%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n021.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue May 5 07:03:11 PM UTC 2026
% Result : Theorem 21.24s 21.43s
% Output : Proof 21.24s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_d_mu,type,
d_mu: $tType ).
thf(ty_d_unsorted,type,
d_unsorted: $tType ).
thf(ty_mreflexive,type,
mreflexive: ( $i > $i > $o ) > $o ).
thf(ty_mweakly_directed,type,
mweakly_directed: ( $i > $i > $o ) > $o ).
thf(ty_mand,type,
mand: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_d_mu_0,type,
d_mu_0: d_mu ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_mequiv,type,
mequiv: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_eigen__1,type,
eigen__1: $i > $o ).
thf(ty_d2mu,type,
d2mu: d_mu > mu ).
thf(ty_mcountersatisfiable,type,
mcountersatisfiable: ( $i > $o ) > $o ).
thf(ty_mnot,type,
mnot: ( $i > $o ) > $i > $o ).
thf(ty_mweakly_connected,type,
mweakly_connected: ( $i > $i > $o ) > $o ).
thf(ty_mserial,type,
mserial: ( $i > $i > $o ) > $o ).
thf(ty_d_unsorted_0,type,
d_unsorted_0: d_unsorted ).
thf(ty_mweakly_dense,type,
mweakly_dense: ( $i > $i > $o ) > $o ).
thf(ty_mor,type,
mor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i > $o ).
thf(ty_mfalse,type,
mfalse: $i > $o ).
thf(ty_mdia,type,
mdia: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_msatisfiable,type,
msatisfiable: ( $i > $o ) > $o ).
thf(ty_mforall_prop,type,
mforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(ty_mimplied,type,
mimplied: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_mxor,type,
mxor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_mfunctional,type,
mfunctional: ( $i > $i > $o ) > $o ).
thf(ty_d2unsorted,type,
d2unsorted: d_unsorted > $i ).
thf(ty_msymmetric,type,
msymmetric: ( $i > $i > $o ) > $o ).
thf(ty_meuclidean,type,
meuclidean: ( $i > $i > $o ) > $o ).
thf(ty_mtrue,type,
mtrue: $i > $o ).
thf(ty_meq_ind,type,
meq_ind: mu > mu > $i > $o ).
thf(ty_mtransitive,type,
mtransitive: ( $i > $i > $o ) > $o ).
thf(ty_mimplies,type,
mimplies: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_mforall_ind,type,
mforall_ind: ( mu > $i > $o ) > $i > $o ).
thf(ty_minvalid,type,
minvalid: ( $i > $o ) > $o ).
thf(ty_mexists_prop,type,
mexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(ty_mexists_ind,type,
mexists_ind: ( mu > $i > $o ) > $i > $o ).
thf(ty_mvalid,type,
mvalid: ( $i > $o ) > $o ).
thf(ty_meq_prop,type,
meq_prop: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(ty_mpartially_functional,type,
mpartially_functional: ( $i > $i > $o ) > $o ).
