TPTP Problem File: SWW688_1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW688_1 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Software Verification
% Problem : WP parameter bottom up mergesort 33
% Version : Especial.
% English :
% Refs : [Pas16] Paskevich (2016), Email to Geoff Sutcliffe
% Source : [Pas16]
% Names : WP_parameter_bottom_up_mergesort33.p [Pas16]
% Status : Theorem
% Rating : ? v7.5.0
% Syntax : Number of formulae : 90 ( 17 unt; 35 typ; 0 def)
% Number of atoms : 173 ( 55 equ)
% Maximal formula atoms : 15 ( 1 avg)
% Number of connectives : 119 ( 7 ~; 2 |; 50 &)
% ( 12 <=>; 48 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of X terms : 6 ( 0 []; 3 ite; 3 let)
% Number arithmetic : 217 ( 76 atm; 12 fun; 31 num; 98 var)
% Number of types : 5 ( 3 usr; 1 ari)
% Number of type conns : 72 ( 29 >; 43 *; 0 +; 0 <<)
% Number of predicates : 15 ( 12 usr; 0 prp; 1-7 aty)
% Number of functors : 29 ( 23 usr; 8 con; 0-5 aty)
% Number of variables : 239 ( 215 !; 2 ?; 239 :)
% ( 22 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TX1_THM_EQU_ARI
% Comments :
% Bugfixes : v7.1.0 - Updated to new $ite and $let syntax.
% : v7.5.0 - Fixed $let expression in conjecture.
%------------------------------------------------------------------------------
tff(bool,type,
bool: $tType ).
tff(true1,type,
true1: bool ).
tff(false1,type,
false1: bool ).
tff(match_bool,type,
match_bool:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(match_bool_True,axiom,
! [A: $tType,Z: A,Z1: A] : match_bool(A,true1,Z,Z1) = Z ).
tff(match_bool_False,axiom,
! [A: $tType,Z: A,Z1: A] : match_bool(A,false1,Z,Z1) = Z1 ).
tff(true_False,axiom,
true1 != false1 ).
tff(bool_inversion,axiom,
! [U: bool] :
( ( U = true1 )
| ( U = false1 ) ) ).
tff(tuple0,type,
tuple0: $tType ).
tff(tuple01,type,
tuple01: tuple0 ).
tff(tuple0_inversion,axiom,
! [U: tuple0] : U = tuple01 ).
tff(map,type,
map: ( $tType * $tType ) > $tType ).
tff(get,type,
get:
!>[A: $tType,B: $tType] : ( ( map(A,B) * A ) > B ) ).
tff(set,type,
set:
!>[A: $tType,B: $tType] : ( ( map(A,B) * A * B ) > map(A,B) ) ).
tff(select_eq,axiom,
! [A: $tType,B: $tType,M: map(A,B),A1: A,A2: A,B1: B] :
( ( A1 = A2 )
=> ( get(A,B,set(A,B,M,A1,B1),A2) = B1 ) ) ).
tff(select_neq,axiom,
! [A: $tType,B: $tType,M: map(A,B),A1: A,A2: A,B1: B] :
( ( A1 != A2 )
=> ( get(A,B,set(A,B,M,A1,B1),A2) = get(A,B,M,A2) ) ) ).
tff(array,type,
array: $tType > $tType ).
tff(mk_array,type,
mk_array:
!>[A: $tType] : ( ( $int * map($int,A) ) > array(A) ) ).
tff(length,type,
length:
!>[A: $tType] : ( array(A) > $int ) ).
tff(length_def,axiom,
! [A: $tType,U: $int,U1: map($int,A)] : length(A,mk_array(A,U,U1)) = U ).
tff(elts,type,
elts:
!>[A: $tType] : ( array(A) > map($int,A) ) ).
tff(elts_def,axiom,
! [A: $tType,U: $int,U1: map($int,A)] : elts(A,mk_array(A,U,U1)) = U1 ).
