ITP001 Axioms: ITP017_5.ax
%------------------------------------------------------------------------------
% File : ITP017_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : poset_2.ax [Gau20]
% : HL4017_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 79 ( 17 unt; 23 typ; 0 def)
% Number of atoms : 1269 ( 15 equ)
% Maximal formula atoms : 24 ( 16 avg)
% Number of connectives : 261 ( 0 ~; 1 |; 75 &)
% ( 16 <=>; 169 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 952 ( 952 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 46 ( 23 >; 23 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 4 prp; 0-2 aty)
% Number of functors : 23 ( 23 usr; 0 con; 1-6 aty)
% Number of variables : 219 ( 207 !; 12 ?; 219 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eposet_2Ebottom,type,
c_2Eposet_2Ebottom: del > $i ).
tff(mem_c_2Eposet_2Ebottom,axiom,
! [A_27a: del] : mem(c_2Eposet_2Ebottom(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(A_27a,bool))) ).
tff(tp_c_2Eposet_2Ecarrier,type,
c_2Eposet_2Ecarrier: del > $i ).
tff(mem_c_2Eposet_2Ecarrier,axiom,
! [A_27a: del] : mem(c_2Eposet_2Ecarrier(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(A_27a,bool))) ).
tff(tp_c_2Eposet_2Echain,type,
c_2Eposet_2Echain: del > $i ).
tff(mem_c_2Eposet_2Echain,axiom,
! [A_27a: del] : mem(c_2Eposet_2Echain(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Eposet_2Ecomplete,type,
c_2Eposet_2Ecomplete: del > $i ).
tff(mem_c_2Eposet_2Ecomplete,axiom,
! [A_27a: del] : mem(c_2Eposet_2Ecomplete(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),bool)) ).
tff(tp_c_2Eposet_2Econtinuous,type,
c_2Eposet_2Econtinuous: del > $i ).
tff(mem_c_2Eposet_2Econtinuous,axiom,
! [A_27a: del] : mem(c_2Eposet_2Econtinuous(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,A_27a),bool))) ).
tff(tp_c_2Eposet_2Edown__continuous,type,
c_2Eposet_2Edown__continuous: del > $i ).
tff(mem_c_2Eposet_2Edown__continuous,axiom,
! [A_27a: del] : mem(c_2Eposet_2Edown__continuous(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,A_27a),bool))) ).
tff(tp_c_2Eposet_2Efunction,type,
c_2Eposet_2Efunction: ( del * del ) > $i ).
tff(mem_c_2Eposet_2Efunction,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eposet_2Efunction(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),arr(arr(A_27a,A_27b),bool)))) ).
tff(tp_c_2Eposet_2Egfp,type,
c_2Eposet_2Egfp: del > $i ).
tff(mem_c_2Eposet_2Egfp,axiom,
! [A_27a: del] : mem(c_2Eposet_2Egfp(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,A_27a),arr(A_27a,bool)))) ).
tff(tp_c_2Eposet_2Eglb,type,
c_2Eposet_2Eglb: del > $i ).
tff(mem_c_2Eposet_2Eglb,axiom,
! [A_27a: del] : mem(c_2Eposet_2Eglb(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,bool),arr(A_27a,bool)))) ).
tff(tp_c_2Eposet_2Elfp,type,
c_2Eposet_2Elfp: del > $i ).
tff(mem_c_2Eposet_2Elfp,axiom,
! [A_27a: del] : mem(c_2Eposet_2Elfp(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,A_27a),arr(A_27a,bool)))) ).
tff(tp_c_2Eposet_2Elub,type,
c_2Eposet_2Elub: del > $i ).
tff(mem_c_2Eposet_2Elub,axiom,
! [A_27a: del] : mem(c_2Eposet_2Elub(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,bool),arr(A_27a,bool)))) ).
tff(tp_c_2Eposet_2Emonotonic,type,
c_2Eposet_2Emonotonic: del > $i ).
tff(mem_c_2Eposet_2Emonotonic,axiom,
! [A_27a: del] : mem(c_2Eposet_2Emonotonic(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,A_27a),bool))) ).
tff(tp_c_2Eposet_2Epointwise__lift,type,
c_2Eposet_2Epointwise__lift: ( del * del ) > $i ).
