ITP001 Axioms: ITP087^7.ax
%------------------------------------------------------------------------------
% File : ITP087^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy 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 : quote.ax [Gau19]
% : HL4087^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 81 ( 7 unt; 45 typ; 0 def)
% Number of atoms : 120 ( 77 equ; 5 cnn)
% Maximal formula atoms : 14 ( 1 avg)
% Number of connectives : 713 ( 5 ~; 7 |; 51 &; 616 @)
% ( 15 <=>; 19 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 10 avg; 616 nst)
% Number of types : 5 ( 4 usr)
% Number of type conns : 134 ( 134 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 41 usr; 8 con; 0-5 aty)
% Number of variables : 246 ( 22 ^ 189 !; 15 ?; 246 :)
% ( 20 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Eind__type_2Erecspace,type,
tyop_2Eind__type_2Erecspace: $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Equote_2Eindex,type,
tyop_2Equote_2Eindex: $tType ).
thf(tyop_2Equote_2Evarmap,type,
tyop_2Equote_2Evarmap: $tType > $tType ).
thf(tyop_2EternaryComparisons_2Eordering,type,
tyop_2EternaryComparisons_2Eordering: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eind__type_2EBOTTOM,type,
c_2Eind__type_2EBOTTOM:
!>[A_27a: $tType] : ( tyop_2Eind__type_2Erecspace @ A_27a ) ).
thf(c_2Eind__type_2ECONSTR,type,
c_2Eind__type_2ECONSTR:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2EternaryComparisons_2EEQUAL,type,
c_2EternaryComparisons_2EEQUAL: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Equote_2EEmpty__vm,type,
c_2Equote_2EEmpty__vm:
!>[A_27a: $tType] : ( tyop_2Equote_2Evarmap @ A_27a ) ).
thf(c_2Equote_2EEnd__idx,type,
c_2Equote_2EEnd__idx: tyop_2Equote_2Eindex ).
thf(c_2Eind__type_2EFCONS,type,
c_2Eind__type_2EFCONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2EternaryComparisons_2EGREATER,type,
c_2EternaryComparisons_2EGREATER: tyop_2EternaryComparisons_2Eordering ).
thf(c_2EternaryComparisons_2ELESS,type,
c_2EternaryComparisons_2ELESS: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Equote_2ELeft__idx,type,
c_2Equote_2ELeft__idx: tyop_2Equote_2Eindex > tyop_2Equote_2Eindex ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Equote_2ENode__vm,type,
c_2Equote_2ENode__vm:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) ) ).
thf(c_2Equote_2ERight__idx,type,
c_2Equote_2ERight__idx: tyop_2Equote_2Eindex > tyop_2Equote_2Eindex ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Equote_2Eindex__CASE,type,
c_2Equote_2Eindex__CASE:
!>[A_27a: $tType] : ( tyop_2Equote_2Eindex > ( tyop_2Equote_2Eindex > A_27a ) > ( tyop_2Equote_2Eindex > A_27a ) > A_27a > A_27a ) ).
thf(c_2Equote_2Eindex__compare,type,
c_2Equote_2Eindex__compare: tyop_2Equote_2Eindex > tyop_2Equote_2Eindex > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Equote_2Eindex__lt,type,
c_2Equote_2Eindex__lt: tyop_2Equote_2Eindex > tyop_2Equote_2Eindex > $o ).
thf(c_2Equote_2Eindex__size,type,
c_2Equote_2Eindex__size: tyop_2Equote_2Eindex > tyop_2Enum_2Enum ).
thf(c_2EternaryComparisons_2Elist__compare,type,
c_2EternaryComparisons_2Elist__compare:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Equote_2Evarmap__CASE,type,
c_2Equote_2Evarmap__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b > ( A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b ) > A_27b ) ).
thf(c_2Equote_2Evarmap__find,type,
c_2Equote_2Evarmap__find:
!>[A_27a: $tType] : ( tyop_2Equote_2Eindex > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27a ) ).
