TPTP Problem File: ITP007^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP007^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2EpatternMatches_2EPMATCH__FLATTEN__FUN__PMATCH__ROW.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_2EpatternMatches_2EPMATCH__FLATTEN__FUN__PMATCH__ROW.p [Gau19]
% : HL403001^2.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 113 ( 5 unt; 29 typ; 0 def)
% Number of atoms : 776 ( 37 equ; 0 cnn)
% Maximal formula atoms : 21 ( 9 avg)
% Number of connectives : 1511 ( 53 ~; 45 |; 57 &;1102 @)
% ( 78 <=>; 176 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 34 ( 34 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 34 usr; 16 con; 0-3 aty)
% Number of variables : 208 ( 9 ^; 194 !; 5 ?; 208 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
thf(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
thf(mem_c_2Ebool_2ET,axiom,
mem @ c_2Ebool_2ET @ bool ).
thf(ax_true_p,axiom,
p @ c_2Ebool_2ET ).
thf(tp_c_2Emin_2E_40,type,
c_2Emin_2E_40: del > $i ).
thf(mem_c_2Emin_2E_40,axiom,
! [A_27a: del] : ( mem @ ( c_2Emin_2E_40 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ A_27a ) ) ).
thf(tp_ty_2Eoption_2Eoption,type,
ty_2Eoption_2Eoption: del > del ).
thf(tp_c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME: del > $i ).
thf(mem_c_2Eoption_2ESOME,axiom,
! [A_27a: del] : ( mem @ ( c_2Eoption_2ESOME @ A_27a ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
thf(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_3F @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_ex_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A ) @ Q ) )
<=> ? [X: $i] :
( ( mem @ X @ A )
& ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(tp_c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND: del > $i ).
thf(mem_c_2Ebool_2ECOND,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2ECOND @ A_27a ) @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Eoption_2EOPTION__MAP,type,
c_2Eoption_2EOPTION__MAP: del > del > $i ).
thf(mem_c_2Eoption_2EOPTION__MAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) ) ).
thf(tp_c_2EpatternMatches_2EPMATCH__ROW,type,
c_2EpatternMatches_2EPMATCH__ROW: del > del > del > $i ).
thf(mem_c_2EpatternMatches_2EPMATCH__ROW,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EpatternMatches_2EPMATCH__ROW @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE: del > $i ).
thf(mem_c_2Eoption_2ENONE,axiom,
! [A_27a: del] : ( mem @ ( c_2Eoption_2ENONE @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ).
thf(tp_c_2EpatternMatches_2EPMATCH__ROW__COND,type,
c_2EpatternMatches_2EPMATCH__ROW__COND: del > del > $i ).
thf(mem_c_2EpatternMatches_2EPMATCH__ROW__COND,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EpatternMatches_2EPMATCH__ROW__COND @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Eoption_2Esome,type,
c_2Eoption_2Esome: del > $i ).
thf(mem_c_2Eoption_2Esome,axiom,
! [A_27a: del] : ( mem @ ( c_2Eoption_2Esome @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(tp_c_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE: del > del > $i ).
thf(mem_c_2Eoption_2Eoption__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27a @ A_27b ) @ A_27b ) ) ) ) ).
thf(tp_c_2EpatternMatches_2EPMATCH__FLATTEN__FUN,type,
c_2EpatternMatches_2EPMATCH__FLATTEN__FUN: del > del > del > $i ).
thf(mem_c_2EpatternMatches_2EPMATCH__FLATTEN__FUN,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EpatternMatches_2EPMATCH__FLATTEN__FUN @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
thf(mem_c_2Ebool_2EF,axiom,
mem @ c_2Ebool_2EF @ bool ).
thf(ax_false_p,axiom,
~ ( p @ c_2Ebool_2EF ) ).
thf(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
thf(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem @ c_2Emin_2E_3D_3D_3E @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_imp_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ Q ) @ R ) )
<=> ( ( p @ Q )
=> ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
thf(mem_c_2Ebool_2E_5C_2F,axiom,
mem @ c_2Ebool_2E_5C_2F @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_or_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ Q ) @ R ) )
<=> ( ( p @ Q )
| ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
thf(mem_c_2Ebool_2E_2F_5C,axiom,
mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_and_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ Q ) @ R ) )
<=> ( ( p @ Q )
& ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
thf(mem_c_2Ebool_2E_7E,axiom,
mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).
thf(ax_neg_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
<=> ~ ( p @ Q ) ) ) ).
thf(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
thf(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : ( mem @ ( c_2Emin_2E_3D @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).
thf(ax_eq_p,axiom,
! [A: del,X: $i] :
( ( mem @ X @ A )
=> ! [Y: $i] :
( ( mem @ Y @ A )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A ) @ X ) @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
thf(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_all_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A ) @ Q ) )
<=> ! [X: $i] :
( ( mem @ X @ A )
=> ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
<=> $true )
| ( ( p @ V0t )
<=> $false ) ) ) ).