tff(mem_c_2Eposet_2Epointwise__lift,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eposet_2Epointwise__lift(A_27a,A_27b),arr(arr(A_27a,bool),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(A_27b,arr(A_27b,bool))),ty_2Epair_2Eprod(arr(arr(A_27a,A_27b),bool),arr(arr(A_27a,A_27b),arr(arr(A_27a,A_27b),bool)))))) ).
tff(tp_c_2Eposet_2Eposet,type,
c_2Eposet_2Eposet: del > $i ).
tff(mem_c_2Eposet_2Eposet,axiom,
! [A_27a: del] : mem(c_2Eposet_2Eposet(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),bool)) ).
tff(tp_c_2Eposet_2Erelation,type,
c_2Eposet_2Erelation: del > $i ).
tff(mem_c_2Eposet_2Erelation,axiom,
! [A_27a: del] : mem(c_2Eposet_2Erelation(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Eposet_2Etop,type,
c_2Eposet_2Etop: del > $i ).
tff(mem_c_2Eposet_2Etop,axiom,
! [A_27a: del] : mem(c_2Eposet_2Etop(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(A_27a,bool))) ).
tff(tp_c_2Eposet_2Eup__continuous,type,
c_2Eposet_2Eup__continuous: del > $i ).
tff(mem_c_2Eposet_2Eup__continuous,axiom,
! [A_27a: del] : mem(c_2Eposet_2Eup__continuous(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),arr(arr(A_27a,A_27a),bool))) ).
tff(ax_thm_2Eposet_2Efunction__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,arr(A_27a,bool))
=> ! [V1b: $i] :
( mem(V1b,arr(A_27b,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Eposet_2Efunction(A_27a,A_27b),V0a),V1b),V2f))
<=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(V0a,V3x))
=> p(ap(V1b,ap(V2f,V3x))) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Eposet__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Eposet_2Eposet(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)))
<=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V0s,V2x)) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(V0s,V3x))
=> p(ap(ap(V1r,V3x),V3x)) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ( p(ap(V0s,V4x))
& p(ap(V0s,V5y))
& p(ap(ap(V1r,V4x),V5y))
& p(ap(ap(V1r,V5y),V4x)) )
=> ( V4x = V5y ) ) ) )
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7y: $i] :
( mem(V7y,A_27a)
=> ! [V8z: $i] :
( mem(V8z,A_27a)
=> ( ( p(ap(V0s,V6x))
& p(ap(V0s,V7y))
& p(ap(V0s,V8z))
& p(ap(ap(V1r,V6x),V7y))
& p(ap(ap(V1r,V7y),V8z)) )
=> p(ap(ap(V1r,V6x),V8z)) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Ecarrier__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Eposet_2Ecarrier(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)) = V0s ) ) ) ).
tff(ax_thm_2Eposet_2Erelation__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Eposet_2Erelation(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)) = V1r ) ) ) ).
tff(ax_thm_2Eposet_2Etop__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Eposet_2Etop(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2x))
<=> ( p(ap(V0s,V2x))
& ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(V0s,V3y))
=> p(ap(ap(V1r,V3y),V2x)) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Ebottom__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Eposet_2Ebottom(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2x))
<=> ( p(ap(V0s,V2x))
& ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(V0s,V3y))
=> p(ap(ap(V1r,V2x),V3y)) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Echain__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2c: $i] :
( mem(V2c,arr(A_27a,bool))
=> ( p(ap(ap(c_2Eposet_2Echain(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2c))
<=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0s,V3x))
& p(ap(V0s,V4y))
& p(ap(V2c,V3x))
& p(ap(V2c,V4y)) )
=> ( p(ap(ap(V1r,V3x),V4y))
| p(ap(ap(V1r,V4y),V3x)) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Elub__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,bool))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(ap(c_2Eposet_2Elub(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2p),V3x))
<=> ( p(ap(V0s,V3x))
& ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0s,V4y))
& p(ap(V2p,V4y)) )
=> p(ap(ap(V1r,V4y),V3x)) ) )
& ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(V0s,V5z))
& ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( ( p(ap(V0s,V6y))
& p(ap(V2p,V6y)) )
=> p(ap(ap(V1r,V6y),V5z)) ) ) )
=> p(ap(ap(V1r,V3x),V5z)) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Eglb__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,bool))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(ap(c_2Eposet_2Eglb(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2p),V3x))
