ITP001 Axioms: ITP058_5.ax
%------------------------------------------------------------------------------
% File : ITP058_5 : TPTP v9.0.0. 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 : wot_2.ax [Gau20]
% : HL4058_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 47 ( 17 unt; 17 typ; 0 def)
% Number of atoms : 311 ( 6 equ)
% Maximal formula atoms : 10 ( 6 avg)
% Number of connectives : 44 ( 1 ~; 1 |; 9 &)
% ( 8 <=>; 25 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 238 ( 238 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 18 ( 17 >; 1 *; 0 +; 0 <<)
% Number of predicates : 20 ( 19 usr; 2 prp; 0-2 aty)
% Number of functors : 17 ( 17 usr; 0 con; 1-2 aty)
% Number of variables : 53 ( 51 !; 2 ?; 53 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Ewot_2EStrongWellOrder,type,
c_2Ewot_2EStrongWellOrder: del > $i ).
tff(mem_c_2Ewot_2EStrongWellOrder,axiom,
! [A_27a: del] : mem(c_2Ewot_2EStrongWellOrder(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Ewot_2EU,type,
c_2Ewot_2EU: del > $i ).
tff(mem_c_2Ewot_2EU,axiom,
! [A_27x: del] : mem(c_2Ewot_2EU(A_27x),arr(arr(A_27x,bool),arr(arr(A_27x,bool),bool))) ).
tff(tp_c_2Ewot_2EWeakWellOrder,type,
c_2Ewot_2EWeakWellOrder: del > $i ).
tff(mem_c_2Ewot_2EWeakWellOrder,axiom,
! [A_27a: del] : mem(c_2Ewot_2EWeakWellOrder(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Ewot_2Echain,type,
c_2Ewot_2Echain: del > $i ).
tff(mem_c_2Ewot_2Echain,axiom,
! [A_27x: del] : mem(c_2Ewot_2Echain(A_27x),arr(arr(arr(A_27x,bool),bool),bool)) ).
tff(tp_c_2Ewot_2Ecomparable,type,
c_2Ewot_2Ecomparable: del > $i ).
tff(mem_c_2Ewot_2Ecomparable,axiom,
! [A_27x: del] : mem(c_2Ewot_2Ecomparable(A_27x),arr(arr(A_27x,bool),bool)) ).
tff(tp_c_2Ewot_2Ecpl,type,
c_2Ewot_2Ecpl: del > $i ).
tff(mem_c_2Ewot_2Ecpl,axiom,
! [A_27x: del] : mem(c_2Ewot_2Ecpl(A_27x),arr(arr(A_27x,bool),arr(arr(A_27x,bool),bool))) ).
tff(tp_c_2Ewot_2Elub__sub,type,
c_2Ewot_2Elub__sub: del > $i ).
tff(mem_c_2Ewot_2Elub__sub,axiom,
! [A_27x: del] : mem(c_2Ewot_2Elub__sub(A_27x),arr(arr(arr(A_27x,bool),bool),arr(A_27x,bool))) ).
tff(tp_c_2Ewot_2Emex,type,
c_2Ewot_2Emex: del > $i ).
tff(mem_c_2Ewot_2Emex,axiom,
! [A_27x: del] : mem(c_2Ewot_2Emex(A_27x),arr(arr(A_27x,bool),A_27x)) ).
tff(tp_c_2Ewot_2Emex__less,type,
c_2Ewot_2Emex__less: del > $i ).
tff(mem_c_2Ewot_2Emex__less,axiom,
! [A_27x: del] : mem(c_2Ewot_2Emex__less(A_27x),arr(A_27x,arr(A_27x,bool))) ).
tff(tp_c_2Ewot_2Emex__less__eq,type,
c_2Ewot_2Emex__less__eq: del > $i ).
tff(mem_c_2Ewot_2Emex__less__eq,axiom,
! [A_27x: del] : mem(c_2Ewot_2Emex__less__eq(A_27x),arr(A_27x,arr(A_27x,bool))) ).
tff(tp_c_2Ewot_2Epreds,type,
c_2Ewot_2Epreds: del > $i ).