thf(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
thf(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
=> ( p @ V1t2 ) )
=> ( ( ( p @ V1t2 )
=> ( p @ V0t1 ) )
=> ( ( p @ V0t1 )
<=> ( p @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( $false
=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( p @ V0t )
| ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__F,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> $false )
=> ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ( p @ V0t )
=> ( ( p @ V0t )
=> $false ) ) ) ).
thf(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
& ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
& $true )
<=> ( p @ V0t ) )
& ( ( $false
& ( p @ V0t ) )
<=> $false )
& ( ( ( p @ V0t )
& $false )
<=> $false )
& ( ( ( p @ V0t )
& ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
| ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
| $true )
<=> $true )
& ( ( $false
| ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
| $false )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
| ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
=> $true )
<=> $true )
& ( ( $false
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( V0x = V0x )
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
thf(conj_thm_2Ebool_2EFUN__EQ__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ( ( V0f = V1g )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ V0f @ V2x )
= ( ap @ V1g @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
<=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $true )
<=> ( p @ V0t ) )
& ( ( $false
<=> ( p @ V0t ) )
<=> ~ ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ESELECT__ELIM__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) )
=> ( p @ ( ap @ V1Q @ ( ap @ ( c_2Emin_2E_40 @ A_27a ) @ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ~ ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ V0P @ V1x ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ~ ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ~ ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ V0P @ V1x ) ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ~ ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) )
<=> ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ( p @ V1B )
| ( p @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__COMM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ( p @ V1B )
| ( p @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
& ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
| ~ ( p @ V1B ) ) )
& ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
& ~ ( p @ V1B ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
=> ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
| ( p @ V1B ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ bool )
=> ( ( ( p @ V0t1 )
=> ( ( p @ V1t2 )
=> ( p @ V2t3 ) ) )
<=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
=> ( p @ V2t3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1x_27: $i] :
( ( mem @ V1x_27 @ bool )
=> ! [V2y: $i] :
( ( mem @ V2y @ bool )
=> ! [V3y_27: $i] :
( ( mem @ V3y_27 @ bool )
=> ( ( ( ( p @ V0x )
<=> ( p @ V1x_27 ) )
& ( ( p @ V1x_27 )
=> ( ( p @ V2y )
<=> ( p @ V3y_27 ) ) ) )
=> ( ( ( p @ V0x )
=> ( p @ V2y ) )
<=> ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EAND__CONG,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ( p @ V2Q )
=> ( ( p @ V0P )
<=> ( p @ V1P_27 ) ) )
& ( ( p @ V1P_27 )
=> ( ( p @ V2Q )
<=> ( p @ V3Q_27 ) ) ) )
=> ( ( ( p @ V0P )
& ( p @ V2Q ) )
<=> ( ( p @ V1P_27 )
& ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EUNWIND__FORALL__THM2,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( V2x = V1v )
=> ( p @ ( ap @ V0f @ V2x ) ) ) )
<=> ( p @ ( ap @ V0f @ V1v ) ) ) ) ) ).
thf(conj_thm_2Eoption_2Eoption__nchotomy,axiom,
! [A_27a: del,V0opt: $i] :
( ( mem @ V0opt @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ( ( V0opt
= ( c_2Eoption_2ENONE @ A_27a ) )
| ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V0opt
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Eoption_2Eoption__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0v: $i] :
( ( mem @ V0v @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ V0v ) @ V1f )
= V0v ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2x ) ) @ V3v ) @ V4f )
= ( ap @ V4f @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eoption_2ESOME__11,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0x )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1y ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(ax_thm_2Eoption_2EOPTION__MAP__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ ( ap @ V0f @ V1x ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V2f ) @ ( c_2Eoption_2ENONE @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ).
thf(conj_thm_2Eoption_2EIF__EQUALS__OPTION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V0P ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) ) @ ( c_2Eoption_2ENONE @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ~ ( p @ V0P ) )
& ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V0P ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ( p @ V0P ) )
& ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V0P ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) ) @ ( c_2Eoption_2ENONE @ A_27a ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2y ) )
<=> ( ( p @ V0P )
& ( V1x = V2y ) ) )
& ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V0P ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2y ) )
<=> ( ~ ( p @ V0P )
& ( V1x = V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Eoption_2EOPTION__MAP__EQ__NONE__both__ways,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a ) @ V0f ) @ V1x )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ( V1x
= ( c_2Eoption_2ENONE @ A_27b ) ) )
& ( ( ( c_2Eoption_2ENONE @ A_27a )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a ) @ V0f ) @ V1x ) )
<=> ( V1x
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ).
