TPTP Problem File: ITP011_3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP011_3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Equotient__option_2EOPTION__REL__def.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Equotient__option_2EOPTION__REL__def.p [Gau19]
% : HL405001_3.p [TPAP]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 87 ( 20 unt; 41 typ; 0 def)
% Number of atoms : 149 ( 59 equ)
% Maximal formula atoms : 27 ( 1 avg)
% Number of connectives : 115 ( 12 ~; 9 |; 45 &)
% ( 36 <=>; 13 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 5 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 30 ( 19 >; 11 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 39 ( 39 usr; 6 con; 0-5 aty)
% Number of variables : 174 ( 133 !; 4 ?; 174 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Eoption_2EIS__NONE_2E0,type,
c_2Eoption_2EIS__NONE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool) ).
tff(c_2Eoption_2EIS__NONE_2E1,type,
c_2Eoption_2EIS__NONE_2E1:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption(A_27a) > tyop_2Emin_2Ebool ) ).
tff(c_2Eoption_2EIS__SOME_2E0,type,
c_2Eoption_2EIS__SOME_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool) ).
tff(c_2Eoption_2EIS__SOME_2E1,type,
c_2Eoption_2EIS__SOME_2E1:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption(A_27a) > tyop_2Emin_2Ebool ) ).
tff(c_2Eoption_2ENONE_2E0,type,
c_2Eoption_2ENONE_2E0:
!>[A_27a: $tType] : tyop_2Eoption_2Eoption(A_27a) ).
tff(c_2Eoption_2EOPTION__JOIN_2E0,type,
c_2Eoption_2EOPTION__JOIN_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a)) ).
tff(c_2Eoption_2EOPTION__JOIN_2E1,type,
c_2Eoption_2EOPTION__JOIN_2E1:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption(tyop_2Eoption_2Eoption(A_27a)) > tyop_2Eoption_2Eoption(A_27a) ) ).
tff(c_2Eoption_2EOPTION__MAP_2E0,type,
c_2Eoption_2EOPTION__MAP_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b))) ).
tff(c_2Eoption_2EOPTION__MAP_2E2,type,
c_2Eoption_2EOPTION__MAP_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Eoption_2Eoption(A_27a) ) > tyop_2Eoption_2Eoption(A_27b) ) ).
tff(c_2Eoption_2EOPTREL_2E0,type,
c_2Eoption_2EOPTREL_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Ebool))) ).
tff(c_2Eoption_2EOPTREL_2E3,type,
c_2Eoption_2EOPTREL_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) * tyop_2Eoption_2Eoption(A_27a) * tyop_2Eoption_2Eoption(A_27b) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Eoption_2ESOME_2E0,type,
c_2Eoption_2ESOME_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)) ).
tff(c_2Eoption_2ESOME_2E1,type,
c_2Eoption_2ESOME_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Eoption_2Eoption(A_27a) ) ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Eoption_2ETHE_2E0,type,
c_2Eoption_2ETHE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),A_27a) ).
tff(c_2Eoption_2ETHE_2E1,type,
c_2Eoption_2ETHE_2E1:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption(A_27a) > A_27a ) ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Eoption_2Eoption__CASE_2E0,type,
c_2Eoption_2Eoption__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b))) ).
tff(c_2Eoption_2Eoption__CASE_2E3,type,
c_2Eoption_2Eoption__CASE_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption(A_27a) * A_27b * tyop_2Emin_2Efun(A_27a,A_27b) ) > A_27b ) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ).
tff(arityeq1_2Ec_2Eoption_2EIS__NONE_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2EIS__NONE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool,c_2Eoption_2EIS__NONE_2E0(A_27a),X0_2E0) ).
tff(arityeq1_2Ec_2Eoption_2EIS__SOME_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2EIS__SOME_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool,c_2Eoption_2EIS__SOME_2E0(A_27a),X0_2E0) ).
tff(arityeq1_2Ec_2Eoption_2EOPTION__JOIN_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(tyop_2Eoption_2Eoption(A_27a))] : c_2Eoption_2EOPTION__JOIN_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2EOPTION__JOIN_2E0(A_27a),X0_2E0) ).
tff(arityeq2_2Ec_2Eoption_2EOPTION__MAP_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2EOPTION__MAP_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b)),c_2Eoption_2EOPTION__MAP_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ).