thf(c_2Equote_2Evarmap__size,type,
c_2Equote_2Evarmap__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Equote_2Evarmap @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Equote_2Eindex__TY__DEF,axiom,
? [V0rep: tyop_2Equote_2Eindex > ( tyop_2Eind__type_2Erecspace @ $o )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ $o ) @ tyop_2Equote_2Eindex
@ ^ [V1a0: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ $o ) > $o )
@ ^ [V2_27index_27: ( tyop_2Eind__type_2Erecspace @ $o ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ $o )
@ ^ [V3a0: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ $o )
@ ^ [V4a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
@ ( ^ [V5a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Eind__type_2ECONSTR @ $o @ c_2Enum_2E0 @ ( c_2Ebool_2EARB @ $o )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V5a
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
@ V4a ) )
@ ( V2_27index_27 @ V4a ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ $o )
@ ^ [V7a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
@ ( ^ [V8a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Eind__type_2ECONSTR @ $o @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ ( c_2Ebool_2EARB @ $o )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V8a
@ ^ [V9n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
@ V7a ) )
@ ( V2_27index_27 @ V7a ) ) )
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
@ ( c_2Eind__type_2ECONSTR @ $o @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ $o )
@ ^ [V10n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) ) ) )
@ ( V2_27index_27 @ V3a0 ) ) )
@ ( V2_27index_27 @ V1a0 ) ) )
@ V0rep ) ).
thf(thm_2Equote_2Eindex__case__def,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Equote_2Eindex,V1f: tyop_2Equote_2Eindex > A_27a,V2f1: tyop_2Equote_2Eindex > A_27a,V3v: A_27a] :
( ( c_2Equote_2Eindex__CASE @ A_27a @ ( c_2Equote_2ELeft__idx @ V0a ) @ V1f @ V2f1 @ V3v )
= ( V1f @ V0a ) )
& ! [V4a: tyop_2Equote_2Eindex,V5f: tyop_2Equote_2Eindex > A_27a,V6f1: tyop_2Equote_2Eindex > A_27a,V7v: A_27a] :
( ( c_2Equote_2Eindex__CASE @ A_27a @ ( c_2Equote_2ERight__idx @ V4a ) @ V5f @ V6f1 @ V7v )
= ( V6f1 @ V4a ) )
& ! [V8f: tyop_2Equote_2Eindex > A_27a,V9f1: tyop_2Equote_2Eindex > A_27a,V10v: A_27a] :
( ( c_2Equote_2Eindex__CASE @ A_27a @ c_2Equote_2EEnd__idx @ V8f @ V9f1 @ V10v )
= V10v ) ) ).
thf(thm_2Equote_2Eindex__size__def,axiom,
( ! [V0a: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__size @ ( c_2Equote_2ELeft__idx @ V0a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Equote_2Eindex__size @ V0a ) ) )
& ! [V1a: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__size @ ( c_2Equote_2ERight__idx @ V1a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Equote_2Eindex__size @ V1a ) ) )
& ( ( c_2Equote_2Eindex__size @ c_2Equote_2EEnd__idx )
= c_2Enum_2E0 ) ) ).
thf(thm_2Equote_2Eindex__lt__def,axiom,
! [V0i1: tyop_2Equote_2Eindex,V1i2: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__lt @ V0i1 @ V1i2 )
<=> ( ( c_2Equote_2Eindex__compare @ V0i1 @ V1i2 )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Equote_2Evarmap__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Eind__type_2Erecspace @ A_27a )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ ( tyop_2Equote_2Evarmap @ A_27a )
@ ^ [V1a0_27: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o )
@ ^ [V2_27varmap_27: ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0_27: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0_27
@ ( c_2Eind__type_2ECONSTR @ A_27a @ c_2Enum_2E0 @ ( c_2Ebool_2EARB @ A_27a )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V5a0: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V6a1: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V7a2: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0_27
@ ( ^ [V8a0: A_27a,V9a1: tyop_2Eind__type_2Erecspace @ A_27a,V10a2: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ V8a0
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V9a1
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V10a2
@ ^ [V11n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V5a0
@ V6a1
@ V7a2 ) )
@ ( c_2Ebool_2E_2F_5C @ ( V2_27varmap_27 @ V6a1 ) @ ( V2_27varmap_27 @ V7a2 ) ) ) ) ) ) )
@ ( V2_27varmap_27 @ V3a0_27 ) ) )
@ ( V2_27varmap_27 @ V1a0_27 ) ) )
@ V0rep ) ).
