TSTP Solution File: ALG263^1 by Satallax---3.5
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- Process Solution
%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : ALG263^1 : TPTP v8.1.0. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n018.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 : 600s
% DateTime : Thu Jul 14 17:57:49 EDT 2022
% Result : Theorem 44.62s 44.98s
% Output : Proof 44.62s
% Verified :
% SZS Type : Refutation
% Derivation depth : 4
% Number of leaves : 157
% Syntax : Number of formulae : 164 ( 50 unt; 13 typ; 41 def)
% Number of atoms : 589 ( 205 equ; 0 cnn)
% Maximal formula atoms : 35 ( 3 avg)
% Number of connectives : 2079 ( 107 ~; 53 |; 0 &;1462 @)
% ( 51 <=>; 398 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 4 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 111 ( 111 >; 0 *; 0 +; 0 <<)
% Number of symbols : 106 ( 103 usr; 98 con; 0-3 aty)
% ( 8 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 555 ( 10 ^ 545 !; 0 ?; 555 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_subst,type,
subst: $tType ).
thf(ty_term,type,
term: $tType ).
thf(ty_push,type,
push: term > subst > subst ).
thf(ty_eigen__1,type,
eigen__1: term ).
thf(ty_eigen__0,type,
eigen__0: subst > term > subst > $o ).
thf(ty_ap,type,
ap: term > term > term ).
thf(ty_id,type,
id: subst ).
thf(ty_one,type,
one: term ).
thf(ty_sub,type,
sub: term > subst > term ).
thf(ty_lam,type,
lam: term > term ).
thf(ty_sh,type,
sh: subst ).
thf(ty_var,type,
var: term > $o ).
thf(ty_comp,type,
comp: subst > subst > subst ).
thf(h0,assumption,
! [X1: term > $o,X2: term] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: term] :
~ ( ( var @ X1 )
=> ( eigen__0 @ id @ X1 @ id ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(h1,assumption,
! [X1: ( subst > term > subst > $o ) > $o,X2: subst > term > subst > $o] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__1 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__0,definition,
( eigen__0
= ( eps__1
@ ^ [X1: subst > term > subst > $o] :
~ ( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) )
=> ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 ) )
=> ( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 )
=> ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) ) )
=> ( ! [X2: term] :
( ( var @ ( sub @ X2 @ id ) )
=> ( X1 @ id @ X2 @ id ) )
=> ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ id @ X2 @ id ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__0])]) ).
thf(sP1,plain,
( sP1
<=> ( ! [X1: term,X2: term] :
( ( ( lam @ X1 )
= ( lam @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term] :
( ( ( sub @ X1 @ sh )
= ( sub @ X2 @ sh ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term,X3: term] :
( ( lam @ X1 )
!= ( ap @ X2 @ X3 ) )
=> ( ! [X1: term,X2: term] :
~ ( var @ ( ap @ X1 @ X2 ) )
=> ( ! [X1: term] :
~ ( var @ ( lam @ X1 ) )
=> ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ( X1 @ X2 )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) )
=> ( ! [X1: term > $o,X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( ( X1 @ X2 )
=> ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ ( push @ X2 @ X3 ) ) ) ) ) )
=> ( ! [X1: term > $o] :
( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( X1 @ ( sub @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) )
=> ( ~ ( ~ ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 ) )
=> ~ ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 ) )
=> ( ~ ( ! [X1: term,X2: subst,X3: subst] :
( ( sub @ ( sub @ X1 @ X2 ) @ X3 )
= ( sub @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: term] :
( ( sub @ X1 @ id )
= X1 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X3 = X4 ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: subst > term > term] :
