TPTP Problem File: LCL955^24.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL955^24 : TPTP v8.2.0. Released v8.1.0.
% Domain : Logic Calculi
% Problem : Goedel translation of SWV087+1 (from TPTP-v5.0.0)
% Version : [BP13] axioms.
% English :
% Refs : [RO12] Raths & Otten (2012), The QMLTP Problem Library for Fi
% : [BP13] Benzmueller & Paulson (2013), Quantified Multimodal Lo
% : [Ste22] Steen (2022), An Extensible Logic Embedding Tool for L
% Source : [TPTP]
% Names : GSV087+1 [QMLTP]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.31 v8.1.0
% Syntax : Number of formulae : 198 ( 12 unt; 51 typ; 10 def)
% Number of atoms : 1919 ( 10 equ; 0 cnn)
% Maximal formula atoms : 97 ( 13 avg)
% Number of connectives : 3079 ( 1 ~; 1 |; 4 &;3068 @)
% ( 1 <=>; 4 =>; 0 <=; 0 <~>)
% Maximal formula depth : 51 ( 15 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 113 ( 113 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 50 usr; 14 con; 0-4 aty)
% Number of variables : 362 ( 351 ^; 8 !; 3 ?; 362 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This output was generated by embedproblem, version 1.7.1 (library
% version 1.3). Generated on Thu Apr 28 13:18:18 EDT 2022 using
% 'modal' embedding, version 1.5.2. Logic specification used:
% $modal == [$constants == $rigid,$quantification == $decreasing,
% $modalities == $modal_system_S5U].
%------------------------------------------------------------------------------
thf(mworld,type,
mworld: $tType ).
thf(mrel_type,type,
mrel: mworld > mworld > $o ).
thf(mactual_type,type,
mactual: mworld ).
thf(mlocal_type,type,
mlocal: ( mworld > $o ) > $o ).
thf(mlocal_def,definition,
( mlocal
= ( ^ [Phi: mworld > $o] : ( Phi @ mactual ) ) ) ).
thf(mnot_type,type,
mnot: ( mworld > $o ) > mworld > $o ).
thf(mand_type,type,
mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mor_type,type,
mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mimplies_type,type,
mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mequiv_type,type,
mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mnot_def,definition,
( mnot
= ( ^ [A: mworld > $o,W: mworld] :
~ ( A @ W ) ) ) ).
thf(mand_def,definition,
( mand
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
& ( B @ W ) ) ) ) ).
thf(mor_def,definition,
( mor
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
| ( B @ W ) ) ) ) ).
thf(mimplies_def,definition,
( mimplies
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
=> ( B @ W ) ) ) ) ).
thf(mequiv_def,definition,
( mequiv
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
<=> ( B @ W ) ) ) ) ).
thf(mbox_type,type,
mbox: ( mworld > $o ) > mworld > $o ).
thf(mbox_def,definition,
( mbox
= ( ^ [Phi: mworld > $o,W: mworld] :
! [V: mworld] :
( ( mrel @ W @ V )
=> ( Phi @ V ) ) ) ) ).
thf(mdia_type,type,
mdia: ( mworld > $o ) > mworld > $o ).
thf(mdia_def,definition,
( mdia
= ( ^ [Phi: mworld > $o,W: mworld] :
? [V: mworld] :
( ( mrel @ W @ V )
& ( Phi @ V ) ) ) ) ).
thf(mrel_universal,axiom,
! [W: mworld,V: mworld] : ( mrel @ W @ V ) ).
thf(eiw_di_type,type,
eiw_di: $i > mworld > $o ).
thf(eiw_di_nonempty,axiom,
! [W: mworld] :
? [X: $i] : ( eiw_di @ X @ W ) ).
thf(eiw_di_decr,axiom,
! [W: mworld,V: mworld,X: $i] :
( ( ( eiw_di @ X @ W )
& ( mrel @ V @ W ) )
=> ( eiw_di @ X @ V ) ) ).
thf(mforall_di_type,type,
mforall_di: ( $i > mworld > $o ) > mworld > $o ).
thf(mforall_di_def,definition,
( mforall_di
= ( ^ [A: $i > mworld > $o,W: mworld] :
! [X: $i] :
( ( eiw_di @ X @ W )
=> ( A @ X @ W ) ) ) ) ).
thf(mexists_di_type,type,
mexists_di: ( $i > mworld > $o ) > mworld > $o ).
thf(mexists_di_def,definition,
( mexists_di
= ( ^ [A: $i > mworld > $o,W: mworld] :
? [X: $i] :
( ( eiw_di @ X @ W )
& ( A @ X @ W ) ) ) ) ).
