TPTP Problem File: SWW616_2.p
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%------------------------------------------------------------------------------
% File : SWW616_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Maximum subarray-T-WP parameter maximum subarray rec
% Version : Especial : Let and conditional terms encoded away.
% English :
% Refs : [Fil14] Filliatre (2014), Email to Geoff Sutcliffe
% : [BF+] Bobot et al. (URL), Toccata: Certified Programs and Cert
% Source : [Fil14]
% Names : maximum_subarray-T-WP_parameter_maximum_subarray_rec [Fil14]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.10 v7.4.0, 0.00 v6.4.0, 0.33 v6.3.0, 0.14 v6.2.0, 0.38 v6.1.0
% Syntax : Number of formulae : 108 ( 32 unt; 44 typ; 0 def)
% Number of atoms : 150 ( 53 equ)
% Maximal formula atoms : 26 ( 1 avg)
% Number of connectives : 95 ( 9 ~; 1 |; 34 &)
% ( 3 <=>; 48 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 267 ( 71 atm; 54 fun; 62 num; 80 var)
% Number of types : 8 ( 6 usr; 1 ari)
% Number of type conns : 66 ( 30 >; 36 *; 0 +; 0 <<)
% Number of predicates : 6 ( 3 usr; 0 prp; 2-3 aty)
% Number of functors : 42 ( 35 usr; 11 con; 0-5 aty)
% Number of variables : 170 ( 170 !; 0 ?; 170 :)
% SPC : TF0_THM_EQU_ARI
% Comments :
%------------------------------------------------------------------------------
tff(uni,type,
uni: $tType ).
tff(ty,type,
ty: $tType ).
tff(sort,type,
sort: ( ty * uni ) > $o ).
tff(witness,type,
witness: ty > uni ).
tff(witness_sort,axiom,
! [A: ty] : sort(A,witness(A)) ).
tff(int,type,
int: ty ).
tff(real,type,
real: ty ).
tff(bool,type,
bool: $tType ).
tff(bool1,type,
bool1: ty ).
tff(true,type,
true: bool ).
tff(false,type,
false: bool ).
tff(match_bool,type,
match_bool: ( ty * bool * uni * uni ) > uni ).
tff(match_bool_sort,axiom,
! [A: ty,X: bool,X1: uni,X2: uni] : sort(A,match_bool(A,X,X1,X2)) ).
tff(match_bool_True,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort(A,Z)
=> ( match_bool(A,true,Z,Z1) = Z ) ) ).
tff(match_bool_False,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort(A,Z1)
=> ( match_bool(A,false,Z,Z1) = Z1 ) ) ).
tff(true_False,axiom,
true != false ).
tff(bool_inversion,axiom,
! [U: bool] :
( ( U = true )
| ( U = false ) ) ).
tff(tuple0,type,
tuple0: $tType ).
tff(tuple01,type,
tuple01: ty ).
tff(tuple02,type,
tuple02: tuple0 ).
tff(tuple0_inversion,axiom,
! [U: tuple0] : U = tuple02 ).
tff(qtmark,type,
qtmark: ty ).
tff(compatOrderMult,axiom,
! [X: $int,Y: $int,Z: $int] :
( $lesseq(X,Y)
=> ( $lesseq(0,Z)
=> $lesseq($product(X,Z),$product(Y,Z)) ) ) ).
tff(ref,type,
ref: ty > ty ).
tff(mk_ref,type,
mk_ref: ( ty * uni ) > uni ).
tff(mk_ref_sort,axiom,
! [A: ty,X: uni] : sort(ref(A),mk_ref(A,X)) ).
tff(contents,type,
contents: ( ty * uni ) > uni ).
tff(contents_sort,axiom,
! [A: ty,X: uni] : sort(A,contents(A,X)) ).
tff(contents_def,axiom,
! [A: ty,U: uni] :
( sort(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).
tff(ref_inversion,axiom,
! [A: ty,U: uni] :
( sort(ref(A),U)
=> ( U = mk_ref(A,contents(A,U)) ) ) ).
tff(abs,type,
abs: $int > $int ).
tff(abs_def,axiom,
! [X: $int] :
( ( $lesseq(0,X)
=> ( abs(X) = X ) )
& ( ~ $lesseq(0,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(div,type,
div: ( $int * $int ) > $int ).
tff(mod,type,
mod: ( $int * $int ) > $int ).
tff(div_mod,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( X = $sum($product(Y,div(X,Y)),mod(X,Y)) ) ) ).
tff(div_bound,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( $lesseq(0,div(X,Y))
& $lesseq(div(X,Y),X) ) ) ).
tff(mod_bound,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( $less($uminus(abs(Y)),mod(X,Y))
& $less(mod(X,Y),abs(Y)) ) ) ).
tff(div_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> $lesseq(0,div(X,Y)) ) ).