( ! [X3: subst,X4: term,X5: subst] :
( ( sub @ ( X2 @ X3 @ X4 ) @ X5 )
= ( X2 @ ( comp @ X3 @ X5 ) @ ( sub @ X4 @ X5 ) ) )
=> ( ( ( lam @ ( X1 @ sh @ one ) )
= ( lam @ ( X2 @ sh @ one ) ) )
=> ! [X3: subst,X4: term] :
( ( X1 @ X3 @ X4 )
= ( X2 @ X3 @ X4 ) ) ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: term,X3: term] :
( ( lam @ ( X1 @ sh @ one ) )
!= ( ap @ ( sub @ X2 @ id ) @ X3 ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ~ ( var @ ( sub @ ( lam @ ( X1 @ sh @ one ) ) @ id ) ) )
=> ( ! [X1: term,X2: term] :
~ ( var @ ( sub @ ( ap @ ( sub @ X1 @ id ) @ X2 ) @ id ) )
=> ! [X1: subst > term > subst > $o] :
( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) )
=> ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 ) )
=> ( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 )
=> ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) ) )
=> ( ! [X2: term] :
( ( var @ ( sub @ X2 @ id ) )
=> ( X1 @ id @ X2 @ id ) )
=> ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ id @ X2 @ id ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ! [X1: term,X2: term] :
~ ( var @ ( sub @ ( ap @ ( sub @ X1 @ id ) @ X2 ) @ id ) )
=> ! [X1: subst > term > subst > $o] :
( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) )
=> ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 ) )
=> ( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 )
=> ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) ) )
=> ( ! [X2: term] :
( ( var @ ( sub @ X2 @ id ) )
=> ( X1 @ id @ X2 @ id ) )
=> ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ id @ X2 @ id ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ! [X1: term > $o] :
( ( X1 @ one )
=> ( ! [X2: term] :
( ( var @ X2 )
=> ( ( X1 @ X2 )
=> ( X1 @ ( sub @ X2 @ sh ) ) ) )
=> ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) ) ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ X1 @ X3 )
= ( ap @ X2 @ X4 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ X1 @ X3 )
= ( ap @ X2 @ X4 ) )
=> ( X3 = X4 ) )
=> sP1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ! [X1: term,X2: subst] :
( ( sub @ ( lam @ X1 ) @ X2 )
= ( lam @ ( sub @ X1 @ ( push @ one @ ( comp @ X2 @ sh ) ) ) ) )
=> ( ! [X1: term,X2: subst,X3: subst] :
( ( sub @ ( sub @ X1 @ X2 ) @ X3 )
= ( sub @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
=> ( ! [X1: term,X2: subst] :
( ( comp @ sh @ ( push @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ( ! [X1: term,X2: subst,X3: subst] :
( ( comp @ ( push @ X1 @ X2 ) @ X3 )
= ( push @ ( sub @ X1 @ X3 ) @ ( comp @ X2 @ X3 ) ) )
=> ( ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 )
=> ( ( ( push @ one @ sh )
= id )
=> ( ! [X1: subst] :
( ( push @ ( sub @ one @ X1 ) @ ( comp @ sh @ X1 ) )
= X1 )
=> ( ( var @ one )
=> ( ! [X1: term] :
( ( var @ X1 )
=> ( var @ ( sub @ X1 @ sh ) ) )
=> sP3 ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: subst > term > subst > $o] :
( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) )
=> ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 ) )
=> ( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 )
=> ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) ) )
=> ( ! [X2: term] :
( ( var @ ( sub @ X2 @ id ) )
=> ( X1 @ id @ X2 @ id ) )
=> ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ id @ X2 @ id ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ X1 @ X3 )
= ( ap @ X2 @ X4 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ X1 @ X3 )
= ( ap @ X2 @ X4 ) )
=> ( X3 = X4 ) )
=> sP1 ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( var @ ( sub @ eigen__1 @ id ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: term] :
( ( var @ X1 )
=> ( eigen__0 @ id @ X1 @ id ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ X1 @ X3 )
= ( ap @ X2 @ X4 ) )