%%% This output was generated by tptputils, version 1.2.
%%% Generated on Wed Apr 27 15:34:04 CEST 2022 using command 'downgrade(tff)'.
thf(typedecl_customqmltpeq,type,
customqmltpeq: $i > $i > mworld > $o ).
thf(true_decl,type,
true: mworld > $o ).
thf(def_decl,type,
def: $i ).
thf(tptp_minus_1_decl,type,
tptp_minus_1: $i ).
thf(tptp_float_0_001_decl,type,
tptp_float_0_001: $i ).
thf(tptp_float_0_0_decl,type,
tptp_float_0_0: $i ).
thf(loopcounter_decl,type,
loopcounter: $i ).
thf(pv65_decl,type,
pv65: $i ).
thf(n0_decl,type,
n0: $i ).
thf(n1_decl,type,
n1: $i ).
thf(n2_decl,type,
n2: $i ).
thf(n3_decl,type,
n3: $i ).
thf(n4_decl,type,
n4: $i ).
thf(use_decl,type,
use: $i ).
thf(n5_decl,type,
n5: $i ).
thf(geq_decl,type,
geq: $i > $i > mworld > $o ).
thf(lt_decl,type,
lt: $i > $i > mworld > $o ).
thf(leq_decl,type,
leq: $i > $i > mworld > $o ).
thf(gt_decl,type,
gt: $i > $i > mworld > $o ).
thf(minus_decl,type,
minus: $i > $i > $i ).
thf(dim_decl,type,
dim: $i > $i > $i ).
thf(sum_decl,type,
sum: $i > $i > $i > $i ).
thf(plus_decl,type,
plus: $i > $i > $i ).
thf(inv_decl,type,
inv: $i > $i ).
thf(uniform_int_rnd_decl,type,
uniform_int_rnd: $i > $i > $i ).
thf(succ_decl,type,
succ: $i > $i ).
thf(pred_decl,type,
pred: $i > $i ).
thf(tptp_update3_decl,type,
tptp_update3: $i > $i > $i > $i > $i ).
thf(tptp_update2_decl,type,
tptp_update2: $i > $i > $i > $i ).
thf(tptp_const_array1_decl,type,
tptp_const_array1: $i > $i > $i ).
thf(tptp_mmul_decl,type,
tptp_mmul: $i > $i > $i ).
thf(a_select3_decl,type,
a_select3: $i > $i > $i > $i ).
thf(tptp_const_array2_decl,type,
tptp_const_array2: $i > $i > $i > $i ).
thf(a_select2_decl,type,
a_select2: $i > $i > $i ).
thf(tptp_madd_decl,type,
tptp_madd: $i > $i > $i ).
thf(tptp_msub_decl,type,
tptp_msub: $i > $i > $i ).
thf(trans_decl,type,
trans: $i > $i ).
thf(reflexivity,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ X @ X ) ) ) ) ) ).
thf(symmetry,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ X @ Y ) ) @ ( mbox @ ( customqmltpeq @ Y @ X ) ) ) ) ) ) ) ) ) ).
thf(transitivity,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ X @ Y ) ) @ ( mbox @ ( customqmltpeq @ Y @ Z ) ) ) @ ( mbox @ ( customqmltpeq @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ).
thf(a_select2_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( a_select2 @ A @ C ) @ ( a_select2 @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(a_select2_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( a_select2 @ C @ A ) @ ( a_select2 @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(a_select3_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ A @ C @ D ) @ ( a_select3 @ B @ C @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(a_select3_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ C @ A @ D ) @ ( a_select3 @ C @ B @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(a_select3_substitution_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ C @ D @ A ) @ ( a_select3 @ C @ D @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(dim_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( dim @ A @ C ) @ ( dim @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(dim_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( dim @ C @ A ) @ ( dim @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(inv_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( inv @ A ) @ ( inv @ B ) ) ) ) ) ) ) ) ) ) ).