thf(thm_2Equote_2Evarmap__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0v: A_27b,V1f: A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b] :
( ( c_2Equote_2Evarmap__CASE @ A_27a @ A_27b @ ( c_2Equote_2EEmpty__vm @ A_27a ) @ V0v @ V1f )
= V0v )
& ! [V2a0: A_27a,V3a1: tyop_2Equote_2Evarmap @ A_27a,V4a2: tyop_2Equote_2Evarmap @ A_27a,V5v: A_27b,V6f: A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b] :
( ( c_2Equote_2Evarmap__CASE @ A_27a @ A_27b @ ( c_2Equote_2ENode__vm @ A_27a @ V2a0 @ V3a1 @ V4a2 ) @ V5v @ V6f )
= ( V6f @ V2a0 @ V3a1 @ V4a2 ) ) ) ).
thf(thm_2Equote_2Evarmap__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Equote_2Evarmap__size @ A_27a @ V0f @ ( c_2Equote_2EEmpty__vm @ A_27a ) )
= c_2Enum_2E0 )
& ! [V1f: A_27a > tyop_2Enum_2Enum,V2a0: A_27a,V3a1: tyop_2Equote_2Evarmap @ A_27a,V4a2: tyop_2Equote_2Evarmap @ A_27a] :
( ( c_2Equote_2Evarmap__size @ A_27a @ V1f @ ( c_2Equote_2ENode__vm @ A_27a @ V2a0 @ V3a1 @ V4a2 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ ( V1f @ V2a0 ) @ ( c_2Earithmetic_2E_2B @ ( c_2Equote_2Evarmap__size @ A_27a @ V1f @ V3a1 ) @ ( c_2Equote_2Evarmap__size @ A_27a @ V1f @ V4a2 ) ) ) ) ) ) ).
thf(thm_2Equote_2Edatatype__index,axiom,
! [V0index: ( tyop_2Equote_2Eindex > tyop_2Equote_2Eindex ) > ( tyop_2Equote_2Eindex > tyop_2Equote_2Eindex ) > tyop_2Equote_2Eindex > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0index @ c_2Equote_2ELeft__idx @ c_2Equote_2ERight__idx @ c_2Equote_2EEnd__idx ) ) ).
thf(thm_2Equote_2Eindex__11,axiom,
( ! [V0a: tyop_2Equote_2Eindex,V1a_27: tyop_2Equote_2Eindex] :
( ( ( c_2Equote_2ELeft__idx @ V0a )
= ( c_2Equote_2ELeft__idx @ V1a_27 ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a: tyop_2Equote_2Eindex,V3a_27: tyop_2Equote_2Eindex] :
( ( ( c_2Equote_2ERight__idx @ V2a )
= ( c_2Equote_2ERight__idx @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) ).
thf(thm_2Equote_2Eindex__distinct,axiom,
( ! [V0a_27: tyop_2Equote_2Eindex,V1a: tyop_2Equote_2Eindex] :
( (~)
@ ( ( c_2Equote_2ELeft__idx @ V1a )
= ( c_2Equote_2ERight__idx @ V0a_27 ) ) )
& ! [V2a: tyop_2Equote_2Eindex] :
( (~)
@ ( ( c_2Equote_2ELeft__idx @ V2a )
= c_2Equote_2EEnd__idx ) )
& ! [V3a: tyop_2Equote_2Eindex] :
( (~)
@ ( ( c_2Equote_2ERight__idx @ V3a )
= c_2Equote_2EEnd__idx ) ) ) ).