=> ( X3 = X4 ) )
=> sP1 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ! [X1: term] :
( ( var @ ( sub @ X1 @ id ) )
=> ( eigen__0 @ id @ X1 @ id ) )
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ! [X1: term] :
( ( var @ X1 )
=> ( var @ ( sub @ X1 @ sh ) ) )
=> sP3 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( ! [X1: term] :
( ( sub @ X1 @ id )
= X1 )
=> sP4 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ! [X1: subst,X2: term,X3: subst,X4: subst] :
( ( eigen__0 @ ( comp @ X1 @ X4 ) @ ( sub @ X2 @ X4 ) @ X3 )
=> ( eigen__0 @ X1 @ X2 @ ( comp @ X4 @ X3 ) ) )
=> sP10 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ! [X1: term,X2: term,X3: term] :
( ( lam @ X1 )
!= ( ap @ X2 @ X3 ) )
=> ( ! [X1: term,X2: term] :
~ ( var @ ( ap @ X1 @ X2 ) )
=> ( ! [X1: term] :
~ ( var @ ( lam @ X1 ) )
=> ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ( X1 @ X2 )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) )
=> ( ! [X1: term > $o,X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( ( X1 @ X2 )
=> ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ ( push @ X2 @ X3 ) ) ) ) ) )
=> ( ! [X1: term > $o] :
( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( X1 @ ( sub @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) )
=> ( ~ ( ~ ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 ) )
=> ~ ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 ) )
=> ( ~ ( ! [X1: term,X2: subst,X3: subst] :
( ( sub @ ( sub @ X1 @ X2 ) @ X3 )
= ( sub @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: term] :
( ( sub @ X1 @ id )
= X1 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X3 = X4 ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: subst > term > term] :
( ! [X3: subst,X4: term,X5: subst] :
( ( sub @ ( X2 @ X3 @ X4 ) @ X5 )
= ( X2 @ ( comp @ X3 @ X5 ) @ ( sub @ X4 @ X5 ) ) )
=> ( ( ( lam @ ( X1 @ sh @ one ) )
= ( lam @ ( X2 @ sh @ one ) ) )
=> ! [X3: subst,X4: term] :
( ( X1 @ X3 @ X4 )
= ( X2 @ X3 @ X4 ) ) ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: term,X3: term] :
( ( lam @ ( X1 @ sh @ one ) )
!= ( ap @ ( sub @ X2 @ id ) @ X3 ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ~ ( var @ ( sub @ ( lam @ ( X1 @ sh @ one ) ) @ id ) ) )
=> sP2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ~ ( ! [X1: term,X2: subst,X3: subst] :
( ( sub @ ( sub @ X1 @ X2 ) @ X3 )
= ( sub @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: term] :
( ( sub @ X1 @ id )
= X1 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X3 = X4 ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: subst > term > term] :
( ! [X3: subst,X4: term,X5: subst] :
( ( sub @ ( X2 @ X3 @ X4 ) @ X5 )
= ( X2 @ ( comp @ X3 @ X5 ) @ ( sub @ X4 @ X5 ) ) )
=> ( ( ( lam @ ( X1 @ sh @ one ) )
= ( lam @ ( X2 @ sh @ one ) ) )
=> ! [X3: subst,X4: term] :
( ( X1 @ X3 @ X4 )
= ( X2 @ X3 @ X4 ) ) ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: term,X3: term] :
( ( lam @ ( X1 @ sh @ one ) )
!= ( ap @ ( sub @ X2 @ id ) @ X3 ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ~ ( var @ ( sub @ ( lam @ ( X1 @ sh @ one ) ) @ id ) ) )
=> sP2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) )
=> ( ~ ( ~ ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 ) )
=> ~ ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 ) )
=> sP15 ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ( X1 @ X2 )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) )
=> ( ! [X1: term > $o,X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( ( X1 @ X2 )
=> ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ ( push @ X2 @ X3 ) ) ) ) ) )
=> ( ! [X1: term > $o] :
( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( X1 @ ( sub @ X2 @ X3 ) ) ) ) )