tff(arityeq3_2Ec_2Eoption_2EOPTREL_2E3_2Emono_2EA_27a_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2EOPTREL_2E3(A_27a,A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool)),c_2Eoption_2EOPTREL_2E0(A_27a,A_27a),X0_2E0),X1_2E0),X2_2E0) ).
tff(arityeq3_2Ec_2Eoption_2EOPTREL_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Eoption_2Eoption(A_27b)] : c_2Eoption_2EOPTREL_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Ebool)),c_2Eoption_2EOPTREL_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ).
tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a] : c_2Eoption_2ESOME_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E0(A_27a),X0_2E0) ).
tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: A_27b] : c_2Eoption_2ESOME_2E1(A_27b,X0_2E0) = app_2E2(A_27b,tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E0(A_27b),X0_2E0) ).
tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2ESOME_2E1(tyop_2Eoption_2Eoption(A_27a),X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(tyop_2Eoption_2Eoption(A_27a)),c_2Eoption_2ESOME_2E0(tyop_2Eoption_2Eoption(A_27a)),X0_2E0) ).
tff(arityeq1_2Ec_2Eoption_2ETHE_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2ETHE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),A_27a,c_2Eoption_2ETHE_2E0(A_27a),X0_2E0) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a),X1_2E0: A_27b,X2_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : c_2Eoption_2Eoption__CASE_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b)),c_2Eoption_2Eoption__CASE_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ).
tff(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2EA_27a_20mono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a),X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a))] : c_2Eoption_2Eoption__CASE_2E3(A_27a,tyop_2Eoption_2Eoption(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a))),c_2Eoption_2Eoption__CASE_2E0(A_27a,tyop_2Eoption_2Eoption(A_27a)),X0_2E0),X1_2E0),X2_2E0) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Ebool_2ETRUTH,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1_2E0: tyop_2Emin_2Ebool,V1t2_2E0: tyop_2Emin_2Ebool] :
( ( p(V0t1_2E0)
=> p(V1t2_2E0) )
=> ( ( p(V1t2_2E0)
=> p(V0t1_2E0) )
=> ( V0t1_2E0 = V1t2_2E0 ) ) ) ).
tff(thm_2Ebool_2EFALSITY,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2EF_2E0)
=> p(V0t_2E0) ) ).
tff(thm_2Ebool_2EEXISTS__SIMP,axiom,
! [A_27a: $tType,V0t_2E0: tyop_2Emin_2Ebool] :
( ? [V1x_2E0: A_27a] : p(V0t_2E0)
<=> p(V0t_2E0) ) ).
tff(thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( ( p(c_2Ebool_2ET_2E0)
& p(V0t_2E0) )
<=> p(V0t_2E0) )
& ( ( p(V0t_2E0)
& p(c_2Ebool_2ET_2E0) )
<=> p(V0t_2E0) )
& ( ( p(c_2Ebool_2EF_2E0)
& p(V0t_2E0) )
<=> p(c_2Ebool_2EF_2E0) )
& ( ( p(V0t_2E0)
& p(c_2Ebool_2EF_2E0) )
<=> p(c_2Ebool_2EF_2E0) )
& ( ( p(V0t_2E0)
& p(V0t_2E0) )
<=> p(V0t_2E0) ) ) ).
tff(thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( ( p(c_2Ebool_2ET_2E0)
| p(V0t_2E0) )
<=> p(c_2Ebool_2ET_2E0) )
& ( ( p(V0t_2E0)
| p(c_2Ebool_2ET_2E0) )
<=> p(c_2Ebool_2ET_2E0) )
& ( ( p(c_2Ebool_2EF_2E0)
| p(V0t_2E0) )
<=> p(V0t_2E0) )
& ( ( p(V0t_2E0)
| p(c_2Ebool_2EF_2E0) )
<=> p(V0t_2E0) )
& ( ( p(V0t_2E0)
| p(V0t_2E0) )
<=> p(V0t_2E0) ) ) ).
tff(thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t_2E0: tyop_2Emin_2Ebool] :
( ~ ~ p(V0t_2E0)
<=> p(V0t_2E0) )
& ( ~ p(c_2Ebool_2ET_2E0)
<=> p(c_2Ebool_2EF_2E0) )
& ( ~ p(c_2Ebool_2EF_2E0)
<=> p(c_2Ebool_2ET_2E0) ) ) ).
tff(thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] :
( ( V0x_2E0 = V0x_2E0 )
<=> p(c_2Ebool_2ET_2E0) ) ).