thf(thm_2Equote_2Eindex__nchotomy,axiom,
! [V0ii: tyop_2Equote_2Eindex] :
( ? [V1i: tyop_2Equote_2Eindex] :
( V0ii
= ( c_2Equote_2ELeft__idx @ V1i ) )
| ? [V2i: tyop_2Equote_2Eindex] :
( V0ii
= ( c_2Equote_2ERight__idx @ V2i ) )
| ( V0ii = c_2Equote_2EEnd__idx ) ) ).
thf(thm_2Equote_2Eindex__Axiom,axiom,
! [A_27a: $tType,V0f0: tyop_2Equote_2Eindex > A_27a > A_27a,V1f1: tyop_2Equote_2Eindex > A_27a > A_27a,V2f2: A_27a] :
? [V3fn: tyop_2Equote_2Eindex > A_27a] :
( ! [V4a: tyop_2Equote_2Eindex] :
( ( V3fn @ ( c_2Equote_2ELeft__idx @ V4a ) )
= ( V0f0 @ V4a @ ( V3fn @ V4a ) ) )
& ! [V5a: tyop_2Equote_2Eindex] :
( ( V3fn @ ( c_2Equote_2ERight__idx @ V5a ) )
= ( V1f1 @ V5a @ ( V3fn @ V5a ) ) )
& ( ( V3fn @ c_2Equote_2EEnd__idx )
= V2f2 ) ) ).
thf(thm_2Equote_2Eindex__induction,axiom,
! [V0P: tyop_2Equote_2Eindex > $o] :
( ( ! [V1i: tyop_2Equote_2Eindex] :
( ( V0P @ V1i )
=> ( V0P @ ( c_2Equote_2ELeft__idx @ V1i ) ) )
& ! [V2i: tyop_2Equote_2Eindex] :
( ( V0P @ V2i )
=> ( V0P @ ( c_2Equote_2ERight__idx @ V2i ) ) )
& ( V0P @ c_2Equote_2EEnd__idx ) )
=> ! [V3i: tyop_2Equote_2Eindex] : ( V0P @ V3i ) ) ).
thf(thm_2Equote_2Eindex__case__cong,axiom,
! [A_27a: $tType,V0v_27: A_27a,V1f1_27: tyop_2Equote_2Eindex > A_27a,V2f_27: tyop_2Equote_2Eindex > A_27a,V3M: tyop_2Equote_2Eindex,V4M_27: tyop_2Equote_2Eindex,V5f: tyop_2Equote_2Eindex > A_27a,V6f1: tyop_2Equote_2Eindex > A_27a,V7v: A_27a] :
( ( ( V3M = V4M_27 )
& ! [V8a: tyop_2Equote_2Eindex] :
( ( V4M_27
= ( c_2Equote_2ELeft__idx @ V8a ) )
=> ( ( V5f @ V8a )
= ( V2f_27 @ V8a ) ) )
& ! [V9a: tyop_2Equote_2Eindex] :
( ( V4M_27
= ( c_2Equote_2ERight__idx @ V9a ) )
=> ( ( V6f1 @ V9a )
= ( V1f1_27 @ V9a ) ) )
& ( ( V4M_27 = c_2Equote_2EEnd__idx )
=> ( V7v = V0v_27 ) ) )
=> ( ( c_2Equote_2Eindex__CASE @ A_27a @ V3M @ V5f @ V6f1 @ V7v )
= ( c_2Equote_2Eindex__CASE @ A_27a @ V4M_27 @ V2f_27 @ V1f1_27 @ V0v_27 ) ) ) ).