thf(ty_mbox,type,
mbox: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(sP1,plain,
( sP1
<=> ( mand
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
~ ( ( X1 @ X3 )
=> ~ ( X2 @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ( ( eigen__0 @ eigen__2 )
=> ( eigen__1 @ eigen__2 ) )
=> ~ ( ( eigen__1 @ eigen__2 )
=> ( eigen__0 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( mand @ ( mimplies @ eigen__0 @ eigen__1 ) @ ( mimplies @ eigen__1 @ eigen__0 ) @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ( mequiv @ eigen__0 )
= ( ^ [X1: $i > $o,X2: $i] :
~ ( ( ( eigen__0 @ X2 )
=> ( X1 @ X2 ) )
=> ~ ( ( X1 @ X2 )
=> ( eigen__0 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ( mimplies @ eigen__1 @ eigen__0 @ eigen__2 )
= ( ( eigen__1 @ eigen__2 )
=> ( eigen__0 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i > $o] :
( ( mand @ X1 )
= ( ^ [X2: $i > $o,X3: $i] :
~ ( ( X1 @ X3 )
=> ~ ( X2 @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( mimplies
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: $i] :
( ( mimplies @ eigen__0 @ eigen__1 @ X1 )
= ( ( eigen__0 @ X1 )
=> ( eigen__1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ( mequiv @ eigen__0 @ eigen__1 @ eigen__2 )
= ( ~ sP2 ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: $i > $o] :
( ( mimplies @ eigen__1 @ X1 )
= ( ^ [X2: $i] :
( ( eigen__1 @ X2 )
=> ( X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ( mand @ ( mimplies @ eigen__0 @ eigen__1 ) )
= ( ^ [X1: $i > $o,X2: $i] :
~ ( ( mimplies @ eigen__0 @ eigen__1 @ X2 )
=> ~ ( X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( mequiv @ eigen__0 @ eigen__1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( mimplies @ eigen__0 @ eigen__1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ( mimplies @ eigen__0 @ eigen__1 )
= ( ^ [X1: $i] :
( ( eigen__0 @ X1 )
=> ( eigen__1 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( mimplies @ eigen__1 @ eigen__0 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ( eigen__1 @ eigen__2 )
=> ( eigen__0 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( ( mimplies @ eigen__1 @ eigen__0 )
= ( ^ [X1: $i] :
( ( eigen__1 @ X1 )
=> ( eigen__0 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( mequiv
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
~ ( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ! [X1: $i > $o] :
( ( mimplies @ eigen__0 @ X1 )
= ( ^ [X2: $i] :
( ( eigen__0 @ X2 )
=> ( X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ! [X1: $i > $o] :
( ( mequiv @ X1 )
= ( ^ [X2: $i > $o,X3: $i] :
~ ( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( ( mimplies @ eigen__0 )
= ( ^ [X1: $i > $o,X2: $i] :
( ( eigen__0 @ X2 )
=> ( X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( ( mand @ ( mimplies @ eigen__0 @ eigen__1 ) @ ( mimplies @ eigen__1 @ eigen__0 ) )
= ( ^ [X1: $i] :
~ ( ( mimplies @ eigen__0 @ eigen__1 @ X1 )
=> ~ ( mimplies @ eigen__1 @ eigen__0 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ! [X1: $i] :
( ( mequiv @ eigen__0 @ eigen__1 @ X1 )
= ( ~ ( ( ( eigen__0 @ X1 )
=> ( eigen__1 @ X1 ) )
=> ~ ( ( eigen__1 @ X1 )
=> ( eigen__0 @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ! [X1: $i] :
( ( mand @ ( mimplies @ eigen__0 @ eigen__1 ) @ ( mimplies @ eigen__1 @ eigen__0 ) @ X1 )
= ( ~ ( ( mimplies @ eigen__0 @ eigen__1 @ X1 )
=> ~ ( mimplies @ eigen__1 @ eigen__0 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( ( mequiv @ eigen__0 @ eigen__1 )
= ( ^ [X1: $i] :
~ ( ( ( eigen__0 @ X1 )
=> ( eigen__1 @ X1 ) )
=> ~ ( ( eigen__1 @ X1 )
=> ( eigen__0 @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( sP12 = sP3 ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( sP13
= ( ( eigen__0 @ eigen__2 )
=> ( eigen__1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ! [X1: $i > $o] :
( ( mand @ ( mimplies @ eigen__0 @ eigen__1 ) @ X1 )
= ( ^ [X2: $i] :
~ ( ( mimplies @ eigen__0 @ eigen__1 @ X2 )
=> ~ ( X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ! [X1: $i > $o] :
( ( mequiv @ eigen__0 @ X1 )
= ( ^ [X2: $i] :
~ ( ( ( eigen__0 @ X2 )
=> ( X1 @ X2 ) )
=> ~ ( ( X1 @ X2 )
=> ( eigen__0 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ! [X1: $i] :
( ( mimplies @ eigen__1 @ eigen__0 @ X1 )
= ( ( eigen__1 @ X1 )
=> ( eigen__0 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( sP13
=> ~ sP15 ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ! [X1: $i > $o] :
( ( mimplies @ X1 )
= ( ^ [X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ( ( eigen__0 @ eigen__2 )
=> ( eigen__1 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ( ( mimplies @ eigen__1 )
= ( ^ [X1: $i > $o,X2: $i] :
( ( eigen__1 @ X2 )
=> ( X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( sP3 = ( ~ sP31 ) ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(mequiv,conjecture,
( mequiv
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mand @ ( mimplies @ X1 @ X2 ) @ ( mimplies @ X2 @ X1 ) ) ) ) ).