=> sP16 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( eigen__0 @ id @ eigen__1 @ id ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ~ ( var @ ( sub @ ( lam @ ( X1 @ sh @ one ) ) @ id ) ) )
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( sP7
=> sP18 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: term] :
( ( sub @ X1 @ id )
= X1 ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( ( var @ eigen__1 )
=> sP18 ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( ! [X1: term > $o,X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( ( X1 @ X2 )
=> ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ ( push @ X2 @ X3 ) ) ) ) ) )
=> ( ! [X1: term > $o] :
( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( X1 @ ( sub @ X2 @ X3 ) ) ) ) )
=> sP16 ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ! [X1: term] :
( ( ( sub @ eigen__1 @ id )
= X1 )
=> ( X1
= ( sub @ eigen__1 @ id ) ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( ~ ( ~ ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 ) )
=> ~ ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 ) )
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( ( sub @ eigen__1 @ id )
= eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( ! [X1: term,X2: subst,X3: subst] :
( ( comp @ ( push @ X1 @ X2 ) @ X3 )
= ( push @ ( sub @ X1 @ X3 ) @ ( comp @ X2 @ X3 ) ) )
=> ( ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 )
=> ( ( ( push @ one @ sh )
= id )
=> ( ! [X1: subst] :
( ( push @ ( sub @ one @ X1 ) @ ( comp @ sh @ X1 ) )
= X1 )
=> ( ( var @ one )
=> sP11 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ( ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 )
=> ( ( ( push @ one @ sh )
= id )
=> ( ! [X1: subst] :
( ( push @ ( sub @ one @ X1 ) @ ( comp @ sh @ X1 ) )
= X1 )
=> ( ( var @ one )
=> sP11 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( ! [X1: term] :
~ ( var @ ( lam @ X1 ) )
=> sP17 ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ( ! [X1: term,X2: subst] :
( ( comp @ sh @ ( push @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> sP27 ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( ! [X1: term > $o] :
( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( X1 @ ( sub @ X2 @ X3 ) ) ) ) )
=> sP16 ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ( ! [X1: term,X2: subst,X3: subst] :
( ( sub @ ( sub @ X1 @ X2 ) @ X3 )
= ( sub @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
=> sP30 ) ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 )
=> sP30 ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X3 = X4 ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: subst > term > term] :
( ! [X3: subst,X4: term,X5: subst] :
( ( sub @ ( X2 @ X3 @ X4 ) @ X5 )
= ( X2 @ ( comp @ X3 @ X5 ) @ ( sub @ X4 @ X5 ) ) )
=> ( ( ( lam @ ( X1 @ sh @ one ) )
= ( lam @ ( X2 @ sh @ one ) ) )
=> ! [X3: subst,X4: term] :
( ( X1 @ X3 @ X4 )
= ( X2 @ X3 @ X4 ) ) ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: term,X3: term] :
( ( lam @ ( X1 @ sh @ one ) )
!= ( ap @ ( sub @ X2 @ id ) @ X3 ) ) )
=> sP19 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( ( ( push @ one @ sh )
= id )
=> ( ! [X1: subst] :
( ( push @ ( sub @ one @ X1 ) @ ( comp @ sh @ X1 ) )
= X1 )
=> ( ( var @ one )
=> sP11 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ! [X1: term] :
( ( var @ ( sub @ X1 @ id ) )
=> ( eigen__0 @ id @ X1 @ id ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( ! [X1: term,X2: subst] :
( ( sub @ one @ ( push @ X1 @ X2 ) )
= X1 )
=> sP12 ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ( ! [X1: subst] :
( ( push @ ( sub @ one @ X1 ) @ ( comp @ sh @ X1 ) )
= X1 )
=> ( ( var @ one )
=> sP11 ) ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ( ! [X1: term,X2: term,X3: subst] :
( ( sub @ ( ap @ X1 @ X2 ) @ X3 )
= ( ap @ ( sub @ X1 @ X3 ) @ ( sub @ X2 @ X3 ) ) )