tff(array_inversion,axiom,
! [A: $tType,U: array(A)] : U = mk_array(A,length(A,U),elts(A,U)) ).
tff(get1,type,
get1:
!>[A: $tType] : ( ( array(A) * $int ) > A ) ).
tff(get_def,axiom,
! [A: $tType,A1: array(A),I: $int] : get1(A,A1,I) = get($int,A,elts(A,A1),I) ).
tff(set1,type,
set1:
!>[A: $tType] : ( ( array(A) * $int * A ) > array(A) ) ).
tff(set_def,axiom,
! [A: $tType,A1: array(A),I: $int,V: A] : set1(A,A1,I,V) = mk_array(A,length(A,A1),set($int,A,elts(A,A1),I,V)) ).
tff(elt2,type,
elt2: $tType ).
tff(le2,type,
le2: ( elt2 * elt2 ) > $o ).
tff(refl2,axiom,
! [X: elt2] : le2(X,X) ).
tff(trans2,axiom,
! [X: elt2,Y: elt2,Z: elt2] :
( le2(X,Y)
=> ( le2(Y,Z)
=> le2(X,Z) ) ) ).
tff(total2,axiom,
! [X: elt2,Y: elt2] :
( le2(X,Y)
| le2(Y,X) ) ).
tff(sorted_sub2,type,
sorted_sub2: ( array(elt2) * $int * $int ) > $o ).
tff(sorted_sub_def2,axiom,
! [A: array(elt2),L: $int,U: $int] :
( sorted_sub2(A,L,U)
<=> ! [I1: $int,I2: $int] :
( ( $lesseq(L,I1)
& $lesseq(I1,I2)
& $less(I2,U) )
=> le2(get1(elt2,A,I1),get1(elt2,A,I2)) ) ) ).
tff(sorted2,type,
sorted2: array(elt2) > $o ).
tff(sorted_def2,axiom,
! [A: array(elt2)] :
( sorted2(A)
<=> ! [I1: $int,I2: $int] :
( ( $lesseq(0,I1)
& $lesseq(I1,I2)
& $less(I2,length(elt2,A)) )
=> le2(get1(elt2,A,I1),get1(elt2,A,I2)) ) ) ).
tff(ref,type,
ref: $tType > $tType ).
tff(mk_ref,type,
mk_ref:
!>[A: $tType] : ( A > ref(A) ) ).
tff(contents,type,
contents:
!>[A: $tType] : ( ref(A) > A ) ).
tff(contents_def2,axiom,
! [A: $tType,U: A] : contents(A,mk_ref(A,U)) = U ).
tff(ref_inversion2,axiom,
! [A: $tType,U: ref(A)] : U = mk_ref(A,contents(A,U)) ).
tff(occ,type,
occ:
!>[A: $tType] : ( ( A * map($int,A) * $int * $int ) > $int ) ).
tff(occ_empty,axiom,
! [A: $tType,V: A,M: map($int,A),L: $int,U: $int] :
( $lesseq(U,L)
=> ( occ(A,V,M,L,U) = 0 ) ) ).
tff(occ_right_no_add,axiom,
! [A: $tType,V: A,M: map($int,A),L: $int,U: $int] :
( $less(L,U)
=> ( ( get($int,A,M,$difference(U,1)) != V )
=> ( occ(A,V,M,L,U) = occ(A,V,M,L,$difference(U,1)) ) ) ) ).
tff(occ_right_add,axiom,
! [A: $tType,V: A,M: map($int,A),L: $int,U: $int] :
( $less(L,U)
=> ( ( get($int,A,M,$difference(U,1)) = V )
=> ( occ(A,V,M,L,U) = $sum(1,occ(A,V,M,L,$difference(U,1))) ) ) ) ).