thf(h0,negated_conjecture,
( mequiv
!= ( ^ [X1: $i > $o,X2: $i > $o] : ( mand @ ( mimplies @ X1 @ X2 ) @ ( mimplies @ X2 @ X1 ) ) ) ),
inference(assume_negation,[status(cth)],[mequiv]) ).
thf(h1,assumption,
~ ! [X1: $i > $o] :
( ( mequiv @ X1 )
= ( ^ [X2: $i > $o] : ( mand @ ( mimplies @ X1 @ X2 ) @ ( mimplies @ X2 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h2,assumption,
( ( mequiv @ eigen__0 )
!= ( ^ [X1: $i > $o] : ( mand @ ( mimplies @ eigen__0 @ X1 ) @ ( mimplies @ X1 @ eigen__0 ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ! [X1: $i > $o] :
( ( mequiv @ eigen__0 @ X1 )
= ( mand @ ( mimplies @ eigen__0 @ X1 ) @ ( mimplies @ X1 @ eigen__0 ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
( ( mequiv @ eigen__0 @ eigen__1 )
!= ( mand @ ( mimplies @ eigen__0 @ eigen__1 ) @ ( mimplies @ eigen__1 @ eigen__0 ) ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
~ ! [X1: $i] :
( ( mequiv @ eigen__0 @ eigen__1 @ X1 )
= ( mand @ ( mimplies @ eigen__0 @ eigen__1 ) @ ( mimplies @ eigen__1 @ eigen__0 ) @ X1 ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ sP26,
introduced(assumption,[]) ).
thf(h7,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) )
=> ( mvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ( minvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( msatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mcountersatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ~ ( meq_ind @ ( d2mu @ d_mu_0 ) @ ( d2mu @ d_mu_0 ) @ ( d2unsorted @ d_unsorted_0 ) ) )
=> ~ ( mtrue @ ( d2unsorted @ d_unsorted_0 ) ) ),
introduced(assumption,[]) ).
thf(h8,assumption,
~ ( mfalse @ ( d2unsorted @ d_unsorted_0 ) ),
introduced(assumption,[]) ).
thf(h9,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) )
=> ( mvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ( minvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( msatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mcountersatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ~ ( meq_ind @ ( d2mu @ d_mu_0 ) @ ( d2mu @ d_mu_0 ) @ ( d2unsorted @ d_unsorted_0 ) ) ),
introduced(assumption,[]) ).
thf(h10,assumption,
mtrue @ ( d2unsorted @ d_unsorted_0 ),
introduced(assumption,[]) ).
thf(h11,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) )
=> ( mvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ( minvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( msatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mcountersatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] : ( X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h12,assumption,
meq_ind @ ( d2mu @ d_mu_0 ) @ ( d2mu @ d_mu_0 ) @ ( d2unsorted @ d_unsorted_0 ),
introduced(assumption,[]) ).
thf(h13,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) )
=> ( mvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ( minvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( msatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h14,assumption,
( mcountersatisfiable
= ( ^ [X1: $i > $o] :
~ ! [X2: $i] : ( X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h15,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) )
=> ( mvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ( minvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] :
~ ( X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h16,assumption,
( msatisfiable
= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h17,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) )
=> ( mvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h18,assumption,
( minvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] :
~ ( X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h19,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h20,assumption,
( mvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h21,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h22,assumption,
( mweakly_directed
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h23,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h24,assumption,
( mweakly_connected
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h25,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h26,assumption,
( mweakly_dense
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h27,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h28,assumption,
( mfunctional
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h29,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h30,assumption,
( mpartially_functional
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h31,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h32,assumption,
( meuclidean
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h33,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h34,assumption,
( mtransitive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h35,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h36,assumption,
( mserial
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h37,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h38,assumption,
( msymmetric
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h39,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h40,assumption,
( mreflexive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h41,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h42,assumption,
( mdia
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h43,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h44,assumption,
( mbox
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h45,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h46,assumption,
( mexists_prop
= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h47,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h48,assumption,
( mexists_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h49,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h50,assumption,
( mforall_prop
= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h51,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h52,assumption,
( mforall_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h53,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 ),
introduced(assumption,[]) ).