tff(thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( ( c_2Ebool_2ET_2E0 = V0t_2E0 )
<=> p(V0t_2E0) )
& ( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
<=> p(V0t_2E0) )
& ( ( c_2Ebool_2EF_2E0 = V0t_2E0 )
<=> ~ p(V0t_2E0) )
& ( ( V0t_2E0 = c_2Ebool_2EF_2E0 )
<=> ~ p(V0t_2E0) ) ) ).
tff(thm_2Eoption_2Eoption__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1e_2E0: tyop_2Eoption_2Eoption(A_27a),V2e_2E0: A_27b] :
( ! [V3x_2E0: A_27a,V4y_2E0: A_27a] :
( ( c_2Eoption_2ESOME_2E1(A_27a,V3x_2E0) = c_2Eoption_2ESOME_2E1(A_27a,V4y_2E0) )
<=> ( V3x_2E0 = V4y_2E0 ) )
& ! [V5x_2E0: A_27a] : c_2Eoption_2ETHE_2E1(A_27a,c_2Eoption_2ESOME_2E1(A_27a,V5x_2E0)) = V5x_2E0
& ! [V6x_2E0: A_27a] : c_2Eoption_2ENONE_2E0(A_27a) != c_2Eoption_2ESOME_2E1(A_27a,V6x_2E0)
& ! [V7x_2E0: A_27a] : c_2Eoption_2ESOME_2E1(A_27a,V7x_2E0) != c_2Eoption_2ENONE_2E0(A_27a)
& ! [V8x_2E0: A_27a] : c_2Eoption_2EIS__SOME_2E1(A_27a,c_2Eoption_2ESOME_2E1(A_27a,V8x_2E0)) = c_2Ebool_2ET_2E0
& ( c_2Eoption_2EIS__SOME_2E1(A_27a,c_2Eoption_2ENONE_2E0(A_27a)) = c_2Ebool_2EF_2E0 )
& ! [V9x_2E0: tyop_2Eoption_2Eoption(A_27a)] :
( p(c_2Eoption_2EIS__NONE_2E1(A_27a,V9x_2E0))
<=> ( V9x_2E0 = c_2Eoption_2ENONE_2E0(A_27a) ) )
& ! [V10x_2E0: tyop_2Eoption_2Eoption(A_27a)] :
( ~ p(c_2Eoption_2EIS__SOME_2E1(A_27a,V10x_2E0))
<=> ( V10x_2E0 = c_2Eoption_2ENONE_2E0(A_27a) ) )
& ! [V11x_2E0: tyop_2Eoption_2Eoption(A_27a)] :
( p(c_2Eoption_2EIS__SOME_2E1(A_27a,V11x_2E0))
=> ( c_2Eoption_2ESOME_2E1(A_27a,c_2Eoption_2ETHE_2E1(A_27a,V11x_2E0)) = V11x_2E0 ) )
& ! [V12x_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2Eoption__CASE_2E3(A_27a,tyop_2Eoption_2Eoption(A_27a),V12x_2E0,c_2Eoption_2ENONE_2E0(A_27a),c_2Eoption_2ESOME_2E0(A_27a)) = V12x_2E0
& ! [V13x_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2Eoption__CASE_2E3(A_27a,tyop_2Eoption_2Eoption(A_27a),V13x_2E0,V13x_2E0,c_2Eoption_2ESOME_2E0(A_27a)) = V13x_2E0
& ! [V14x_2E0: tyop_2Eoption_2Eoption(A_27a)] :
( p(c_2Eoption_2EIS__NONE_2E1(A_27a,V14x_2E0))
=> ( c_2Eoption_2Eoption__CASE_2E3(A_27a,A_27b,V14x_2E0,V2e_2E0,V0f_2E0) = V2e_2E0 ) )
& ! [V15x_2E0: tyop_2Eoption_2Eoption(A_27a)] :
( p(c_2Eoption_2EIS__SOME_2E1(A_27a,V15x_2E0))
=> ( c_2Eoption_2Eoption__CASE_2E3(A_27a,A_27b,V15x_2E0,V2e_2E0,V0f_2E0) = app_2E2(A_27a,A_27b,V0f_2E0,c_2Eoption_2ETHE_2E1(A_27a,V15x_2E0)) ) )
& ! [V16x_2E0: tyop_2Eoption_2Eoption(A_27a)] :
( p(c_2Eoption_2EIS__SOME_2E1(A_27a,V16x_2E0))
=> ( c_2Eoption_2Eoption__CASE_2E3(A_27a,tyop_2Eoption_2Eoption(A_27a),V16x_2E0,V1e_2E0,c_2Eoption_2ESOME_2E0(A_27a)) = V16x_2E0 ) )