<=> ( p(ap(V0s,V3x))
& ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0s,V4y))
& p(ap(V2p,V4y)) )
=> p(ap(ap(V1r,V3x),V4y)) ) )
& ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(V0s,V5z))
& ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( ( p(ap(V0s,V6y))
& p(ap(V2p,V6y)) )
=> p(ap(ap(V1r,V5z),V6y)) ) ) )
=> p(ap(ap(V1r,V5z),V3x)) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Ecomplete__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ( p(ap(c_2Eposet_2Ecomplete(A_27a),V0p))
<=> ! [V1c: $i] :
( mem(V1c,arr(A_27a,bool))
=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Elub(A_27a),V0p),V1c),V2x)) )
& ? [V3x: $i] :
( mem(V3x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Eglb(A_27a),V0p),V1c),V3x)) ) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eposet__nonempty,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Eposet_2Eposet(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)))
=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V0s,V2x)) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eposet__refl,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)))
& p(ap(V0s,V2x)) )
=> p(ap(ap(V1r,V2x),V2x)) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eposet__antisym,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)))
& p(ap(V0s,V2x))
& p(ap(V0s,V3y))
& p(ap(ap(V1r,V2x),V3y))
& p(ap(ap(V1r,V3y),V2x)) )
=> ( V2x = V3y ) ) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eposet__trans,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4z: $i] :
( mem(V4z,A_27a)
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)))
& p(ap(V0s,V2x))
& p(ap(V0s,V3y))
& p(ap(V0s,V4z))
& p(ap(ap(V1r,V2x),V3y))
& p(ap(ap(V1r,V3y),V4z)) )
=> p(ap(ap(V1r,V2x),V4z)) ) ) ) ) ) ) ).
tff(lamtp_f166,type,
f166: ( del * $i * $i ) > $i ).
tff(lameq_f166,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,bool))
=> ! [V4j: $i] : ( ap(f166(A_27a,V0s,V2p),V4j) = ap(ap(c_2Ebool_2E_2F_5C,ap(V0s,V4j)),ap(V2p,V4j)) ) ) ) ).
tff(conj_thm_2Eposet_2Elub__pred,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,bool))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(ap(c_2Eposet_2Elub(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),f166(A_27a,V0s,V2p)),V3x))
<=> p(ap(ap(ap(c_2Eposet_2Elub(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2p),V3x)) ) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eglb__pred,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,bool))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(ap(c_2Eposet_2Eglb(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),f166(A_27a,V0s,V2p)),V3x))
<=> p(ap(ap(ap(c_2Eposet_2Eglb(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2p),V3x)) ) ) ) ) ) ).
tff(conj_thm_2Eposet_2Ecomplete__up,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1c: $i] :
( mem(V1c,arr(A_27a,bool))
=> ( p(ap(c_2Eposet_2Ecomplete(A_27a),V0p))
=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Elub(A_27a),V0p),V1c),V2x)) ) ) ) ) ).
tff(conj_thm_2Eposet_2Ecomplete__down,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1c: $i] :
( mem(V1c,arr(A_27a,bool))
=> ( p(ap(c_2Eposet_2Ecomplete(A_27a),V0p))
=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Eglb(A_27a),V0p),V1c),V2x)) ) ) ) ) ).
tff(conj_thm_2Eposet_2Ecomplete__top,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),V0p))
& p(ap(c_2Eposet_2Ecomplete(A_27a),V0p)) )
=> ? [V1x: $i] :
( mem(V1x,A_27a)
& p(ap(ap(c_2Eposet_2Etop(A_27a),V0p),V1x)) ) ) ) ).
tff(conj_thm_2Eposet_2Ecomplete__bottom,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),V0p))
& p(ap(c_2Eposet_2Ecomplete(A_27a),V0p)) )
=> ? [V1x: $i] :
( mem(V1x,A_27a)
& p(ap(ap(c_2Eposet_2Ebottom(A_27a),V0p),V1x)) ) ) ) ).
tff(lamtp_f167,type,
f167: ( del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f167,axiom,
! [A_27b: del,A_27a: del,V4g: $i] :
( mem(V4g,arr(A_27a,A_27b))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V2r: $i] :
( mem(V2r,arr(A_27b,arr(A_27b,bool)))
=> ! [V0t: $i] :
( mem(V0t,arr(A_27a,bool))
=> ! [V5x: $i] : ( ap(f167(A_27b,A_27a,V4g,V3f,V2r,V0t),V5x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(V0t,V5x)),ap(ap(V2r,ap(V3f,V5x)),ap(V4g,V5x))) ) ) ) ) ) ).