thf(minus_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( minus @ A @ C ) @ ( minus @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(minus_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( minus @ C @ A ) @ ( minus @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(plus_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( plus @ A @ C ) @ ( plus @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(plus_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( plus @ C @ A ) @ ( plus @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(pred_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( pred @ A ) @ ( pred @ B ) ) ) ) ) ) ) ) ) ) ).
thf(succ_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( succ @ A ) @ ( succ @ B ) ) ) ) ) ) ) ) ) ) ).
thf(sum_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( sum @ A @ C @ D ) @ ( sum @ B @ C @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(sum_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( sum @ C @ A @ D ) @ ( sum @ C @ B @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(sum_substitution_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( sum @ C @ D @ A ) @ ( sum @ C @ D @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_const_array1_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_const_array1 @ A @ C ) @ ( tptp_const_array1 @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_const_array1_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_const_array1 @ C @ A ) @ ( tptp_const_array1 @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_const_array2_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_const_array2 @ A @ C @ D ) @ ( tptp_const_array2 @ B @ C @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_const_array2_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_const_array2 @ C @ A @ D ) @ ( tptp_const_array2 @ C @ B @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_const_array2_substitution_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_const_array2 @ C @ D @ A ) @ ( tptp_const_array2 @ C @ D @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_madd_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_madd @ A @ C ) @ ( tptp_madd @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_madd_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_madd @ C @ A ) @ ( tptp_madd @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_mmul_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_mmul @ A @ C ) @ ( tptp_mmul @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_mmul_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_mmul @ C @ A ) @ ( tptp_mmul @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_msub_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_msub @ A @ C ) @ ( tptp_msub @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_msub_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_msub @ C @ A ) @ ( tptp_msub @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_update2_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_update2 @ A @ C @ D ) @ ( tptp_update2 @ B @ C @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_update2_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_update2 @ C @ A @ D ) @ ( tptp_update2 @ C @ B @ D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_update2_substitution_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_update2 @ C @ D @ A ) @ ( tptp_update2 @ C @ D @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_update3_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] :
( mbox
@ ( mforall_di
@ ^ [E: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_update3 @ A @ C @ D @ E ) @ ( tptp_update3 @ B @ C @ D @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_update3_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] :
( mbox
@ ( mforall_di
@ ^ [E: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_update3 @ C @ A @ D @ E ) @ ( tptp_update3 @ C @ B @ D @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_update3_substitution_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] :
( mbox
@ ( mforall_di
@ ^ [E: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_update3 @ C @ D @ A @ E ) @ ( tptp_update3 @ C @ D @ B @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(tptp_update3_substitution_4,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] :
( mbox
@ ( mforall_di
@ ^ [E: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( tptp_update3 @ C @ D @ E @ A ) @ ( tptp_update3 @ C @ D @ E @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(trans_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( trans @ A ) @ ( trans @ B ) ) ) ) ) ) ) ) ) ) ).
thf(uniform_int_rnd_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( uniform_int_rnd @ A @ C ) @ ( uniform_int_rnd @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(uniform_int_rnd_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( customqmltpeq @ ( uniform_int_rnd @ C @ A ) @ ( uniform_int_rnd @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(geq_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( geq @ A @ C ) ) ) @ ( mbox @ ( geq @ B @ C ) ) ) ) ) ) ) ) ) ) ) ).
thf(geq_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( geq @ C @ A ) ) ) @ ( mbox @ ( geq @ C @ B ) ) ) ) ) ) ) ) ) ) ) ).
thf(gt_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( gt @ A @ C ) ) ) @ ( mbox @ ( gt @ B @ C ) ) ) ) ) ) ) ) ) ) ) ).
thf(gt_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( gt @ C @ A ) ) ) @ ( mbox @ ( gt @ C @ B ) ) ) ) ) ) ) ) ) ) ) ).
thf(leq_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( leq @ A @ C ) ) ) @ ( mbox @ ( leq @ B @ C ) ) ) ) ) ) ) ) ) ) ) ).