thf(h54,assumption,
( mxor
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h55,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h56,assumption,
sP18,
introduced(assumption,[]) ).
thf(h57,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 ),
introduced(assumption,[]) ).
thf(h58,assumption,
( mimplied
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h59,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 ),
introduced(assumption,[]) ).
thf(h60,assumption,
sP7,
introduced(assumption,[]) ).
thf(h61,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h62,assumption,
sP1,
introduced(assumption,[]) ).
thf(h63,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h64,assumption,
( mor
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h65,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h66,assumption,
( mnot
= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h67,assumption,
~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h68,assumption,
( meq_prop
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h69,assumption,
~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) ),
introduced(assumption,[]) ).
thf(h70,assumption,
! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h71,assumption,
~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h72,assumption,
! [X1: d_mu] : ( X1 = d_mu_0 ),
introduced(assumption,[]) ).
thf(h73,assumption,
~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h74,assumption,
! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ),
introduced(assumption,[]) ).
thf(h75,assumption,
~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) ),
introduced(assumption,[]) ).
thf(h76,assumption,
! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h77,assumption,
! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) ),
introduced(assumption,[]) ).
thf(h78,assumption,
! [X1: d_unsorted] : ( X1 = d_unsorted_0 ),
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP31
| ~ sP13
| ~ sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( sP31
| sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( sP31
| sP13 ),
inference(prop_rule,[status(thm)],[]) ).
thf(4,plain,
( sP2
| sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(5,plain,
( sP2
| sP33 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP2
| ~ sP33
| ~ sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( ~ sP35
| ~ sP3
| ~ sP31 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( ~ sP35
| sP3
| sP31 ),
inference(prop_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP5
| ~ sP15
| sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP5
| sP15
| ~ sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP27
| ~ sP13
| sP33 ),
inference(prop_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP27
| sP13
| ~ sP33 ),
inference(prop_rule,[status(thm)],[]) ).
thf(13,plain,
( ~ sP9
| ~ sP12
| ~ sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP9
| sP12
| sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP24
| sP35 ),
inference(all_rule,[status(thm)],[]) ).
thf(16,plain,
( ~ sP30
| sP5 ),
inference(all_rule,[status(thm)],[]) ).
thf(17,plain,
( ~ sP8
| sP27 ),
inference(all_rule,[status(thm)],[]) ).
thf(18,plain,
( ~ sP23
| sP9 ),
inference(all_rule,[status(thm)],[]) ).
thf(19,plain,
( ~ sP22
| sP24 ),
inference(prop_rule,[status(thm)],[]) ).
thf(20,plain,
( ~ sP17
| sP30 ),
inference(prop_rule,[status(thm)],[]) ).
thf(21,plain,
( ~ sP14
| sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(22,plain,
( ~ sP25
| sP23 ),
inference(prop_rule,[status(thm)],[]) ).
thf(23,plain,
( ~ sP28
| sP22 ),
inference(all_rule,[status(thm)],[]) ).
thf(24,plain,
( ~ sP10
| sP17 ),
inference(all_rule,[status(thm)],[]) ).
thf(25,plain,
( ~ sP19
| sP14 ),
inference(all_rule,[status(thm)],[]) ).
thf(26,plain,
( ~ sP29
| sP25 ),
inference(all_rule,[status(thm)],[]) ).
thf(27,plain,
( ~ sP11
| sP28 ),
inference(prop_rule,[status(thm)],[]) ).
thf(28,plain,
( ~ sP34
| sP10 ),
inference(prop_rule,[status(thm)],[]) ).
thf(29,plain,
( ~ sP21
| sP19 ),
inference(prop_rule,[status(thm)],[]) ).
thf(30,plain,
( ~ sP4
| sP29 ),
inference(prop_rule,[status(thm)],[]) ).
thf(31,plain,
( ~ sP6
| sP11 ),
inference(all_rule,[status(thm)],[]) ).
thf(32,plain,
( ~ sP32
| sP34 ),
inference(all_rule,[status(thm)],[]) ).
thf(33,plain,
( ~ sP32
| sP21 ),
inference(all_rule,[status(thm)],[]) ).