thf(ax_thm_2Eoption_2Esome__def,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Eoption_2Esome @ A_27a ) @ V0P )
= ( ap
@ ( ap
@ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0P @ V1x ) ) ) )
@ ( ap @ ( c_2Eoption_2ESOME @ A_27a )
@ ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0P @ V2x ) ) ) ) )
@ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(ax_thm_2EpatternMatches_2EPMATCH__ROW__COND__def,axiom,
! [A_27a: del,A_27b: del,V0pat: $i] :
( ( mem @ V0pat @ ( arr @ A_27a @ A_27b ) )
=> ! [V1guard: $i] :
( ( mem @ V1guard @ ( arr @ A_27a @ bool ) )
=> ! [V2inp: $i] :
( ( mem @ V2inp @ A_27b )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EpatternMatches_2EPMATCH__ROW__COND @ A_27a @ A_27b ) @ V0pat ) @ V1guard ) @ V2inp ) @ V3v ) )
<=> ( ( ( ap @ V0pat @ V3v )
= V2inp )
& ( p @ ( ap @ V1guard @ V3v ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EpatternMatches_2EPMATCH__ROW__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0pat: $i] :
( ( mem @ V0pat @ ( arr @ A_27b @ A_27c ) )
=> ! [V1guard: $i] :
( ( mem @ V1guard @ ( arr @ A_27b @ bool ) )
=> ! [V2rhs: $i] :
( ( mem @ V2rhs @ ( arr @ A_27b @ A_27a ) )
=> ! [V3i: $i] :
( ( mem @ V3i @ A_27c )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EpatternMatches_2EPMATCH__ROW @ A_27a @ A_27b @ A_27c ) @ V0pat ) @ V1guard ) @ V2rhs ) @ V3i )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a ) @ V2rhs )
@ ( ap @ ( c_2Eoption_2Esome @ A_27b )
@ ( lam @ A_27b
@ ^ [V4v: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EpatternMatches_2EPMATCH__ROW__COND @ A_27b @ A_27c ) @ V0pat ) @ V1guard ) @ V3i ) @ V4v ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EpatternMatches_2EPMATCH__FLATTEN__FUN__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ bool ) )
=> ! [V2row: $i] :
( ( mem @ V2row @ ( arr @ A_27b @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27c )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EpatternMatches_2EPMATCH__FLATTEN__FUN @ A_27a @ A_27b @ A_27c ) @ V0p ) @ V1g ) @ V2row ) @ V3v )
= ( ap
@ ( ap
@ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( ap @ ( c_2Eoption_2Esome @ A_27b )
@ ( lam @ A_27b
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EpatternMatches_2EPMATCH__ROW__COND @ A_27b @ A_27c ) @ V0p ) @ V1g ) @ V3v ) @ V4x ) ) ) )
@ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ A_27b
@ ^ [V5x: $i] : ( ap @ ( ap @ V2row @ V5x ) @ V5x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( p @ V0A )
=> ( ~ ( p @ V0A )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( ( p @ V0A )
=> $false )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ~ ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( p @ V0A )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( ~ ( p @ V0A )
=> $false )
=> ( ( ( p @ V0A )
=> $false )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
<=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q )
| ( p @ V2r ) )
& ( ( p @ V0p )
| ~ ( p @ V2r )
| ~ ( p @ V1q ) )
& ( ( p @ V1q )
| ~ ( p @ V2r )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
& ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
| ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ~ ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ( ( p @ V0p )
<=> ~ ( p @ V1q ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2EpatternMatches_2EPMATCH__FLATTEN__FUN__PMATCH__ROW,conjecture,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1x1: $i] :
( ( mem @ V1x1 @ A_27a )
=> ! [V2x2: $i] :
( ( mem @ V2x2 @ A_27a )
=> ( ( ( ap @ V0p @ V1x1 )
= ( ap @ V0p @ V2x2 ) )
=> ( V1x1 = V2x2 ) ) ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ bool ) )
=> ! [V4p_27: $i] :
( ( mem @ V4p_27 @ ( arr @ A_27c @ A_27a ) )
=> ! [V5g_27: $i] :
( ( mem @ V5g_27 @ ( arr @ A_27a @ ( arr @ A_27c @ bool ) ) )
=> ! [V6r_27: $i] :
( ( mem @ V6r_27 @ ( arr @ A_27a @ ( arr @ A_27c @ A_27d ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EpatternMatches_2EPMATCH__FLATTEN__FUN @ A_27d @ A_27a @ A_27b ) @ V0p ) @ V3g )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ ( ap @ ( c_2EpatternMatches_2EPMATCH__ROW @ A_27d @ A_27c @ A_27a ) @ V4p_27 ) @ ( ap @ V5g_27 @ V7x ) ) @ ( ap @ V6r_27 @ V7x ) ) ) )
= ( ap
@ ( ap
@ ( ap @ ( c_2EpatternMatches_2EPMATCH__ROW @ A_27d @ A_27c @ A_27b )
@ ( lam @ A_27c
@ ^ [V8x: $i] : ( ap @ V0p @ ( ap @ V4p_27 @ V8x ) ) ) )
@ ( lam @ A_27c
@ ^ [V9x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V3g @ ( ap @ V4p_27 @ V9x ) ) ) @ ( ap @ ( ap @ V5g_27 @ ( ap @ V4p_27 @ V9x ) ) @ V9x ) ) ) )
@ ( lam @ A_27c
@ ^ [V10x: $i] : ( ap @ ( ap @ V6r_27 @ ( ap @ V4p_27 @ V10x ) ) @ V10x ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------