thf(leq_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( leq @ C @ A ) ) ) @ ( mbox @ ( leq @ C @ B ) ) ) ) ) ) ) ) ) ) ) ).
thf(lt_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( lt @ A @ C ) ) ) @ ( mbox @ ( lt @ B @ C ) ) ) ) ) ) ) ) ) ) ) ).
thf(lt_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( customqmltpeq @ A @ B ) ) @ ( mbox @ ( lt @ C @ A ) ) ) @ ( mbox @ ( lt @ C @ B ) ) ) ) ) ) ) ) ) ) ) ).
thf(totality,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mor @ ( mbox @ ( gt @ X @ Y ) ) @ ( mor @ ( mbox @ ( gt @ Y @ X ) ) @ ( mbox @ ( customqmltpeq @ X @ Y ) ) ) ) ) ) ) ) ) ).
thf(transitivity_gt,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( gt @ X @ Y ) ) @ ( mbox @ ( gt @ Y @ Z ) ) ) @ ( mbox @ ( gt @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ).
thf(irreflexivity_gt,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( mnot @ ( mbox @ ( gt @ X @ X ) ) ) ) ) ) ) ).
thf(reflexivity_leq,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( leq @ X @ X ) ) ) ) ) ).
thf(transitivity_leq,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ X @ Y ) ) @ ( mbox @ ( leq @ Y @ Z ) ) ) @ ( mbox @ ( leq @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ).
thf(lt_gt,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mand @ ( mbox @ ( mimplies @ ( mbox @ ( lt @ X @ Y ) ) @ ( mbox @ ( gt @ Y @ X ) ) ) ) @ ( mbox @ ( mimplies @ ( mbox @ ( gt @ Y @ X ) ) @ ( mbox @ ( lt @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
thf(leq_geq,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mand @ ( mbox @ ( mimplies @ ( mbox @ ( geq @ X @ Y ) ) @ ( mbox @ ( leq @ Y @ X ) ) ) ) @ ( mbox @ ( mimplies @ ( mbox @ ( leq @ Y @ X ) ) @ ( mbox @ ( geq @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
thf(leq_gt1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( gt @ Y @ X ) ) @ ( mbox @ ( leq @ X @ Y ) ) ) ) ) ) ) ) ) ).
thf(leq_gt2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ X @ Y ) ) @ ( mbox @ ( mnot @ ( mbox @ ( customqmltpeq @ X @ Y ) ) ) ) ) @ ( mbox @ ( gt @ Y @ X ) ) ) ) ) ) ) ) ) ).
thf(leq_gt_pred,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mand @ ( mbox @ ( mimplies @ ( mbox @ ( leq @ X @ ( pred @ Y ) ) ) @ ( mbox @ ( gt @ Y @ X ) ) ) ) @ ( mbox @ ( mimplies @ ( mbox @ ( gt @ Y @ X ) ) @ ( mbox @ ( leq @ X @ ( pred @ Y ) ) ) ) ) ) ) ) ) ) ) ).
thf(gt_succ,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( gt @ ( succ @ X ) @ X ) ) ) ) ) ).
thf(leq_succ,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( leq @ X @ Y ) ) @ ( mbox @ ( leq @ X @ ( succ @ Y ) ) ) ) ) ) ) ) ) ) ).
thf(leq_succ_gt_equiv,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mand @ ( mbox @ ( mimplies @ ( mbox @ ( leq @ X @ Y ) ) @ ( mbox @ ( gt @ ( succ @ Y ) @ X ) ) ) ) @ ( mbox @ ( mimplies @ ( mbox @ ( gt @ ( succ @ Y ) @ X ) ) @ ( mbox @ ( leq @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
thf(uniform_int_rand_ranges_hi,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ ( uniform_int_rnd @ C @ X ) @ X ) ) ) ) ) ) ) ) ) ).
thf(uniform_int_rand_ranges_lo,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ n0 @ ( uniform_int_rnd @ C @ X ) ) ) ) ) ) ) ) ) ) ).