=> sP37 ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ ( sub @ X1 @ id ) @ X3 )
= ( ap @ ( sub @ X2 @ id ) @ X4 ) )
=> ( X3 = X4 ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: subst > term > term] :
( ! [X3: subst,X4: term,X5: subst] :
( ( sub @ ( X2 @ X3 @ X4 ) @ X5 )
= ( X2 @ ( comp @ X3 @ X5 ) @ ( sub @ X4 @ X5 ) ) )
=> ( ( ( lam @ ( X1 @ sh @ one ) )
= ( lam @ ( X2 @ sh @ one ) ) )
=> ! [X3: subst,X4: term] :
( ( X1 @ X3 @ X4 )
= ( X2 @ X3 @ X4 ) ) ) ) )
=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: term,X3: term] :
( ( lam @ ( X1 @ sh @ one ) )
!= ( ap @ ( sub @ X2 @ id ) @ X3 ) ) )
=> sP19 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ( ! [X1: term,X2: term] :
( ( ( sub @ X1 @ sh )
= ( sub @ X2 @ sh ) )
=> ( X1 = X2 ) )
=> sP14 ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ! [X1: term,X2: term] :
( ( X1 = X2 )
=> ( X2 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ( eigen__1
= ( sub @ eigen__1 @ id ) ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ( ( var @ one )
=> sP11 ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( sP26
=> sP43 ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ( ! [X1: subst,X2: term,X3: subst,X4: subst] :
( ( eigen__0 @ X1 @ X2 @ ( comp @ X4 @ X3 ) )
=> ( eigen__0 @ ( comp @ X1 @ X4 ) @ ( sub @ X2 @ X4 ) @ X3 ) )
=> sP13 ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> sP27 ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: term,X3: term] :
( ( lam @ ( X1 @ sh @ one ) )
!= ( ap @ ( sub @ X2 @ id ) @ X3 ) ) )
=> sP19 ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ( var @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ( ! [X1: term,X2: term] :
~ ( var @ ( ap @ X1 @ X2 ) )
=> sP29 ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: subst > term > term] :
( ! [X3: subst,X4: term,X5: subst] :
( ( sub @ ( X2 @ X3 @ X4 ) @ X5 )
= ( X2 @ ( comp @ X3 @ X5 ) @ ( sub @ X4 @ X5 ) ) )
=> ( ( ( lam @ ( X1 @ sh @ one ) )
= ( lam @ ( X2 @ sh @ one ) ) )
=> ! [X3: subst,X4: term] :
( ( X1 @ X3 @ X4 )
= ( X2 @ X3 @ X4 ) ) ) ) )
=> sP48 ) ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(def_axapp,definition,
( axapp
= ( ! [X1: term,X2: term,X3: subst] :
( ( sub @ ( ap @ X1 @ X2 ) @ X3 )
= ( ap @ ( sub @ X1 @ X3 ) @ ( sub @ X2 @ X3 ) ) ) ) ) ).
thf(def_axvarcons,definition,
( axvarcons
= ( ! [X1: term,X2: subst] :
( ( sub @ one @ ( push @ X1 @ X2 ) )
= X1 ) ) ) ).
thf(def_axvarid,definition,
axvarid = sP21 ).
thf(def_axabs,definition,
( axabs
= ( ! [X1: term,X2: subst] :
( ( sub @ ( lam @ X1 ) @ X2 )
= ( lam @ ( sub @ X1 @ ( push @ one @ ( comp @ X2 @ sh ) ) ) ) ) ) ) ).
thf(def_axclos,definition,
( axclos
= ( ! [X1: term,X2: subst,X3: subst] :
( ( sub @ ( sub @ X1 @ X2 ) @ X3 )
= ( sub @ X1 @ ( comp @ X2 @ X3 ) ) ) ) ) ).
thf(def_axidl,definition,
( axidl
= ( ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 ) ) ) ).
thf(def_axshiftcons,definition,
( axshiftcons
= ( ! [X1: term,X2: subst] :
( ( comp @ sh @ ( push @ X1 @ X2 ) )
= X2 ) ) ) ).
thf(def_axassoc,definition,
( axassoc
= ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) ) ) ) ).
thf(def_axmap,definition,
( axmap
= ( ! [X1: term,X2: subst,X3: subst] :
( ( comp @ ( push @ X1 @ X2 ) @ X3 )
= ( push @ ( sub @ X1 @ X3 ) @ ( comp @ X2 @ X3 ) ) ) ) ) ).
thf(def_axidr,definition,
( axidr
= ( ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 ) ) ) ).
thf(def_axvarshift,definition,
( axvarshift
= ( ( push @ one @ sh )
= id ) ) ).
thf(def_axscons,definition,
( axscons
= ( ! [X1: subst] :
( ( push @ ( sub @ one @ X1 ) @ ( comp @ sh @ X1 ) )
= X1 ) ) ) ).
thf(def_ulamvar1,definition,
( ulamvar1
= ( var @ one ) ) ).
thf(def_ulamvarsh,definition,
( ulamvarsh
= ( ! [X1: term] :
( ( var @ X1 )
=> ( var @ ( sub @ X1 @ sh ) ) ) ) ) ).