thf(thm_2Equote_2Eindex__case__eq,axiom,
! [A_27a: $tType,V0x: tyop_2Equote_2Eindex,V1v_27: A_27a,V2v: A_27a,V3f1: tyop_2Equote_2Eindex > A_27a,V4f: tyop_2Equote_2Eindex > A_27a] :
( ( ( c_2Equote_2Eindex__CASE @ A_27a @ V0x @ V4f @ V3f1 @ V2v )
= V1v_27 )
<=> ( ? [V5i: tyop_2Equote_2Eindex] :
( ( V0x
= ( c_2Equote_2ELeft__idx @ V5i ) )
& ( ( V4f @ V5i )
= V1v_27 ) )
| ? [V6i: tyop_2Equote_2Eindex] :
( ( V0x
= ( c_2Equote_2ERight__idx @ V6i ) )
& ( ( V3f1 @ V6i )
= V1v_27 ) )
| ( ( V0x = c_2Equote_2EEnd__idx )
& ( V2v = V1v_27 ) ) ) ) ).
thf(thm_2Equote_2Eindex__compare__ind,axiom,
! [V0P: tyop_2Equote_2Eindex > tyop_2Equote_2Eindex > $o] :
( ( ( V0P @ c_2Equote_2EEnd__idx @ c_2Equote_2EEnd__idx )
& ! [V1v10: tyop_2Equote_2Eindex] : ( V0P @ c_2Equote_2EEnd__idx @ ( c_2Equote_2ELeft__idx @ V1v10 ) )
& ! [V2v11: tyop_2Equote_2Eindex] : ( V0P @ c_2Equote_2EEnd__idx @ ( c_2Equote_2ERight__idx @ V2v11 ) )
& ! [V3v2: tyop_2Equote_2Eindex] : ( V0P @ ( c_2Equote_2ELeft__idx @ V3v2 ) @ c_2Equote_2EEnd__idx )
& ! [V4v3: tyop_2Equote_2Eindex] : ( V0P @ ( c_2Equote_2ERight__idx @ V4v3 ) @ c_2Equote_2EEnd__idx )
& ! [V5n_27: tyop_2Equote_2Eindex,V6m_27: tyop_2Equote_2Eindex] :
( ( V0P @ V5n_27 @ V6m_27 )
=> ( V0P @ ( c_2Equote_2ELeft__idx @ V5n_27 ) @ ( c_2Equote_2ELeft__idx @ V6m_27 ) ) )
& ! [V7n_27: tyop_2Equote_2Eindex,V8m_27: tyop_2Equote_2Eindex] : ( V0P @ ( c_2Equote_2ELeft__idx @ V7n_27 ) @ ( c_2Equote_2ERight__idx @ V8m_27 ) )
& ! [V9n_27: tyop_2Equote_2Eindex,V10m_27: tyop_2Equote_2Eindex] :
( ( V0P @ V9n_27 @ V10m_27 )
=> ( V0P @ ( c_2Equote_2ERight__idx @ V9n_27 ) @ ( c_2Equote_2ERight__idx @ V10m_27 ) ) )
& ! [V11n_27: tyop_2Equote_2Eindex,V12m_27: tyop_2Equote_2Eindex] : ( V0P @ ( c_2Equote_2ERight__idx @ V11n_27 ) @ ( c_2Equote_2ELeft__idx @ V12m_27 ) ) )
=> ! [V13v: tyop_2Equote_2Eindex,V14v1: tyop_2Equote_2Eindex] : ( V0P @ V13v @ V14v1 ) ) ).