thf(const_array1_select,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [L: $i] :
( mbox
@ ( mforall_di
@ ^ [U: $i] :
( mbox
@ ( mforall_di
@ ^ [Val: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ L @ I ) ) @ ( mbox @ ( leq @ I @ U ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select2 @ ( tptp_const_array1 @ ( dim @ L @ U ) @ Val ) @ I ) @ Val ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(const_array2_select,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [L1: $i] :
( mbox
@ ( mforall_di
@ ^ [U1: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] :
( mbox
@ ( mforall_di
@ ^ [L2: $i] :
( mbox
@ ( mforall_di
@ ^ [U2: $i] :
( mbox
@ ( mforall_di
@ ^ [Val: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ L1 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ U1 ) ) @ ( mand @ ( mbox @ ( leq @ L2 @ J ) ) @ ( mbox @ ( leq @ J @ U2 ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_const_array2 @ ( dim @ L1 @ U1 ) @ ( dim @ L2 @ U2 ) @ Val ) @ I @ J ) @ Val ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_trans,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ A @ I @ J ) @ ( a_select3 @ A @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( trans @ A ) @ I @ J ) @ ( a_select3 @ ( trans @ A ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_inv,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ A @ I @ J ) @ ( a_select3 @ A @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( inv @ A ) @ I @ J ) @ ( a_select3 @ ( inv @ A ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_update_diagonal,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ A @ I @ J ) @ ( a_select3 @ A @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] :
( mbox
@ ( mforall_di
@ ^ [K: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mand @ ( mbox @ ( leq @ J @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ K ) ) @ ( mbox @ ( leq @ K @ N ) ) ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_update3 @ A @ K @ K @ VAL ) @ I @ J ) @ ( a_select3 @ ( tptp_update3 @ A @ K @ K @ VAL ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_add,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mimplies
@ ( mand
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ A @ I @ J ) @ ( a_select3 @ A @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ B @ I @ J ) @ ( a_select3 @ B @ J @ I ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_madd @ A @ B ) @ I @ J ) @ ( a_select3 @ ( tptp_madd @ A @ B ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_sub,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mimplies
@ ( mand
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ A @ I @ J ) @ ( a_select3 @ A @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ B @ I @ J ) @ ( a_select3 @ B @ J @ I ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_msub @ A @ B ) @ I @ J ) @ ( a_select3 @ ( tptp_msub @ A @ B ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_aba1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ B @ I @ J ) @ ( a_select3 @ B @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_mmul @ A @ ( tptp_mmul @ B @ ( trans @ A ) ) ) @ I @ J ) @ ( a_select3 @ ( tptp_mmul @ A @ ( tptp_mmul @ B @ ( trans @ A ) ) ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_aba2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mforall_di
@ ^ [M: $i] :
( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ M ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ M ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ B @ I @ J ) @ ( a_select3 @ B @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_mmul @ A @ ( tptp_mmul @ B @ ( trans @ A ) ) ) @ I @ J ) @ ( a_select3 @ ( tptp_mmul @ A @ ( tptp_mmul @ B @ ( trans @ A ) ) ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(matrix_symm_joseph_update,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] :
( mbox
@ ( mforall_di
@ ^ [D: $i] :
( mbox
@ ( mforall_di
@ ^ [E: $i] :
( mbox
@ ( mforall_di
@ ^ [F: $i] :
( mbox
@ ( mforall_di
@ ^ [N: $i] :
( mbox
@ ( mforall_di
@ ^ [M: $i] :
( mbox
@ ( mimplies
@ ( mand
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ M ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ M ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ D @ I @ J ) @ ( a_select3 @ D @ J @ I ) ) ) ) ) ) ) ) )
@ ( mand
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ A @ I @ J ) @ ( a_select3 @ A @ J @ I ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ F @ I @ J ) @ ( a_select3 @ F @ J @ I ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ N ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ N ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_madd @ A @ ( tptp_mmul @ B @ ( tptp_mmul @ ( tptp_madd @ ( tptp_mmul @ C @ ( tptp_mmul @ D @ ( trans @ C ) ) ) @ ( tptp_mmul @ E @ ( tptp_mmul @ F @ ( trans @ E ) ) ) ) @ ( trans @ B ) ) ) ) @ I @ J ) @ ( a_select3 @ ( tptp_madd @ A @ ( tptp_mmul @ B @ ( tptp_mmul @ ( tptp_madd @ ( tptp_mmul @ C @ ( tptp_mmul @ D @ ( trans @ C ) ) ) @ ( tptp_mmul @ E @ ( tptp_mmul @ F @ ( trans @ E ) ) ) ) @ ( trans @ B ) ) ) ) @ J @ I ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(sum_plus_base,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [Body: $i] : ( mbox @ ( customqmltpeq @ ( sum @ n0 @ tptp_minus_1 @ Body ) @ n0 ) ) ) ) ) ).