tff(occ_bounds,axiom,
! [A: $tType,V: A,M: map($int,A),L: $int,U: $int] :
( $lesseq(L,U)
=> ( $lesseq(0,occ(A,V,M,L,U))
& $lesseq(occ(A,V,M,L,U),$difference(U,L)) ) ) ).
tff(occ_append,axiom,
! [A: $tType,V: A,M: map($int,A),L: $int,Mid: $int,U: $int] :
( ( $lesseq(L,Mid)
& $lesseq(Mid,U) )
=> ( occ(A,V,M,L,U) = $sum(occ(A,V,M,L,Mid),occ(A,V,M,Mid,U)) ) ) ).
tff(occ_neq,axiom,
! [A: $tType,V: A,M: map($int,A),L: $int,U: $int] :
( ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get($int,A,M,I) != V ) )
=> ( occ(A,V,M,L,U) = 0 ) ) ).
tff(occ_exists,axiom,
! [A: $tType,V: A,M: map($int,A),L: $int,U: $int] :
( $less(0,occ(A,V,M,L,U))
=> ? [I: $int] :
( $lesseq(L,I)
& $less(I,U)
& ( get($int,A,M,I) = V ) ) ) ).
tff(occ_pos,axiom,
! [A: $tType,M: map($int,A),L: $int,U: $int,I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> $less(0,occ(A,get($int,A,M,I),M,L,U)) ) ).
tff(occ_eq,axiom,
! [A: $tType,V: A,M1: map($int,A),M2: map($int,A),L: $int,U: $int] :
( ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get($int,A,M1,I) = get($int,A,M2,I) ) )
=> ( occ(A,V,M1,L,U) = occ(A,V,M2,L,U) ) ) ).
tff(permut,type,
permut:
!>[A: $tType] : ( ( map($int,A) * map($int,A) * $int * $int ) > $o ) ).
tff(permut_def,axiom,
! [A: $tType,M1: map($int,A),M2: map($int,A),L: $int,U: $int] :
( permut(A,M1,M2,L,U)
<=> ! [V: A] : occ(A,V,M1,L,U) = occ(A,V,M2,L,U) ) ).
tff(permut_trans,axiom,
! [A: $tType,A1: map($int,A),A2: map($int,A),A3: map($int,A),L: $int,U: $int] :
( permut(A,A1,A2,L,U)
=> ( permut(A,A2,A3,L,U)
=> permut(A,A1,A3,L,U) ) ) ).
tff(permut_exists,axiom,
! [A: $tType,A1: map($int,A),A2: map($int,A),L: $int,U: $int,I: $int] :
( permut(A,A1,A2,L,U)
=> ( ( $lesseq(L,I)
& $less(I,U) )
=> ? [J: $int] :
( $lesseq(L,J)
& $less(J,U)
& ( get($int,A,A1,J) = get($int,A,A2,I) ) ) ) ) ).
tff(map_eq_sub,type,
map_eq_sub:
!>[A: $tType] : ( ( map($int,A) * map($int,A) * $int * $int ) > $o ) ).
tff(map_eq_sub_def,axiom,
! [A: $tType,A1: map($int,A),A2: map($int,A),L: $int,U: $int] :
( map_eq_sub(A,A1,A2,L,U)
<=> ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get($int,A,A1,I) = get($int,A,A2,I) ) ) ) ).
tff(array_eq_sub,type,
array_eq_sub:
!>[A: $tType] : ( ( array(A) * array(A) * $int * $int ) > $o ) ).
tff(array_eq_sub_def,axiom,
! [A: $tType,A1: array(A),A2: array(A),L: $int,U: $int] :
( array_eq_sub(A,A1,A2,L,U)
<=> ( ( length(A,A1) = length(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length(A,A1))
& $lesseq(0,U)
& $lesseq(U,length(A,A1))
& map_eq_sub(A,elts(A,A1),elts(A,A2),L,U) ) ) ).
tff(array_eq,type,
array_eq:
!>[A: $tType] : ( ( array(A) * array(A) ) > $o ) ).