thf(thm_2Equote_2Eindex__compare__def,axiom,
( ( ( c_2Equote_2Eindex__compare @ c_2Equote_2EEnd__idx @ c_2Equote_2EEnd__idx )
= c_2EternaryComparisons_2EEQUAL )
& ! [V0v10: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ c_2Equote_2EEnd__idx @ ( c_2Equote_2ELeft__idx @ V0v10 ) )
= c_2EternaryComparisons_2ELESS )
& ! [V1v11: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ c_2Equote_2EEnd__idx @ ( c_2Equote_2ERight__idx @ V1v11 ) )
= c_2EternaryComparisons_2ELESS )
& ! [V2v2: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ ( c_2Equote_2ELeft__idx @ V2v2 ) @ c_2Equote_2EEnd__idx )
= c_2EternaryComparisons_2EGREATER )
& ! [V3v3: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ ( c_2Equote_2ERight__idx @ V3v3 ) @ c_2Equote_2EEnd__idx )
= c_2EternaryComparisons_2EGREATER )
& ! [V4n_27: tyop_2Equote_2Eindex,V5m_27: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ ( c_2Equote_2ELeft__idx @ V4n_27 ) @ ( c_2Equote_2ELeft__idx @ V5m_27 ) )
= ( c_2Equote_2Eindex__compare @ V4n_27 @ V5m_27 ) )
& ! [V6n_27: tyop_2Equote_2Eindex,V7m_27: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ ( c_2Equote_2ELeft__idx @ V6n_27 ) @ ( c_2Equote_2ERight__idx @ V7m_27 ) )
= c_2EternaryComparisons_2ELESS )
& ! [V8n_27: tyop_2Equote_2Eindex,V9m_27: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ ( c_2Equote_2ERight__idx @ V8n_27 ) @ ( c_2Equote_2ERight__idx @ V9m_27 ) )
= ( c_2Equote_2Eindex__compare @ V8n_27 @ V9m_27 ) )
& ! [V10n_27: tyop_2Equote_2Eindex,V11m_27: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Eindex__compare @ ( c_2Equote_2ERight__idx @ V10n_27 ) @ ( c_2Equote_2ELeft__idx @ V11m_27 ) )
= c_2EternaryComparisons_2EGREATER ) ) ).
thf(thm_2Equote_2Ecompare__index__equal,axiom,
! [V0i1: tyop_2Equote_2Eindex,V1i2: tyop_2Equote_2Eindex] :
( ( ( c_2Equote_2Eindex__compare @ V0i1 @ V1i2 )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V0i1 = V1i2 ) ) ).
thf(thm_2Equote_2Ecompare__list__index,axiom,
! [V0l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V1l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V0l1 @ V1l2 )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V0l1 = V1l2 ) ) ).
thf(thm_2Equote_2Edatatype__varmap,axiom,
! [A_27a: $tType,V0varmap: ( tyop_2Equote_2Evarmap @ A_27a ) > ( A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0varmap @ ( c_2Equote_2EEmpty__vm @ A_27a ) @ ( c_2Equote_2ENode__vm @ A_27a ) ) ) ).
thf(thm_2Equote_2Evarmap__11,axiom,
! [A_27a: $tType,V0a0: A_27a,V1a1: tyop_2Equote_2Evarmap @ A_27a,V2a2: tyop_2Equote_2Evarmap @ A_27a,V3a0_27: A_27a,V4a1_27: tyop_2Equote_2Evarmap @ A_27a,V5a2_27: tyop_2Equote_2Evarmap @ A_27a] :
( ( ( c_2Equote_2ENode__vm @ A_27a @ V0a0 @ V1a1 @ V2a2 )
= ( c_2Equote_2ENode__vm @ A_27a @ V3a0_27 @ V4a1_27 @ V5a2_27 ) )
<=> ( ( V0a0 = V3a0_27 )
& ( V1a1 = V4a1_27 )
& ( V2a2 = V5a2_27 ) ) ) ).
thf(thm_2Equote_2Evarmap__distinct,axiom,
! [A_27a: $tType,V0a2: tyop_2Equote_2Evarmap @ A_27a,V1a1: tyop_2Equote_2Evarmap @ A_27a,V2a0: A_27a] :
( (~)
@ ( ( c_2Equote_2EEmpty__vm @ A_27a )
= ( c_2Equote_2ENode__vm @ A_27a @ V2a0 @ V1a1 @ V0a2 ) ) ) ).