thf(sum_plus_base_float,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [Body: $i] : ( mbox @ ( customqmltpeq @ tptp_float_0_0 @ ( sum @ n0 @ tptp_minus_1 @ Body ) ) ) ) ) ) ).
thf(succ_tptp_minus_1,axiom,
mlocal @ ( mbox @ ( customqmltpeq @ ( succ @ tptp_minus_1 ) @ n0 ) ) ).
thf(succ_plus_1_r,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ X @ n1 ) @ ( succ @ X ) ) ) ) ) ) ).
thf(succ_plus_1_l,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ n1 @ X ) @ ( succ @ X ) ) ) ) ) ) ).
thf(succ_plus_2_r,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ X @ n2 ) @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ).
thf(succ_plus_2_l,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ n2 @ X ) @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ).
thf(succ_plus_3_r,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ X @ n3 ) @ ( succ @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ) ).
thf(succ_plus_3_l,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ n3 @ X ) @ ( succ @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ) ).
thf(succ_plus_4_r,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ X @ n4 ) @ ( succ @ ( succ @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ) ) ).
thf(succ_plus_4_l,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ n4 @ X ) @ ( succ @ ( succ @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ) ) ).
thf(succ_plus_5_r,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ X @ n5 ) @ ( succ @ ( succ @ ( succ @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ) ) ) ).
thf(succ_plus_5_l,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( plus @ n5 @ X ) @ ( succ @ ( succ @ ( succ @ ( succ @ ( succ @ X ) ) ) ) ) ) ) ) ) ) ).
thf(pred_minus_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( minus @ X @ n1 ) @ ( pred @ X ) ) ) ) ) ) ).
thf(pred_succ,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( pred @ ( succ @ X ) ) @ X ) ) ) ) ) ).
thf(succ_pred,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( customqmltpeq @ ( succ @ ( pred @ X ) ) @ X ) ) ) ) ) ).
thf(leq_succ_succ,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mand @ ( mbox @ ( mimplies @ ( mbox @ ( leq @ ( succ @ X ) @ ( succ @ Y ) ) ) @ ( mbox @ ( leq @ X @ Y ) ) ) ) @ ( mbox @ ( mimplies @ ( mbox @ ( leq @ X @ Y ) ) @ ( mbox @ ( leq @ ( succ @ X ) @ ( succ @ Y ) ) ) ) ) ) ) ) ) ) ) ).
thf(leq_succ_gt,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( leq @ ( succ @ X ) @ Y ) ) @ ( mbox @ ( gt @ Y @ X ) ) ) ) ) ) ) ) ) ).
thf(leq_minus,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( leq @ ( minus @ X @ Y ) @ X ) ) @ ( mbox @ ( leq @ n0 @ Y ) ) ) ) ) ) ) ) ) ).
thf(sel3_update_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [U: $i] :
( mbox
@ ( mforall_di
@ ^ [V: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL: $i] : ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_update3 @ X @ U @ V @ VAL ) @ U @ V ) @ VAL ) ) ) ) ) ) ) ) ) ) ) ).