tff(lamtp_f168,type,
f168: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f168,axiom,
! [A_27b: del,A_27a: del,V0t: $i] :
( mem(V0t,arr(A_27a,bool))
=> ! [V2r: $i] :
( mem(V2r,arr(A_27b,arr(A_27b,bool)))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V4g: $i] : ( ap(f168(A_27b,A_27a,V0t,V2r,V3f),V4g) = ap(c_2Ebool_2E_21(A_27a),f167(A_27b,A_27a,V4g,V3f,V2r,V0t)) ) ) ) ) ).
tff(lamtp_f169,type,
f169: ( del * del * $i * $i ) > $i ).
tff(lameq_f169,axiom,
! [A_27b: del,A_27a: del,V0t: $i] :
( mem(V0t,arr(A_27a,bool))
=> ! [V2r: $i] :
( mem(V2r,arr(A_27b,arr(A_27b,bool)))
=> ! [V3f: $i] : ( ap(f169(A_27b,A_27a,V0t,V2r),V3f) = f168(A_27b,A_27a,V0t,V2r,V3f) ) ) ) ).
tff(ax_thm_2Eposet_2Epointwise__lift__def,axiom,
! [A_27a: del,A_27b: del,V0t: $i] :
( mem(V0t,arr(A_27a,bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2r: $i] :
( mem(V2r,arr(A_27b,arr(A_27b,bool)))
=> ( ap(ap(c_2Eposet_2Epointwise__lift(A_27a,A_27b),V0t),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(A_27b,arr(A_27b,bool))),V1s),V2r)) = ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,A_27b),bool),arr(arr(A_27a,A_27b),arr(arr(A_27a,A_27b),bool))),ap(ap(c_2Eposet_2Efunction(A_27a,A_27b),V0t),V1s)),f169(A_27b,A_27a,V0t,V2r)) ) ) ) ) ).
tff(conj_thm_2Eposet_2Ecomplete__pointwise,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27b,bool))
=> ( p(ap(c_2Eposet_2Ecomplete(A_27a),V0p))
=> p(ap(c_2Eposet_2Ecomplete(arr(A_27b,A_27a)),ap(ap(c_2Eposet_2Epointwise__lift(A_27b,A_27a),V1t),V0p))) ) ) ) ).
tff(ax_thm_2Eposet_2Emonotonic__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ( p(ap(ap(c_2Eposet_2Emonotonic(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2f))
<=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0s,V3x))
& p(ap(V0s,V4y))
& p(ap(ap(V1r,V3x),V4y)) )
=> p(ap(ap(V1r,ap(V2f,V3x)),ap(V2f,V4y))) ) ) ) ) ) ) ) ).
tff(lamtp_f170,type,
f170: ( del * $i * $i * $i * $i ) > $i ).
tff(lameq_f170,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V3c: $i] :
( mem(V3c,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6z: $i] : ( ap(f170(A_27a,V0s,V3c,V2f,V5y),V6z) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2E_2F_5C,ap(V0s,V6z)),ap(V3c,V6z))),ap(ap(c_2Emin_2E_3D(A_27a),V5y),ap(V2f,V6z))) ) ) ) ) ) ).
tff(lamtp_f171,type,
f171: ( del * $i * $i * $i ) > $i ).
tff(lameq_f171,axiom,
! [A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ! [V3c: $i] :
( mem(V3c,arr(A_27a,bool))
=> ! [V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V5y: $i] : ( ap(f171(A_27a,V2f,V3c,V0s),V5y) = ap(c_2Ebool_2E_3F(A_27a),f170(A_27a,V0s,V3c,V2f,V5y)) ) ) ) ) ).
tff(ax_thm_2Eposet_2Eup__continuous__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ( p(ap(ap(c_2Eposet_2Eup__continuous(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2f))
<=> ! [V3c: $i] :
( mem(V3c,arr(A_27a,bool))
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( ( p(ap(ap(c_2Eposet_2Echain(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V3c))
& p(ap(ap(ap(c_2Eposet_2Elub(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V3c),V4x)) )
=> p(ap(ap(ap(c_2Eposet_2Elub(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),f171(A_27a,V2f,V3c,V0s)),ap(V2f,V4x))) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Edown__continuous__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ( p(ap(ap(c_2Eposet_2Edown__continuous(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2f))
<=> ! [V3c: $i] :
( mem(V3c,arr(A_27a,bool))
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( ( p(ap(ap(c_2Eposet_2Echain(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V3c))
& p(ap(ap(ap(c_2Eposet_2Eglb(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V3c),V4x)) )
=> p(ap(ap(ap(c_2Eposet_2Eglb(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),f171(A_27a,V2f,V3c,V0s)),ap(V2f,V4x))) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Econtinuous__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( p(ap(ap(c_2Eposet_2Econtinuous(A_27a),V0p),V1f))
<=> ( p(ap(ap(c_2Eposet_2Eup__continuous(A_27a),V0p),V1f))
& p(ap(ap(c_2Eposet_2Edown__continuous(A_27a),V0p),V1f)) ) ) ) ) ).