tff(array_eq_def,axiom,
! [A: $tType,A1: array(A),A2: array(A)] :
( array_eq(A,A1,A2)
<=> ( ( length(A,A1) = length(A,A2) )
& map_eq_sub(A,elts(A,A1),elts(A,A2),0,length(A,A1)) ) ) ).
tff(exchange,type,
exchange:
!>[A: $tType] : ( ( map($int,A) * map($int,A) * $int * $int * $int * $int ) > $o ) ).
tff(exchange_def,axiom,
! [A: $tType,A1: map($int,A),A2: map($int,A),L: $int,U: $int,I: $int,J: $int] :
( exchange(A,A1,A2,L,U,I,J)
<=> ( $lesseq(L,I)
& $less(I,U)
& $lesseq(L,J)
& $less(J,U)
& ( get($int,A,A1,I) = get($int,A,A2,J) )
& ( get($int,A,A1,J) = get($int,A,A2,I) )
& ! [K: $int] :
( ( $lesseq(L,K)
& $less(K,U) )
=> ( ( K != I )
=> ( ( K != J )
=> ( get($int,A,A1,K) = get($int,A,A2,K) ) ) ) ) ) ) ).
tff(exchange_set,axiom,
! [A: $tType,A1: map($int,A),L: $int,U: $int,I: $int,J: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( ( $lesseq(L,J)
& $less(J,U) )
=> exchange(A,A1,set($int,A,set($int,A,A1,I,get($int,A,A1,J)),J,get($int,A,A1,I)),L,U,I,J) ) ) ).
tff(exchange1,type,
exchange1:
!>[A: $tType] : ( ( array(A) * array(A) * $int * $int ) > $o ) ).
tff(exchange_def1,axiom,
! [A: $tType,A1: array(A),A2: array(A),I: $int,J: $int] :
( exchange1(A,A1,A2,I,J)
<=> ( ( length(A,A1) = length(A,A2) )
& exchange(A,elts(A,A1),elts(A,A2),0,length(A,A1),I,J) ) ) ).
tff(permut1,type,
permut1:
!>[A: $tType] : ( ( array(A) * array(A) * $int * $int ) > $o ) ).
tff(permut_def1,axiom,
! [A: $tType,A1: array(A),A2: array(A),L: $int,U: $int] :
( permut1(A,A1,A2,L,U)
<=> ( ( length(A,A1) = length(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length(A,A1))
& $lesseq(0,U)
& $lesseq(U,length(A,A1))
& permut(A,elts(A,A1),elts(A,A2),L,U) ) ) ).
tff(permut_sub,type,
permut_sub:
!>[A: $tType] : ( ( array(A) * array(A) * $int * $int ) > $o ) ).
tff(permut_sub_def,axiom,
! [A: $tType,A1: array(A),A2: array(A),L: $int,U: $int] :
( permut_sub(A,A1,A2,L,U)
<=> ( map_eq_sub(A,elts(A,A1),elts(A,A2),0,L)
& permut1(A,A1,A2,L,U)
& map_eq_sub(A,elts(A,A1),elts(A,A2),U,length(A,A1)) ) ) ).
tff(permut_all,type,
permut_all:
!>[A: $tType] : ( ( array(A) * array(A) ) > $o ) ).
tff(permut_all_def,axiom,
! [A: $tType,A1: array(A),A2: array(A)] :
( permut_all(A,A1,A2)
<=> ( ( length(A,A1) = length(A,A2) )
& permut(A,elts(A,A1),elts(A,A2),0,length(A,A1)) ) ) ).
tff(exchange_permut_sub,axiom,
! [A: $tType,A1: array(A),A2: array(A),I: $int,J: $int,L: $int,U: $int] :
( exchange1(A,A1,A2,I,J)
=> ( ( $lesseq(L,I)
& $less(I,U) )
=> ( ( $lesseq(L,J)
& $less(J,U) )
=> ( $lesseq(0,L)
=> ( $lesseq(U,length(A,A1))
=> permut_sub(A,A1,A2,L,U) ) ) ) ) ) ).