thf(def_ulamvarind,definition,
( ulamvarind
= ( ! [X1: term > $o] :
( ( X1 @ one )
=> ( ! [X2: term] :
( ( var @ X2 )
=> ( ( X1 @ X2 )
=> ( X1 @ ( sub @ X2 @ sh ) ) ) )
=> ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_apinj1,definition,
( apinj1
= ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ X1 @ X3 )
= ( ap @ X2 @ X4 ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_apinj2,definition,
( apinj2
= ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( ap @ X1 @ X3 )
= ( ap @ X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) ).
thf(def_laminj,definition,
( laminj
= ( ! [X1: term,X2: term] :
( ( ( lam @ X1 )
= ( lam @ X2 ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_shinj,definition,
( shinj
= ( ! [X1: term,X2: term] :
( ( ( sub @ X1 @ sh )
= ( sub @ X2 @ sh ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_lamnotap,definition,
( lamnotap
= ( ! [X1: term,X2: term,X3: term] :
( ( lam @ X1 )
!= ( ap @ X2 @ X3 ) ) ) ) ).
thf(def_apnotvar,definition,
( apnotvar
= ( ! [X1: term,X2: term] :
~ ( var @ ( ap @ X1 @ X2 ) ) ) ) ).
thf(def_lamnotvar,definition,
( lamnotvar
= ( ! [X1: term] :
~ ( var @ ( lam @ X1 ) ) ) ) ).
thf(def_induction,definition,
( induction
= ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ( X1 @ X2 )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) ) ) ) ).
thf(def_pushprop,definition,
( pushprop
= ( ! [X1: term > $o,X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( ( X1 @ X2 )
=> ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ ( push @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_induction2lem,definition,
( induction2lem
= ( ! [X1: term > $o] :
( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( X1 @ ( sub @ X2 @ X3 ) ) ) ) ) ) ) ).
thf(def_induction2,definition,
( induction2
= ( ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ( !! @ X1 ) ) ) ) ) ) ).
thf(def_substmonoid,definition,
( substmonoid
= ( ~ ( ~ ( ! [X1: subst,X2: subst,X3: subst] :
( ( comp @ ( comp @ X1 @ X2 ) @ X3 )
= ( comp @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ ! [X1: subst] :
( ( comp @ id @ X1 )
= X1 ) )
=> ~ ! [X1: subst] :
( ( comp @ X1 @ id )
= X1 ) ) ) ) ).
thf(def_termmset,definition,
( termmset
= ( ~ ( ! [X1: term,X2: subst,X3: subst] :
( ( sub @ ( sub @ X1 @ X2 ) @ X3 )
= ( sub @ X1 @ ( comp @ X2 @ X3 ) ) )
=> ~ sP21 ) ) ) ).
thf(def_hoasap,definition,
( hoasap
= ( ^ [X1: subst,X2: term,X3: subst] : ( ap @ ( sub @ X2 @ X3 ) ) ) ) ).
thf(def_hoaslam,definition,
( hoaslam
= ( ^ [X1: subst,X2: subst > term > term] : ( lam @ ( X2 @ sh @ one ) ) ) ) ).
thf(def_hoasvar,definition,
( hoasvar
= ( ^ [X1: subst,X2: term,X3: subst] : ( var @ ( sub @ X2 @ X3 ) ) ) ) ).
thf(def_hoasapinj1,definition,
( hoasapinj1
= ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( hoasap @ id @ X1 @ id @ X3 )
= ( hoasap @ id @ X2 @ id @ X4 ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_hoasapinj2,definition,
( hoasapinj2
= ( ! [X1: term,X2: term,X3: term,X4: term] :
( ( ( hoasap @ id @ X1 @ id @ X3 )
= ( hoasap @ id @ X2 @ id @ X4 ) )
=> ( X3 = X4 ) ) ) ) ).
thf(def_hoaslaminj,definition,
( hoaslaminj
= ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: subst > term > term] :
( ! [X3: subst,X4: term,X5: subst] :
( ( sub @ ( X2 @ X3 @ X4 ) @ X5 )
= ( X2 @ ( comp @ X3 @ X5 ) @ ( sub @ X4 @ X5 ) ) )
=> ( ( ( hoaslam @ id @ X1 )
= ( hoaslam @ id @ X2 ) )
=> ! [X3: subst,X4: term] :
( ( X1 @ X3 @ X4 )
= ( X2 @ X3 @ X4 ) ) ) ) ) ) ) ).