thf(thm_2Equote_2Evarmap__nchotomy,axiom,
! [A_27a: $tType,V0vv: tyop_2Equote_2Evarmap @ A_27a] :
( ( V0vv
= ( c_2Equote_2EEmpty__vm @ A_27a ) )
| ? [V1a: A_27a,V2v: tyop_2Equote_2Evarmap @ A_27a,V3v0: tyop_2Equote_2Evarmap @ A_27a] :
( V0vv
= ( c_2Equote_2ENode__vm @ A_27a @ V1a @ V2v @ V3v0 ) ) ) ).
thf(thm_2Equote_2Evarmap__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0: A_27b,V1f1: A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b > A_27b > A_27b] :
? [V2fn: ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b] :
( ( ( V2fn @ ( c_2Equote_2EEmpty__vm @ A_27a ) )
= V0f0 )
& ! [V3a0: A_27a,V4a1: tyop_2Equote_2Evarmap @ A_27a,V5a2: tyop_2Equote_2Evarmap @ A_27a] :
( ( V2fn @ ( c_2Equote_2ENode__vm @ A_27a @ V3a0 @ V4a1 @ V5a2 ) )
= ( V1f1 @ V3a0 @ V4a1 @ V5a2 @ ( V2fn @ V4a1 ) @ ( V2fn @ V5a2 ) ) ) ) ).
thf(thm_2Equote_2Evarmap__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Equote_2Evarmap @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Equote_2EEmpty__vm @ A_27a ) )
& ! [V1v: tyop_2Equote_2Evarmap @ A_27a,V2v0: tyop_2Equote_2Evarmap @ A_27a] :
( ( ( V0P @ V1v )
& ( V0P @ V2v0 ) )
=> ! [V3a: A_27a] : ( V0P @ ( c_2Equote_2ENode__vm @ A_27a @ V3a @ V1v @ V2v0 ) ) ) )
=> ! [V4v: tyop_2Equote_2Evarmap @ A_27a] : ( V0P @ V4v ) ) ).
thf(thm_2Equote_2Evarmap__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0v_27: A_27b,V1f_27: A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b,V2M: tyop_2Equote_2Evarmap @ A_27a,V3M_27: tyop_2Equote_2Evarmap @ A_27a,V4v: A_27b,V5f: A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b] :
( ( ( V2M = V3M_27 )
& ( ( V3M_27
= ( c_2Equote_2EEmpty__vm @ A_27a ) )
=> ( V4v = V0v_27 ) )
& ! [V6a0: A_27a,V7a1: tyop_2Equote_2Evarmap @ A_27a,V8a2: tyop_2Equote_2Evarmap @ A_27a] :
( ( V3M_27
= ( c_2Equote_2ENode__vm @ A_27a @ V6a0 @ V7a1 @ V8a2 ) )
=> ( ( V5f @ V6a0 @ V7a1 @ V8a2 )
= ( V1f_27 @ V6a0 @ V7a1 @ V8a2 ) ) ) )
=> ( ( c_2Equote_2Evarmap__CASE @ A_27a @ A_27b @ V2M @ V4v @ V5f )
= ( c_2Equote_2Evarmap__CASE @ A_27a @ A_27b @ V3M_27 @ V0v_27 @ V1f_27 ) ) ) ).
thf(thm_2Equote_2Evarmap__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Equote_2Evarmap @ A_27a,V1v_27: A_27b,V2v: A_27b,V3f: A_27a > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27b] :
( ( ( c_2Equote_2Evarmap__CASE @ A_27a @ A_27b @ V0x @ V2v @ V3f )
= V1v_27 )
<=> ( ( ( V0x
= ( c_2Equote_2EEmpty__vm @ A_27a ) )
& ( V2v = V1v_27 ) )
| ? [V4a: A_27a,V5v_27_27: tyop_2Equote_2Evarmap @ A_27a,V6v0: tyop_2Equote_2Evarmap @ A_27a] :
( ( V0x
= ( c_2Equote_2ENode__vm @ A_27a @ V4a @ V5v_27_27 @ V6v0 ) )
& ( ( V3f @ V4a @ V5v_27_27 @ V6v0 )
= V1v_27 ) ) ) ) ).