thf(sel3_update_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] :
( mbox
@ ( mforall_di
@ ^ [U: $i] :
( mbox
@ ( mforall_di
@ ^ [V: $i] :
( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL2: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( mnot @ ( mbox @ ( customqmltpeq @ I @ U ) ) ) ) @ ( mand @ ( mbox @ ( customqmltpeq @ J @ V ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ X @ U @ V ) @ VAL ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_update3 @ X @ I @ J @ VAL2 ) @ U @ V ) @ VAL ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(sel3_update_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [J: $i] :
( mbox
@ ( mforall_di
@ ^ [U: $i] :
( mbox
@ ( mforall_di
@ ^ [V: $i] :
( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL: $i] :
( mbox
@ ( mimplies
@ ( mand
@ ( mbox
@ ( mforall_di
@ ^ [I0: $i] :
( mbox
@ ( mforall_di
@ ^ [J0: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I0 ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J0 ) ) @ ( mand @ ( mbox @ ( leq @ I0 @ U ) ) @ ( mbox @ ( leq @ J0 @ V ) ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select3 @ X @ I0 @ J0 ) @ VAL ) ) ) ) ) ) ) )
@ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mand @ ( mbox @ ( leq @ I @ U ) ) @ ( mand @ ( mbox @ ( leq @ n0 @ J ) ) @ ( mbox @ ( leq @ J @ V ) ) ) ) ) )
@ ( mbox @ ( customqmltpeq @ ( a_select3 @ ( tptp_update3 @ X @ U @ V @ VAL ) @ I @ J ) @ VAL ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(sel2_update_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [U: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL: $i] : ( mbox @ ( customqmltpeq @ ( a_select2 @ ( tptp_update2 @ X @ U @ VAL ) @ U ) @ VAL ) ) ) ) ) ) ) ) ) ).
thf(sel2_update_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [U: $i] :
( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL2: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( mnot @ ( mbox @ ( customqmltpeq @ I @ U ) ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select2 @ X @ U ) @ VAL ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select2 @ ( tptp_update2 @ X @ I @ VAL2 ) @ U ) @ VAL ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(sel2_update_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [I: $i] :
( mbox
@ ( mforall_di
@ ^ [U: $i] :
( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [VAL: $i] :
( mbox
@ ( mimplies
@ ( mand
@ ( mbox
@ ( mforall_di
@ ^ [I0: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ I0 ) ) @ ( mbox @ ( leq @ I0 @ U ) ) ) @ ( mbox @ ( customqmltpeq @ ( a_select2 @ X @ I0 ) @ VAL ) ) ) ) ) )
@ ( mand @ ( mbox @ ( leq @ n0 @ I ) ) @ ( mbox @ ( leq @ I @ U ) ) ) )
@ ( mbox @ ( customqmltpeq @ ( a_select2 @ ( tptp_update2 @ X @ U @ VAL ) @ I ) @ VAL ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ttrue,axiom,
mlocal @ ( mbox @ true ) ).
thf(defuse,axiom,
mlocal @ ( mbox @ ( mnot @ ( mbox @ ( customqmltpeq @ def @ use ) ) ) ) ).
thf(cl5_nebula_array_0028,conjecture,
mlocal @ ( mbox @ ( mimplies @ ( mbox @ ( geq @ pv65 @ tptp_float_0_001 ) ) @ ( mand @ ( mbox @ ( mimplies @ ( mbox @ ( mnot @ ( mbox @ ( gt @ ( plus @ n1 @ loopcounter ) @ n1 ) ) ) ) @ ( mbox @ true ) ) ) @ ( mbox @ ( mimplies @ ( mbox @ ( gt @ ( plus @ n1 @ loopcounter ) @ n1 ) ) @ ( mbox @ true ) ) ) ) ) ) ).
thf(gt_5_4,axiom,
mlocal @ ( mbox @ ( gt @ n5 @ n4 ) ) ).
thf(gt_4_tptp_minus_1,axiom,
mlocal @ ( mbox @ ( gt @ n4 @ tptp_minus_1 ) ) ).
thf(gt_5_tptp_minus_1,axiom,
mlocal @ ( mbox @ ( gt @ n5 @ tptp_minus_1 ) ) ).
thf(gt_0_tptp_minus_1,axiom,
mlocal @ ( mbox @ ( gt @ n0 @ tptp_minus_1 ) ) ).
thf(gt_1_tptp_minus_1,axiom,
mlocal @ ( mbox @ ( gt @ n1 @ tptp_minus_1 ) ) ).
thf(gt_2_tptp_minus_1,axiom,
mlocal @ ( mbox @ ( gt @ n2 @ tptp_minus_1 ) ) ).