thf(34,plain,
( ~ sP20
| sP4 ),
inference(all_rule,[status(thm)],[]) ).
thf(35,plain,
( ~ sP1
| sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(36,plain,
( ~ sP7
| sP32 ),
inference(prop_rule,[status(thm)],[]) ).
thf(37,plain,
( ~ sP18
| sP20 ),
inference(prop_rule,[status(thm)],[]) ).
thf(38,plain,
( sP26
| ~ sP12
| ~ sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(39,plain,
( sP26
| sP12
| sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(40,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,h62,h60,h56,h6]) ).
thf(41,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h75,h76,h73,h74,h71,h72,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h77,h78])],[h75,40,h77,h78]) ).
thf(42,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h73,h74,h71,h72,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h75,h76])],[h73,41,h75,h76]) ).
thf(43,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h71,h72,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h73,h74])],[h71,42,h73,h74]) ).
thf(44,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h71,h72])],[h69,43,h71,h72]) ).
thf(45,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h69,h70])],[h67,44,h69,h70]) ).
thf(46,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h67,h68])],[h65,45,h67,h68]) ).
thf(47,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h65,h66])],[h63,46,h65,h66]) ).
thf(48,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h63,h64])],[h61,47,h63,h64]) ).
thf(49,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h61,h62])],[h59,48,h61,h62]) ).
thf(50,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h59,h60])],[h57,49,h59,h60]) ).
thf(51,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h57,h58])],[h55,50,h57,h58]) ).
thf(52,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h53,h54,h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h55,h56])],[h53,51,h55,h56]) ).
thf(53,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h51,h52,h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h53,h54])],[h51,52,h53,h54]) ).
thf(54,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h49,h50,h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h51,h52])],[h49,53,h51,h52]) ).
thf(55,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h47,h48,h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h49,h50])],[h47,54,h49,h50]) ).
thf(56,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h45,h46,h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h47,h48])],[h45,55,h47,h48]) ).
thf(57,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h43,h44,h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h45,h46])],[h43,56,h45,h46]) ).
thf(58,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h41,h42,h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h43,h44])],[h41,57,h43,h44]) ).
thf(59,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h39,h40,h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h41,h42])],[h39,58,h41,h42]) ).
thf(60,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h37,h38,h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h39,h40])],[h37,59,h39,h40]) ).
thf(61,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h35,h36,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h37,h38])],[h35,60,h37,h38]) ).
thf(62,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h35,h36])],[h33,61,h35,h36]) ).
thf(63,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h33,h34])],[h31,62,h33,h34]) ).
thf(64,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h31,h32])],[h29,63,h31,h32]) ).
thf(65,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h29,h30])],[h27,64,h29,h30]) ).
thf(66,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h27,h28])],[h25,65,h27,h28]) ).
thf(67,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h25,h26])],[h23,66,h25,h26]) ).
thf(68,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h23,h24])],[h21,67,h23,h24]) ).
thf(69,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h21,h22])],[h19,68,h21,h22]) ).
thf(70,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h19,h20])],[h17,69,h19,h20]) ).
thf(71,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h17,h18])],[h15,70,h17,h18]) ).
thf(72,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h13,h14,h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h15,h16])],[h13,71,h15,h16]) ).
thf(73,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h11,h12,h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h13,h14])],[h11,72,h13,h14]) ).
thf(74,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h9,h10,h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h11,h12])],[h9,73,h11,h12]) ).
thf(75,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h7,h8,h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h9,h10])],[h7,74,h9,h10]) ).