thf(def_hoaslamnotap,definition,
( hoaslamnotap
= ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ! [X2: term,X3: term] :
( ( hoaslam @ id @ X1 )
!= ( hoasap @ id @ X2 @ id @ X3 ) ) ) ) ) ).
thf(def_hoaslamnotvar,definition,
( hoaslamnotvar
= ( ! [X1: subst > term > term] :
( ! [X2: subst,X3: term,X4: subst] :
( ( sub @ ( X1 @ X2 @ X3 ) @ X4 )
= ( X1 @ ( comp @ X2 @ X4 ) @ ( sub @ X3 @ X4 ) ) )
=> ~ ( hoasvar @ id @ ( hoaslam @ id @ X1 ) @ id ) ) ) ) ).
thf(def_hoasapnotvar,definition,
( hoasapnotvar
= ( ! [X1: term,X2: term] :
~ ( hoasvar @ id @ ( hoasap @ id @ X1 @ id @ X2 ) @ id ) ) ) ).
thf(def_hoasinduction_lem1,definition,
( hoasinduction_lem1
= ( ! [X1: subst > term > subst > $o] :
( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) )
=> ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 ) )
=> ( ! [X2: subst,X3: term,X4: subst,X5: subst] :
( ( X1 @ ( comp @ X2 @ X5 ) @ ( sub @ X3 @ X5 ) @ X4 )
=> ( X1 @ X2 @ X3 @ ( comp @ X5 @ X4 ) ) )
=> ( ! [X2: term] :
( ( hoasvar @ id @ X2 @ id )
=> ( X1 @ id @ X2 @ id ) )
=> ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ id @ X2 @ id ) ) ) ) ) ) ) ).
thf(def_hoasinduction_lem1_gthm,definition,
( hoasinduction_lem1_gthm
= ( axapp
=> ( axvarcons
=> ( axvarid
=> ( axabs
=> ( axclos
=> ( axidl
=> ( axshiftcons
=> ( axassoc
=> ( axmap
=> ( axidr
=> ( axvarshift
=> ( axscons
=> ( ulamvar1
=> ( ulamvarsh
=> ( ulamvarind
=> ( apinj1
=> ( apinj2
=> ( laminj
=> ( shinj
=> ( lamnotap
=> ( apnotvar
=> ( lamnotvar
=> ( induction
=> ( pushprop
=> ( induction2lem
=> ( induction2
=> ( substmonoid
=> ( termmset
=> ( hoasapinj1
=> ( hoasapinj2
=> ( hoaslaminj
=> ( hoaslamnotap
=> ( hoaslamnotvar
=> ( hoasapnotvar
=> hoasinduction_lem1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(thm,conjecture,
sP39 ).
thf(h2,negated_conjecture,
~ sP39,
inference(assume_negation,[status(cth)],[thm]) ).
thf(1,plain,
( ~ sP45
| ~ sP26
| sP43 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP24
| sP45 ),
inference(all_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP42
| sP24 ),
inference(all_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP21
| sP26 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
sP42,
inference(eq_sym,[status(thm)],]) ).
thf(6,plain,
( ~ sP49
| sP7
| ~ sP43 ),
inference(mating_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP36
| sP20 ),
inference(all_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP20
| ~ sP7
| sP18 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( sP22
| ~ sP18 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( sP22
| sP49 ),
inference(prop_rule,[status(thm)],]) ).
thf(11,plain,
( sP8
| ~ sP22 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(12,plain,
( sP10
| ~ sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(13,plain,
( sP10
| sP36 ),
inference(prop_rule,[status(thm)],]) ).
thf(14,plain,
( sP13
| ~ sP10 ),
inference(prop_rule,[status(thm)],]) ).
thf(15,plain,
( sP46
| ~ sP13 ),
inference(prop_rule,[status(thm)],]) ).
thf(16,plain,
( sP5
| ~ sP46 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__0]) ).
thf(17,plain,
( sP2
| ~ sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(18,plain,
( sP19
| ~ sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(19,plain,
( sP48
| ~ sP19 ),
inference(prop_rule,[status(thm)],]) ).
thf(20,plain,
( sP51
| ~ sP48 ),
inference(prop_rule,[status(thm)],]) ).