tff(mem_c_2Ewot_2Epreds,axiom,
! [A_27x: del] : mem(c_2Ewot_2Epreds(A_27x),arr(A_27x,arr(A_27x,bool))) ).
tff(tp_c_2Ewot_2Epreds__image,type,
c_2Ewot_2Epreds__image: del > $i ).
tff(mem_c_2Ewot_2Epreds__image,axiom,
! [A_27x: del] : mem(c_2Ewot_2Epreds__image(A_27x),arr(arr(A_27x,bool),arr(arr(A_27x,bool),bool))) ).
tff(tp_c_2Ewot_2Esetsuc,type,
c_2Ewot_2Esetsuc: del > $i ).
tff(mem_c_2Ewot_2Esetsuc,axiom,
! [A_27x: del] : mem(c_2Ewot_2Esetsuc(A_27x),arr(arr(A_27x,bool),arr(A_27x,bool))) ).
tff(tp_c_2Ewot_2Esuccl,type,
c_2Ewot_2Esuccl: del > $i ).
tff(mem_c_2Ewot_2Esuccl,axiom,
! [A_27x: del] : mem(c_2Ewot_2Esuccl(A_27x),arr(arr(arr(A_27x,bool),bool),bool)) ).
tff(tp_c_2Ewot_2Etower,type,
c_2Ewot_2Etower: del > $i ).
tff(mem_c_2Ewot_2Etower,axiom,
! [A_27x: del] : mem(c_2Ewot_2Etower(A_27x),arr(arr(arr(A_27x,bool),bool),bool)) ).
tff(tp_c_2Ewot_2Euncl,type,
c_2Ewot_2Euncl: del > $i ).
tff(mem_c_2Ewot_2Euncl,axiom,
! [A_27x: del] : mem(c_2Ewot_2Euncl(A_27x),arr(arr(arr(A_27x,bool),bool),bool)) ).
tff(ax_thm_2Ewot_2Ecpl__def,axiom,
! [A_27x: del,V0A: $i] :
( mem(V0A,arr(A_27x,bool))
=> ! [V1B: $i] :
( mem(V1B,arr(A_27x,bool))
=> ( p(ap(ap(c_2Ewot_2Ecpl(A_27x),V0A),V1B))
<=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27x),V0A),V1B))
| p(ap(ap(c_2Epred__set_2ESUBSET(A_27x),V1B),V0A)) ) ) ) ) ).
tff(ax_thm_2Ewot_2Echain__def,axiom,
! [A_27x: del,V0C: $i] :
( mem(V0C,arr(arr(A_27x,bool),bool))
=> ( p(ap(c_2Ewot_2Echain(A_27x),V0C))
<=> ! [V1a: $i] :
( mem(V1a,arr(A_27x,bool))
=> ! [V2b: $i] :
( mem(V2b,arr(A_27x,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27x,bool)),V1a),V0C))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27x,bool)),V2b),V0C)) )
=> p(ap(ap(c_2Ewot_2Ecpl(A_27x),V1a),V2b)) ) ) ) ) ) ).
tff(ax_thm_2Ewot_2Emex__def,axiom,
! [A_27x: del,V0s: $i] :
( mem(V0s,arr(A_27x,bool))
=> ( ap(c_2Ewot_2Emex(A_27x),V0s) = ap(c_2Epred__set_2ECHOICE(A_27x),ap(c_2Epred__set_2ECOMPL(A_27x),V0s)) ) ) ).
tff(ax_thm_2Ewot_2Esetsuc__def,axiom,
! [A_27x: del,V0s: $i] :
( mem(V0s,arr(A_27x,bool))
=> ( ap(c_2Ewot_2Esetsuc(A_27x),V0s) = ap(ap(c_2Epred__set_2EINSERT(A_27x),ap(c_2Ewot_2Emex(A_27x),V0s)),V0s) ) ) ).
tff(ax_thm_2Ewot_2Esuccl__def,axiom,
! [A_27x: del,V0c: $i] :
( mem(V0c,arr(arr(A_27x,bool),bool))
=> ( p(ap(c_2Ewot_2Esuccl(A_27x),V0c))
<=> ! [V1s: $i] :
( mem(V1s,arr(A_27x,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27x,bool)),V1s),V0c))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27x,bool)),ap(c_2Ewot_2Esetsuc(A_27x),V1s)),V0c)) ) ) ) ) ).