tff(permut_sub_weakening,axiom,
! [A: $tType,A1: array(A),A2: array(A),L1: $int,U1: $int,L2: $int,U2: $int] :
( permut_sub(A,A1,A2,L1,U1)
=> ( ( $lesseq(0,L2)
& $lesseq(L2,L1) )
=> ( ( $lesseq(U1,U2)
& $lesseq(U2,length(A,A1)) )
=> permut_sub(A,A1,A2,L2,U2) ) ) ) ).
tff(exchange_permut_all,axiom,
! [A: $tType,A1: array(A),A2: array(A),I: $int,J: $int] :
( exchange1(A,A1,A2,I,J)
=> permut_all(A,A1,A2) ) ).
tff(abs,type,
abs: $int > $int ).
tff(abs_def,axiom,
! [X: $int] :
$ite($lesseq(0,X),abs(X) = X,abs(X) = $uminus(X)) ).
tff(abs_le,axiom,
! [X: $int,Y: $int] :
( $lesseq(abs(X),Y)
<=> ( $lesseq($uminus(Y),X)
& $lesseq(X,Y) ) ) ).
tff(abs_pos,axiom,
! [X: $int] : $lesseq(0,abs(X)) ).
tff(min,type,
min: ( $int * $int ) > $int ).
tff(min_def,axiom,
! [X: $int,Y: $int] :
$ite($lesseq(X,Y),min(X,Y) = X,min(X,Y) = Y) ).
tff(max,type,
max: ( $int * $int ) > $int ).
tff(max_def,axiom,
! [X: $int,Y: $int] :
$ite($lesseq(X,Y),max(X,Y) = Y,max(X,Y) = X) ).
tff(min_r,axiom,
! [X: $int,Y: $int] :
( $lesseq(Y,X)
=> ( min(X,Y) = Y ) ) ).
tff(max_l,axiom,
! [X: $int,Y: $int] :
( $lesseq(Y,X)
=> ( max(X,Y) = X ) ) ).
tff(min_comm,axiom,
! [X: $int,Y: $int] : min(X,Y) = min(Y,X) ).
tff(max_comm,axiom,
! [X: $int,Y: $int] : max(X,Y) = max(Y,X) ).
tff(min_assoc,axiom,
! [X: $int,Y: $int,Z: $int] : min(min(X,Y),Z) = min(X,min(Y,Z)) ).
tff(max_assoc,axiom,
! [X: $int,Y: $int,Z: $int] : max(max(X,Y),Z) = max(X,max(Y,Z)) ).
tff(wP_parameter_bottom_up_mergesort,conjecture,
! [A: $int,A1: map($int,elt2)] :
$let(
a: array(elt2),
a:= mk_array(elt2,A,A1),
( $lesseq(0,A)
=> ! [Tmp: $int,Tmp1: map($int,elt2)] :
( ( $lesseq(0,Tmp)
& ( Tmp = A )
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,Tmp) )
=> ( get($int,elt2,Tmp1,I) = get($int,elt2,A1,I) ) ) )
=> ! [Len: $int,A2: map($int,elt2)] :
$let(
a1: array(elt2),
a1:= mk_array(elt2,A,A2),
( ( $lesseq(1,Len)
& permut_all(elt2,a,a1)
& ! [K: $int] :
$let(
l: $int,
l:= $product(K,Len),
( ( $lesseq(0,l)
& $less(l,A) )
=> sorted_sub2(a1,l,min(A,$sum(l,Len))) ) ) )
=> ( ~ $less(Len,A)
=> ( sorted_sub2(a1,$product(0,Len),min(A,$sum(0,Len)))
=> ( $lesseq(0,A)
=> sorted2(a1) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------