thf(21,plain,
( sP40
| ~ sP51 ),
inference(prop_rule,[status(thm)],]) ).
thf(22,plain,
( sP34
| ~ sP40 ),
inference(prop_rule,[status(thm)],]) ).
thf(23,plain,
( sP15
| ~ sP34 ),
inference(prop_rule,[status(thm)],]) ).
thf(24,plain,
( sP25
| ~ sP15 ),
inference(prop_rule,[status(thm)],]) ).
thf(25,plain,
( sP16
| ~ sP25 ),
inference(prop_rule,[status(thm)],]) ).
thf(26,plain,
( sP31
| ~ sP16 ),
inference(prop_rule,[status(thm)],]) ).
thf(27,plain,
( sP23
| ~ sP31 ),
inference(prop_rule,[status(thm)],]) ).
thf(28,plain,
( sP17
| ~ sP23 ),
inference(prop_rule,[status(thm)],]) ).
thf(29,plain,
( sP29
| ~ sP17 ),
inference(prop_rule,[status(thm)],]) ).
thf(30,plain,
( sP50
| ~ sP29 ),
inference(prop_rule,[status(thm)],]) ).
thf(31,plain,
( sP14
| ~ sP50 ),
inference(prop_rule,[status(thm)],]) ).
thf(32,plain,
( sP41
| ~ sP14 ),
inference(prop_rule,[status(thm)],]) ).
thf(33,plain,
( sP1
| ~ sP41 ),
inference(prop_rule,[status(thm)],]) ).
thf(34,plain,
( sP9
| ~ sP1 ),
inference(prop_rule,[status(thm)],]) ).
thf(35,plain,
( sP6
| ~ sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(36,plain,
( sP3
| ~ sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(37,plain,
( sP11
| ~ sP3 ),
inference(prop_rule,[status(thm)],]) ).
thf(38,plain,
( sP44
| ~ sP11 ),
inference(prop_rule,[status(thm)],]) ).
thf(39,plain,
( sP38
| ~ sP44 ),
inference(prop_rule,[status(thm)],]) ).
thf(40,plain,
( sP35
| ~ sP38 ),
inference(prop_rule,[status(thm)],]) ).
thf(41,plain,
( sP28
| ~ sP35 ),
inference(prop_rule,[status(thm)],]) ).
thf(42,plain,
( sP27
| ~ sP28 ),
inference(prop_rule,[status(thm)],]) ).
thf(43,plain,
( sP47
| ~ sP27 ),
inference(prop_rule,[status(thm)],]) ).
thf(44,plain,
( sP30
| ~ sP47 ),
inference(prop_rule,[status(thm)],]) ).
thf(45,plain,
( sP33
| ~ sP30 ),
inference(prop_rule,[status(thm)],]) ).
thf(46,plain,
( sP32
| ~ sP33 ),
inference(prop_rule,[status(thm)],]) ).
thf(47,plain,
( sP4
| ~ sP32 ),
inference(prop_rule,[status(thm)],]) ).
thf(48,plain,
( sP12
| ~ sP4 ),
inference(prop_rule,[status(thm)],]) ).
thf(49,plain,
( sP12
| sP21 ),
inference(prop_rule,[status(thm)],]) ).
thf(50,plain,
( sP37
| ~ sP12 ),
inference(prop_rule,[status(thm)],]) ).
thf(51,plain,
( sP39
| ~ sP37 ),
inference(prop_rule,[status(thm)],]) ).
thf(52,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,h2]) ).
thf(53,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[52,h1]) ).
thf(54,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[53,h0]) ).
thf(0,theorem,
sP39,
inference(contra,[status(thm),contra(discharge,[h2])],[52,h2]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : ALG263^1 : TPTP v8.1.0. Bugfixed v5.2.0.
% 0.03/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n018.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 600
% 0.13/0.34 % DateTime : Wed Jun 8 22:39:52 EDT 2022
% 0.13/0.34 % CPUTime :
% 44.62/44.98 % SZS status Theorem
% 44.62/44.98 % Mode: mode454:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=3.:SINE_DEPTH=0
% 44.62/44.98 % Inferences: 1464
% 44.62/44.98 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------