thf(gt_3_tptp_minus_1,axiom,
mlocal @ ( mbox @ ( gt @ n3 @ tptp_minus_1 ) ) ).
thf(gt_4_0,axiom,
mlocal @ ( mbox @ ( gt @ n4 @ n0 ) ) ).
thf(gt_5_0,axiom,
mlocal @ ( mbox @ ( gt @ n5 @ n0 ) ) ).
thf(gt_1_0,axiom,
mlocal @ ( mbox @ ( gt @ n1 @ n0 ) ) ).
thf(gt_2_0,axiom,
mlocal @ ( mbox @ ( gt @ n2 @ n0 ) ) ).
thf(gt_3_0,axiom,
mlocal @ ( mbox @ ( gt @ n3 @ n0 ) ) ).
thf(gt_4_1,axiom,
mlocal @ ( mbox @ ( gt @ n4 @ n1 ) ) ).
thf(gt_5_1,axiom,
mlocal @ ( mbox @ ( gt @ n5 @ n1 ) ) ).
thf(gt_2_1,axiom,
mlocal @ ( mbox @ ( gt @ n2 @ n1 ) ) ).
thf(gt_3_1,axiom,
mlocal @ ( mbox @ ( gt @ n3 @ n1 ) ) ).
thf(gt_4_2,axiom,
mlocal @ ( mbox @ ( gt @ n4 @ n2 ) ) ).
thf(gt_5_2,axiom,
mlocal @ ( mbox @ ( gt @ n5 @ n2 ) ) ).
thf(gt_3_2,axiom,
mlocal @ ( mbox @ ( gt @ n3 @ n2 ) ) ).
thf(gt_4_3,axiom,
mlocal @ ( mbox @ ( gt @ n4 @ n3 ) ) ).
thf(gt_5_3,axiom,
mlocal @ ( mbox @ ( gt @ n5 @ n3 ) ) ).
thf(finite_domain_4,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ X @ n4 ) ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n0 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n1 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n2 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n3 ) ) @ ( mbox @ ( customqmltpeq @ X @ n4 ) ) ) ) ) ) ) ) ) ) ) ).
thf(finite_domain_5,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ X @ n5 ) ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n0 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n1 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n2 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n3 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n4 ) ) @ ( mbox @ ( customqmltpeq @ X @ n5 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(finite_domain_0,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ X @ n0 ) ) ) @ ( mbox @ ( customqmltpeq @ X @ n0 ) ) ) ) ) ) ) ).
thf(finite_domain_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ X @ n1 ) ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n0 ) ) @ ( mbox @ ( customqmltpeq @ X @ n1 ) ) ) ) ) ) ) ) ).
thf(finite_domain_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ X @ n2 ) ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n0 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n1 ) ) @ ( mbox @ ( customqmltpeq @ X @ n2 ) ) ) ) ) ) ) ) ) ).
thf(finite_domain_3,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( leq @ n0 @ X ) ) @ ( mbox @ ( leq @ X @ n3 ) ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n0 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n1 ) ) @ ( mor @ ( mbox @ ( customqmltpeq @ X @ n2 ) ) @ ( mbox @ ( customqmltpeq @ X @ n3 ) ) ) ) ) ) ) ) ) ) ).
thf(successor_4,axiom,
mlocal @ ( mbox @ ( customqmltpeq @ ( succ @ ( succ @ ( succ @ ( succ @ n0 ) ) ) ) @ n4 ) ) ).
thf(successor_5,axiom,
mlocal @ ( mbox @ ( customqmltpeq @ ( succ @ ( succ @ ( succ @ ( succ @ ( succ @ n0 ) ) ) ) ) @ n5 ) ) ).
thf(successor_1,axiom,
mlocal @ ( mbox @ ( customqmltpeq @ ( succ @ n0 ) @ n1 ) ) ).
thf(successor_2,axiom,
mlocal @ ( mbox @ ( customqmltpeq @ ( succ @ ( succ @ n0 ) ) @ n2 ) ) ).
thf(successor_3,axiom,
mlocal @ ( mbox @ ( customqmltpeq @ ( succ @ ( succ @ ( succ @ n0 ) ) ) @ n3 ) ) ).
%------------------------------------------------------------------------------