tff(ax_thm_2Ewot_2Euncl__def,axiom,
! [A_27x: del,V0c: $i] :
( mem(V0c,arr(arr(A_27x,bool),bool))
=> ( p(ap(c_2Ewot_2Euncl(A_27x),V0c))
<=> ! [V1w: $i] :
( mem(V1w,arr(arr(A_27x,bool),bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27x,bool)),V1w),V0c))
& p(ap(c_2Ewot_2Echain(A_27x),V1w)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27x,bool)),ap(c_2Epred__set_2EBIGUNION(A_27x),V1w)),V0c)) ) ) ) ) ).
tff(ax_thm_2Ewot_2Etower__def,axiom,
! [A_27x: del,V0A: $i] :
( mem(V0A,arr(arr(A_27x,bool),bool))
=> ( p(ap(c_2Ewot_2Etower(A_27x),V0A))
<=> ( p(ap(c_2Ewot_2Esuccl(A_27x),V0A))
& p(ap(c_2Ewot_2Euncl(A_27x),V0A)) ) ) ) ).
tff(ax_thm_2Ewot_2Emex__less__eq__def,axiom,
! [A_27x: del,V0a: $i] :
( mem(V0a,A_27x)
=> ! [V1b: $i] :
( mem(V1b,A_27x)
=> ( p(ap(ap(c_2Ewot_2Emex__less__eq(A_27x),V0a),V1b))
<=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27x),ap(c_2Ewot_2Epreds(A_27x),V0a)),ap(c_2Ewot_2Epreds(A_27x),V1b))) ) ) ) ).
tff(ax_thm_2Ewot_2Emex__less__def,axiom,
! [A_27x: del] : ( c_2Ewot_2Emex__less(A_27x) = ap(c_2Erelation_2ESTRORD(A_27x),c_2Ewot_2Emex__less__eq(A_27x)) ) ).
tff(ax_thm_2Ewot_2EWeakWellOrder__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Ewot_2EWeakWellOrder(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2EWeakOrder(A_27a),V0R))
& ! [V1B: $i] :
( mem(V1B,arr(A_27a,bool))
=> ( ( V1B != c_2Epred__set_2EEMPTY(A_27a) )
=> ? [V2m: $i] :
( mem(V2m,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2m),V1B))
& ! [V3b: $i] :
( mem(V3b,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3b),V1B))
=> p(ap(ap(V0R,V2m),V3b)) ) ) ) ) ) ) ) ) ).
tff(lamtp_f1151,type,
f1151: ( del * $i ) > $i ).
tff(lameq_f1151,axiom,
! [A_27x: del,V0X: $i] :
( mem(V0X,arr(A_27x,bool))
=> ! [V1x: $i] : ( ap(f1151(A_27x,V0X),V1x) = ap(ap(c_2Epair_2E_2C(arr(A_27x,bool),bool),ap(c_2Ewot_2Epreds(A_27x),V1x)),ap(ap(c_2Ebool_2EIN(A_27x),V1x),V0X)) ) ) ).
tff(ax_thm_2Ewot_2Epreds__image__def,axiom,
! [A_27x: del,V0X: $i] :
( mem(V0X,arr(A_27x,bool))
=> ( ap(c_2Ewot_2Epreds__image(A_27x),V0X) = ap(c_2Epred__set_2EGSPEC(arr(A_27x,bool),A_27x),f1151(A_27x,V0X)) ) ) ).
tff(ax_thm_2Ewot_2EStrongWellOrder__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Ewot_2EStrongWellOrder(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2EStrongLinearOrder(A_27a),V0R))
& p(ap(c_2Erelation_2EWF(A_27a),V0R)) ) ) ) ).
tff(conj_thm_2Ewot_2EStrongWellOrderExists,axiom,
! [A_27a: del] :
? [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
& p(ap(c_2Erelation_2EStrongLinearOrder(A_27a),V0R))
& p(ap(c_2Erelation_2EWF(A_27a),V0R)) ) ).
%------------------------------------------------------------------------------