thf(thm_2Equote_2Evarmap__find__ind,axiom,
! [A_27a: $tType,V0P: tyop_2Equote_2Eindex > ( tyop_2Equote_2Evarmap @ A_27a ) > $o] :
( ( ! [V1x: A_27a,V2v1: tyop_2Equote_2Evarmap @ A_27a,V3v2: tyop_2Equote_2Evarmap @ A_27a] : ( V0P @ c_2Equote_2EEnd__idx @ ( c_2Equote_2ENode__vm @ A_27a @ V1x @ V2v1 @ V3v2 ) )
& ! [V4i1: tyop_2Equote_2Eindex,V5x: A_27a,V6v1: tyop_2Equote_2Evarmap @ A_27a,V7v2: tyop_2Equote_2Evarmap @ A_27a] :
( ( V0P @ V4i1 @ V7v2 )
=> ( V0P @ ( c_2Equote_2ERight__idx @ V4i1 ) @ ( c_2Equote_2ENode__vm @ A_27a @ V5x @ V6v1 @ V7v2 ) ) )
& ! [V8i1: tyop_2Equote_2Eindex,V9x: A_27a,V10v1: tyop_2Equote_2Evarmap @ A_27a,V11v2: tyop_2Equote_2Evarmap @ A_27a] :
( ( V0P @ V8i1 @ V10v1 )
=> ( V0P @ ( c_2Equote_2ELeft__idx @ V8i1 ) @ ( c_2Equote_2ENode__vm @ A_27a @ V9x @ V10v1 @ V11v2 ) ) )
& ! [V12i: tyop_2Equote_2Eindex] : ( V0P @ V12i @ ( c_2Equote_2EEmpty__vm @ A_27a ) ) )
=> ! [V13v: tyop_2Equote_2Eindex,V14v1: tyop_2Equote_2Evarmap @ A_27a] : ( V0P @ V13v @ V14v1 ) ) ).
thf(thm_2Equote_2Evarmap__find__def,axiom,
! [A_27a: $tType] :
( ! [V0x: A_27a,V1v2: tyop_2Equote_2Evarmap @ A_27a,V2v1: tyop_2Equote_2Evarmap @ A_27a] :
( ( c_2Equote_2Evarmap__find @ A_27a @ c_2Equote_2EEnd__idx @ ( c_2Equote_2ENode__vm @ A_27a @ V0x @ V2v1 @ V1v2 ) )
= V0x )
& ! [V3x: A_27a,V4v2: tyop_2Equote_2Evarmap @ A_27a,V5v1: tyop_2Equote_2Evarmap @ A_27a,V6i1: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Evarmap__find @ A_27a @ ( c_2Equote_2ERight__idx @ V6i1 ) @ ( c_2Equote_2ENode__vm @ A_27a @ V3x @ V5v1 @ V4v2 ) )
= ( c_2Equote_2Evarmap__find @ A_27a @ V6i1 @ V4v2 ) )
& ! [V7x: A_27a,V8v2: tyop_2Equote_2Evarmap @ A_27a,V9v1: tyop_2Equote_2Evarmap @ A_27a,V10i1: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Evarmap__find @ A_27a @ ( c_2Equote_2ELeft__idx @ V10i1 ) @ ( c_2Equote_2ENode__vm @ A_27a @ V7x @ V9v1 @ V8v2 ) )
= ( c_2Equote_2Evarmap__find @ A_27a @ V10i1 @ V9v1 ) )
& ! [V11i: tyop_2Equote_2Eindex] :
( ( c_2Equote_2Evarmap__find @ A_27a @ V11i @ ( c_2Equote_2EEmpty__vm @ A_27a ) )
= ( c_2Emin_2E_40 @ A_27a
@ ^ [V12x: A_27a] : c_2Ebool_2ET ) ) ) ).
%------------------------------------------------------------------------------