_ponens,[status(thm)],[20,2]) ).
tff(22,plain,
( ( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ) )
<=> ( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,plain,
( ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) )
<=> ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,plain,
( ( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) ) )
<=> ( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ) ) ),
inference(monotonicity,[status(thm)],[23]) ).
tff(25,plain,
( ( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) ) )
<=> ( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ) ) ),
inference(transitivity,[status(thm)],[24,22]) ).
tff(26,plain,
( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [V_P_2: $i,V_n_2: $i,B_x: $i,B_m: $i] :
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
| ~ hBOOL(hAPP(V_P_2,B_m))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(V_n_2,V_P_2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_x_5(V_n_2,V_P_2))) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(V_n_2,V_P_2)),V_n_2))
| ~ hBOOL(hAPP(V_P_2,tptp_fun_B_m_4(V_n_2,V_P_2))) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
| ~ hBOOL(hAPP(V_P_2,B_x)) ) )
| ~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
~ ( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ),
inference(unit_resolution,[status(thm)],[27,21]) ).
tff(29,plain,
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_x_5(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ),
inference(tautology,[status(thm)],[]) ).
tff(30,plain,
( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ),
inference(unit_resolution,[status(thm)],[29,28]) ).
tff(31,plain,
( ( c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))) )
<=> ( c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,axiom,
c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_surjh) ).
tff(33,plain,
c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))) = hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))),
inference(monotonicity,[status(thm)],[33]) ).
tff(35,plain,
( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))) ),
inference(monotonicity,[status(thm)],[34]) ).
tff(36,plain,
^ [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] :
refl(( hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))
<=> hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) )),
inference(bind,[status(th)],[]) ).
tff(37,plain,
( ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))
<=> ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ),
inference(quant_intro,[status(thm)],[36]) ).
tff(38,plain,
( ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))
<=> ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ),
inference(rewrite,[status(thm)],[]) ).
tff(39,plain,
( ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))
<=> ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ),
inference(elim_unused_vars,[status(thm)],[]) ).
tff(40,plain,
( ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] :
( ( V_ba_2 != hAPP(V_f_2,V_x_2) )
| hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) )
<=> ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ),
inference(destructive_equality_resolution,[status(thm)],[]) ).
tff(41,plain,
( ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] :
( ( V_ba_2 != hAPP(V_f_2,V_x_2) )
| hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) )
<=> ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ),
inference(transitivity,[status(thm)],[40,39]) ).
tff(42,plain,
^ [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] :
rewrite(( ( ( V_ba_2 = hAPP(V_f_2,V_x_2) )
=> hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) )
<=> ( ( V_ba_2 != hAPP(V_f_2,V_x_2) )
| hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ) )),
inference(bind,[status(th)],[]) ).
tff(43,plain,
( ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] :
( ( V_ba_2 = hAPP(V_f_2,V_x_2) )
=> hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) )
<=> ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] :
( ( V_ba_2 != hAPP(V_f_2,V_x_2) )
| hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ) ),
inference(quant_intro,[status(thm)],[42]) ).
tff(44,plain,
( ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] :
( ( V_ba_2 = hAPP(V_f_2,V_x_2) )
=> hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) )
<=> ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ),
inference(transitivity,[status(thm)],[43,41]) ).
tff(45,axiom,
! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i,V_ba_2: $i] :
( ( V_ba_2 = hAPP(V_f_2,V_x_2) )
=> hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_range__eqI) ).
tff(46,plain,
! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))),
inference(modus_ponens,[status(thm)],[45,44]) ).
tff(47,plain,
! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))),
inference(modus_ponens,[status(thm)],[46,38]) ).
tff(48,plain,
! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))),
inference(skolemize,[status(sab)],[47]) ).
tff(49,plain,
! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))),
inference(modus_ponens,[status(thm)],[48,37]) ).
tff(50,plain,
( ~ ! [T_b: $i,T_a: $i,V_x_2: $i,V_f_2: $i] : hBOOL(hAPP(hAPP(c_member(T_a),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))
| hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(51,plain,
hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,v_h____,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))),
inference(unit_resolution,[status(thm)],[50,49]) ).
tff(52,plain,
hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),
inference(modus_ponens,[status(thm)],[51,35]) ).
tff(53,plain,
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))) ),
inference(rewrite,[status(thm)],[]) ).
tff(54,plain,
( ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
=> ( v_Lba____ = v_Lab____ ) )
<=> ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ( v_Lba____ = v_Lab____ ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(55,axiom,
~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
=> ( v_Lba____ = v_Lab____ ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
tff(56,plain,
~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ( v_Lba____ = v_Lab____ ) ),
inference(modus_ponens,[status(thm)],[55,54]) ).
tff(57,plain,
~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))),
inference(or_elim,[status(thm)],[56]) ).
tff(58,plain,
~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))),
inference(modus_ponens,[status(thm)],[57,53]) ).
tff(59,plain,
( class_Orderings_Olinorder(tc_Nat_Onat)
<=> class_Orderings_Olinorder(tc_Nat_Onat) ),
inference(rewrite,[status(thm)],[]) ).
tff(60,axiom,
class_Orderings_Olinorder(tc_Nat_Onat),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Olinorder) ).