& ! [V17v_2E0: A_27b,V18f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : c_2Eoption_2Eoption__CASE_2E3(A_27a,A_27b,c_2Eoption_2ENONE_2E0(A_27a),V17v_2E0,V18f_2E0) = V17v_2E0
& ! [V19x_2E0: A_27a,V20v_2E0: A_27b,V21f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : c_2Eoption_2Eoption__CASE_2E3(A_27a,A_27b,c_2Eoption_2ESOME_2E1(A_27a,V19x_2E0),V20v_2E0,V21f_2E0) = app_2E2(A_27a,A_27b,V21f_2E0,V19x_2E0)
& ! [V22f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V23x_2E0: A_27a] : c_2Eoption_2EOPTION__MAP_2E2(A_27a,A_27b,V22f_2E0,c_2Eoption_2ESOME_2E1(A_27a,V23x_2E0)) = c_2Eoption_2ESOME_2E1(A_27b,app_2E2(A_27a,A_27b,V22f_2E0,V23x_2E0))
& ! [V24f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : c_2Eoption_2EOPTION__MAP_2E2(A_27a,A_27b,V24f_2E0,c_2Eoption_2ENONE_2E0(A_27a)) = c_2Eoption_2ENONE_2E0(A_27b)
& ( c_2Eoption_2EOPTION__JOIN_2E1(A_27a,c_2Eoption_2ENONE_2E0(tyop_2Eoption_2Eoption(A_27a))) = c_2Eoption_2ENONE_2E0(A_27a) )
& ! [V25x_2E0: tyop_2Eoption_2Eoption(A_27a)] : c_2Eoption_2EOPTION__JOIN_2E1(A_27a,c_2Eoption_2ESOME_2E1(tyop_2Eoption_2Eoption(A_27a),V25x_2E0)) = V25x_2E0 ) ).
tff(thm_2Eoption_2EOPTREL__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1x_2E0: tyop_2Eoption_2Eoption(A_27a),V2y_2E0: tyop_2Eoption_2Eoption(A_27b)] :
( p(c_2Eoption_2EOPTREL_2E3(A_27a,A_27b,V0R_2E0,V1x_2E0,V2y_2E0))
<=> ( ( ( V1x_2E0 = c_2Eoption_2ENONE_2E0(A_27a) )
& ( V2y_2E0 = c_2Eoption_2ENONE_2E0(A_27b) ) )
| ? [V3x0_2E0: A_27a,V4y0_2E0: A_27b] :
( ( V1x_2E0 = c_2Eoption_2ESOME_2E1(A_27a,V3x0_2E0) )
& ( V2y_2E0 = c_2Eoption_2ESOME_2E1(A_27b,V4y0_2E0) )
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V3x0_2E0),V4y0_2E0)) ) ) ) ).
tff(thm_2Equotient__option_2EOPTION__REL__def,conjecture,
! [A_27a: $tType,V0y_2E0: A_27a,V1x_2E0: A_27a,V2R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ( c_2Eoption_2EOPTREL_2E3(A_27a,A_27a,V2R_2E0,c_2Eoption_2ENONE_2E0(A_27a),c_2Eoption_2ENONE_2E0(A_27a)) = c_2Ebool_2ET_2E0 )
& ( c_2Eoption_2EOPTREL_2E3(A_27a,A_27a,V2R_2E0,c_2Eoption_2ESOME_2E1(A_27a,V1x_2E0),c_2Eoption_2ENONE_2E0(A_27a)) = c_2Ebool_2EF_2E0 )
& ( c_2Eoption_2EOPTREL_2E3(A_27a,A_27a,V2R_2E0,c_2Eoption_2ENONE_2E0(A_27a),c_2Eoption_2ESOME_2E1(A_27a,V0y_2E0)) = c_2Ebool_2EF_2E0 )
& ( c_2Eoption_2EOPTREL_2E3(A_27a,A_27a,V2R_2E0,c_2Eoption_2ESOME_2E1(A_27a,V1x_2E0),c_2Eoption_2ESOME_2E1(A_27a,V0y_2E0)) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2R_2E0,V1x_2E0),V0y_2E0) ) ) ).
%------------------------------------------------------------------------------