tff(ax_thm_2Eposet_2Elfp__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(ap(c_2Eposet_2Elfp(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2f),V3x))
<=> ( p(ap(V0s,V3x))
& ( ap(V2f,V3x) = V3x )
& ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0s,V4y))
& p(ap(ap(V1r,ap(V2f,V4y)),V4y)) )
=> p(ap(ap(V1r,V3x),V4y)) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Eposet_2Egfp__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1r: $i] :
( mem(V1r,arr(A_27a,arr(A_27a,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(ap(c_2Eposet_2Egfp(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))),V0s),V1r)),V2f),V3x))
<=> ( p(ap(V0s,V3x))
& ( ap(V2f,V3x) = V3x )
& ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0s,V4y))
& p(ap(ap(V1r,V4y),ap(V2f,V4y))) )
=> p(ap(ap(V1r,V4y),V3x)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eposet_2Elfp__unique,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3x_27: $i] :
( mem(V3x_27,A_27a)
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),V0p))
& p(ap(ap(ap(c_2Eposet_2Elfp(A_27a),V0p),V1f),V2x))
& p(ap(ap(ap(c_2Eposet_2Elfp(A_27a),V0p),V1f),V3x_27)) )
=> ( V2x = V3x_27 ) ) ) ) ) ) ).
tff(conj_thm_2Eposet_2Egfp__unique,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3x_27: $i] :
( mem(V3x_27,A_27a)
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),V0p))
& p(ap(ap(ap(c_2Eposet_2Egfp(A_27a),V0p),V1f),V2x))
& p(ap(ap(ap(c_2Eposet_2Egfp(A_27a),V0p),V1f),V3x_27)) )
=> ( V2x = V3x_27 ) ) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eknaster__tarski__lfp,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),V0p))
& p(ap(c_2Eposet_2Ecomplete(A_27a),V0p))
& p(ap(ap(ap(c_2Eposet_2Efunction(A_27a,A_27a),ap(c_2Eposet_2Ecarrier(A_27a),V0p)),ap(c_2Eposet_2Ecarrier(A_27a),V0p)),V1f))
& p(ap(ap(c_2Eposet_2Emonotonic(A_27a),V0p),V1f)) )
=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Elfp(A_27a),V0p),V1f),V2x)) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eknaster__tarski__gfp,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),V0p))
& p(ap(c_2Eposet_2Ecomplete(A_27a),V0p))
& p(ap(ap(ap(c_2Eposet_2Efunction(A_27a,A_27a),ap(c_2Eposet_2Ecarrier(A_27a),V0p)),ap(c_2Eposet_2Ecarrier(A_27a),V0p)),V1f))
& p(ap(ap(c_2Eposet_2Emonotonic(A_27a),V0p),V1f)) )
=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Egfp(A_27a),V0p),V1f),V2x)) ) ) ) ) ).
tff(conj_thm_2Eposet_2Eknaster__tarski,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ( p(ap(c_2Eposet_2Eposet(A_27a),V0p))
& p(ap(c_2Eposet_2Ecomplete(A_27a),V0p))
& p(ap(ap(ap(c_2Eposet_2Efunction(A_27a,A_27a),ap(c_2Eposet_2Ecarrier(A_27a),V0p)),ap(c_2Eposet_2Ecarrier(A_27a),V0p)),V1f))
& p(ap(ap(c_2Eposet_2Emonotonic(A_27a),V0p),V1f)) )
=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Elfp(A_27a),V0p),V1f),V2x)) )
& ? [V3x: $i] :
( mem(V3x,A_27a)
& p(ap(ap(ap(c_2Eposet_2Egfp(A_27a),V0p),V1f),V3x)) ) ) ) ) ) ).
%------------------------------------------------------------------------------