tff(61,plain,
class_Orderings_Olinorder(tc_Nat_Onat),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
^ [V_y: $i,V_x: $i,T_a: $i] :
refl(( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
<=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(63,plain,
( ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
<=> ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,plain,
( ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
<=> ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(65,plain,
^ [V_y: $i,V_x: $i,T_a: $i] :
rewrite(( ( class_Orderings_Olinorder(T_a)
=> ( hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) )
<=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(66,plain,
( ! [V_y: $i,V_x: $i,T_a: $i] :
( class_Orderings_Olinorder(T_a)
=> ( hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) )
<=> ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ) ),
inference(quant_intro,[status(thm)],[65]) ).
tff(67,axiom,
! [V_y: $i,V_x: $i,T_a: $i] :
( class_Orderings_Olinorder(T_a)
=> ( hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__le__less__linear) ).
tff(68,plain,
! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ),
inference(modus_ponens,[status(thm)],[67,66]) ).
tff(69,plain,
! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ),
inference(modus_ponens,[status(thm)],[68,64]) ).
tff(70,plain,
! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ),
inference(skolemize,[status(sab)],[69]) ).
tff(71,plain,
! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) ),
inference(modus_ponens,[status(thm)],[70,63]) ).
tff(72,plain,
( ( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) )
<=> ( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(73,plain,
( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ class_Orderings_Olinorder(tc_Nat_Onat) )
<=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(74,plain,
( ( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ class_Orderings_Olinorder(tc_Nat_Onat) )
<=> ( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) ),
inference(monotonicity,[status(thm)],[73]) ).
tff(75,plain,
( ( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ class_Orderings_Olinorder(tc_Nat_Onat) )
<=> ( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ) ),
inference(transitivity,[status(thm)],[74,72]) ).
tff(76,plain,
( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ class_Orderings_Olinorder(tc_Nat_Onat) ),
inference(quant_inst,[status(thm)],[]) ).
tff(77,plain,
( ~ ! [V_y: $i,V_x: $i,T_a: $i] :
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))
| ~ class_Orderings_Olinorder(T_a) )
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))) ),
inference(modus_ponens,[status(thm)],[76,75]) ).
tff(78,plain,
hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2))),
inference(unit_resolution,[status(thm)],[77,71,61,58]) ).
tff(79,plain,
( ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ),
inference(tautology,[status(thm)],[]) ).
tff(80,plain,
( ~ ( ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),hAPP(v_h____,v_Sledgehammer_058p__2)))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
| ~ hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(v_h____,v_Sledgehammer_058p__2)),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))
| ~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ),
inference(unit_resolution,[status(thm)],[79,78]) ).
tff(81,plain,
~ ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ),
inference(unit_resolution,[status(thm)],[80,52,30]) ).
tff(82,plain,
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))))
| hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ),
inference(tautology,[status(thm)],[]) ).
tff(83,plain,
hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))),
inference(unit_resolution,[status(thm)],[82,81]) ).
tff(84,plain,
( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))))
| hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))) ),
inference(tautology,[status(thm)],[]) ).
tff(85,plain,
hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))),
inference(unit_resolution,[status(thm)],[84,81]) ).
tff(86,plain,
( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) )
| ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
| ~ hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ),
inference(tautology,[status(thm)],[]) ).
tff(87,plain,
( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ),
inference(unit_resolution,[status(thm)],[86,85,83]) ).
tff(88,plain,
^ [V_y_2: $i,V_x_2: $i,T_a: $i] :
refl(( ( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
<=> ( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(89,plain,
( ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
<=> ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ) ),
inference(quant_intro,[status(thm)],[88]) ).
tff(90,plain,
( ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
<=> ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(91,plain,
^ [V_y_2: $i,V_x_2: $i,T_a: $i] :
trans(monotonicity(trans(rewrite(( ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) )
<=> ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )),
rewrite(( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) )
<=> ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )),
( ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) )
<=> ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )),
( ( class_Orderings_Olinorder(T_a)
=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
<=> ( class_Orderings_Olinorder(T_a)
=> ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ) )),
rewrite(( ( class_Orderings_Olinorder(T_a)
=> ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
<=> ( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ) )),
( ( class_Orderings_Olinorder(T_a)
=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
<=> ( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(92,plain,
( ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( class_Orderings_Olinorder(T_a)
=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
<=> ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ) ),
inference(quant_intro,[status(thm)],[91]) ).
tff(93,axiom,
! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( class_Orderings_Olinorder(T_a)
=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__not__less) ).
tff(94,plain,
! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,plain,
! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ),
inference(modus_ponens,[status(thm)],[94,90]) ).
tff(96,plain,
! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ),
inference(skolemize,[status(sab)],[95]) ).
tff(97,plain,
! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) ),
inference(modus_ponens,[status(thm)],[96,89]) ).
tff(98,plain,
( ( ~ ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) )
<=> ( ~ ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(99,plain,
( ~ ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(100,plain,
( ~ ! [V_y_2: $i,V_x_2: $i,T_a: $i] :
( ~ class_Orderings_Olinorder(T_a)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) ) )
| ~ class_Orderings_Olinorder(tc_Nat_Onat)
| ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))))
<=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))),tptp_fun_B_m_4(c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))))) ) ),
inference(modus_ponens,[status(thm)],[99,98]) ).
tff(101,plain,
$false,
inference(unit_resolution,[status(thm)],[100,97,61,87]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : SCT161+1 : TPTP v9.0.0. Released v5.2.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n010.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 03:54:41 EDT 2025
% 0.12/0.34 % CPUTime :
% 126.14/126.35 % SZS status Theorem
% 126.14/126.35 % SZS output start Proof
% See solution above
% 126.60/126.76 % E exiting
%------------------------------------------------------------------------------