thf(lcl698_1,axiom,
~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ~ ( ! [X1: $i] :
~ ! [X2: d_unsorted] :
( X1
!= ( d2unsorted @ X2 ) )
=> ~ ! [X1: d_unsorted] : ( X1 = d_unsorted_0 ) )
=> ~ ! [X1: d_unsorted,X2: d_unsorted] :
( ( ( d2unsorted @ X1 )
= ( d2unsorted @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: mu] :
~ ! [X2: d_mu] :
( X1
!= ( d2mu @ X2 ) ) )
=> ~ ! [X1: d_mu] : ( X1 = d_mu_0 ) )
=> ~ ! [X1: d_mu,X2: d_mu] :
( ( ( d2mu @ X1 )
= ( d2mu @ X2 ) )
=> ( X1 = X2 ) ) )
=> ( meq_prop
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) )
=> ( mnot
!= ( ^ [X1: $i > $o,X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ~ ( X1 @ X3 )
=> ( X2 @ X3 ) ) ) ) )
=> ~ sP1 )
=> ~ sP7 )
=> ( mimplied
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) )
=> ~ sP18 )
=> ( mxor
!= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( ( X1 @ X3 )
=> ( X2 @ X3 ) )
=> ~ ( ( X2 @ X3 )
=> ( X1 @ X3 ) ) ) ) ) )
=> ( mforall_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mforall_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_ind
!= ( ^ [X1: mu > $i > $o,X2: $i] :
~ ! [X3: mu] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mexists_prop
!= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
~ ! [X3: $i > $o] :
~ ( X1 @ X3 @ X2 ) ) ) )
=> ( mbox
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ( X2 @ X4 ) ) ) ) )
=> ( mdia
!= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ( X2 @ X4 ) ) ) ) )
=> ( mreflexive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) )
=> ( msymmetric
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ( X1 @ X3 @ X2 ) ) ) ) )
=> ( mserial
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
~ ( X1 @ X2 @ X3 ) ) ) )
=> ( mtransitive
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X3 @ X4 ) )
=> ( X1 @ X2 @ X4 ) ) ) ) )
=> ( meuclidean
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) ) ) ) )
=> ( mpartially_functional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) )
=> ( mfunctional
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( mweakly_dense
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ( X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X2 @ X4 )
=> ~ ( X1 @ X4 @ X3 ) ) ) ) ) )
=> ( mweakly_connected
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ~ ( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ( X1 @ X3 @ X4 ) )
=> ( X3 = X4 ) )
=> ( X1 @ X4 @ X3 ) ) ) ) )
=> ( mweakly_directed
!= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ~ ( ( X1 @ X2 @ X3 )
=> ~ ( X1 @ X2 @ X4 ) )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ~ ( X1 @ X4 @ X5 ) ) ) ) ) )
=> ( mvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ( minvalid
!= ( ^ [X1: $i > $o] :
! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( msatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) )
=> ( mcountersatisfiable
!= ( ^ [X1: $i > $o] :
~ ! [X2: $i] : ( X1 @ X2 ) ) ) )
=> ~ ( meq_ind @ ( d2mu @ d_mu_0 ) @ ( d2mu @ d_mu_0 ) @ ( d2unsorted @ d_unsorted_0 ) ) )
=> ~ ( mtrue @ ( d2unsorted @ d_unsorted_0 ) ) )
=> ( mfalse @ ( d2unsorted @ d_unsorted_0 ) ) ) ).
thf(76,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h6,h5,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h7,h8])],[lcl698_1,75,h7,h8]) ).
thf(77,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h5,h4,h3,h2,h1,h0]),tab_negall(discharge,[h6]),tab_negall(eigenvar,eigen__2)],[h5,76,h6]) ).
thf(78,plain,
$false,
inference(tab_fe,[status(thm),assumptions([h4,h3,h2,h1,h0]),tab_fe(discharge,[h5])],[h4,77,h5]) ).
thf(79,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h2,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,78,h4]) ).
thf(80,plain,
$false,
inference(tab_fe,[status(thm),assumptions([h2,h1,h0]),tab_fe(discharge,[h3])],[h2,79,h3]) ).
thf(81,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,80,h2]) ).
thf(82,plain,
$false,
inference(tab_fe,[status(thm),assumptions([h0]),tab_fe(discharge,[h1])],[h0,81,h1]) ).
thf(0,theorem,
( mequiv
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mand @ ( mimplies @ X1 @ X2 ) @ ( mimplies @ X2 @ X1 ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[82,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWX153_1 : TPTP v9.3.0. Released v9.3.0.
% 0.00/0.12 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.17/0.34 % Computer : n021.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue May 5 09:28:40 EDT 2026
% 0.17/0.34 % CPUTime :
% 21.24/21.43 % SZS status Theorem
% 21.24/21.43 % Mode: cade22grackle2x798d
% 21.24/21.43 % Steps: 9151
% 21.24/21.43 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------