tff(div_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& $less(0,Y) )
=> $lesseq(div(X,Y),0) ) ).
tff(mod_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& ( Y != 0 ) )
=> $lesseq(0,mod(X,Y)) ) ).
tff(mod_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& ( Y != 0 ) )
=> $lesseq(mod(X,Y),0) ) ).
tff(rounds_toward_zero,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> $lesseq(abs($product(div(X,Y),Y)),abs(X)) ) ).
tff(div_1,axiom,
! [X: $int] : div(X,1) = X ).
tff(mod_1,axiom,
! [X: $int] : mod(X,1) = 0 ).
tff(div_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( div(X,Y) = 0 ) ) ).
tff(mod_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( mod(X,Y) = X ) ) ).
tff(div_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( div($sum($product(X,Y),Z),X) = $sum(Y,div(Z,X)) ) ) ).
tff(mod_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( mod($sum($product(X,Y),Z),X) = mod(Z,X) ) ) ).
tff(map,type,
map: ( ty * ty ) > ty ).
tff(get,type,
get: ( ty * ty * uni * uni ) > uni ).
tff(get_sort,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort(B,get(B,A,X,X1)) ).
tff(set,type,
set: ( ty * ty * uni * uni * uni ) > uni ).
tff(set_sort,axiom,
! [A: ty,B: ty,X: uni,X1: uni,X2: uni] : sort(map(A,B),set(B,A,X,X1,X2)) ).
tff(select_eq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni,B1: uni] :
( sort(B,B1)
=> ( ( A1 = A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = B1 ) ) ) ).
tff(select_neq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni] :
( sort(A,A1)
=> ( sort(A,A2)
=> ! [B1: uni] :
( ( A1 != A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = get(B,A,M,A2) ) ) ) ) ).
tff(const,type,
const: ( ty * ty * uni ) > uni ).
tff(const_sort,axiom,
! [A: ty,B: ty,X: uni] : sort(map(A,B),const(B,A,X)) ).
tff(const1,axiom,
! [A: ty,B: ty,B1: uni,A1: uni] :
( sort(B,B1)
=> ( get(B,A,const(B,A,B1),A1) = B1 ) ) ).
tff(array,type,
array: ty > ty ).
tff(mk_array,type,
mk_array: ( ty * $int * uni ) > uni ).
tff(mk_array_sort,axiom,
! [A: ty,X: $int,X1: uni] : sort(array(A),mk_array(A,X,X1)) ).
tff(length,type,
length: ( ty * uni ) > $int ).
tff(length_def,axiom,
! [A: ty,U: $int,U1: uni] : length(A,mk_array(A,U,U1)) = U ).
tff(elts,type,
elts: ( ty * uni ) > uni ).
tff(elts_sort,axiom,
! [A: ty,X: uni] : sort(map(int,A),elts(A,X)) ).
tff(elts_def,axiom,
! [A: ty,U: $int,U1: uni] :
( sort(map(int,A),U1)
=> ( elts(A,mk_array(A,U,U1)) = U1 ) ) ).
tff(array_inversion,axiom,
! [A: ty,U: uni] : U = mk_array(A,length(A,U),elts(A,U)) ).
tff(get1,type,
get1: ( ty * uni * $int ) > uni ).
tff(get_sort1,axiom,
! [A: ty,X: uni,X1: $int] : sort(A,get1(A,X,X1)) ).
tff(t2tb,type,
t2tb: $int > uni ).
tff(t2tb_sort,axiom,
! [X: $int] : sort(int,t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > $int ).
tff(bridgeL,axiom,
! [I: $int] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] : t2tb(tb2t(J)) = J ).
tff(get_def,axiom,
! [A: ty,A1: uni,I: $int] : get1(A,A1,I) = get(A,int,elts(A,A1),t2tb(I)) ).
tff(set1,type,
set1: ( ty * uni * $int * uni ) > uni ).
tff(set_sort1,axiom,
! [A: ty,X: uni,X1: $int,X2: uni] : sort(array(A),set1(A,X,X1,X2)) ).
tff(set_def,axiom,
! [A: ty,A1: uni,I: $int,V: uni] : set1(A,A1,I,V) = mk_array(A,length(A,A1),set(A,int,elts(A,A1),t2tb(I),V)) ).
tff(make,type,
make: ( ty * $int * uni ) > uni ).
tff(make_sort,axiom,
! [A: ty,X: $int,X1: uni] : sort(array(A),make(A,X,X1)) ).
tff(make_def,axiom,
! [A: ty,N: $int,V: uni] : make(A,N,V) = mk_array(A,N,const(A,int,V)) ).
tff(map_int_int,type,
map_int_int: $tType ).
tff(sum,type,
sum: ( map_int_int * $int * $int ) > $int ).
tff(sum_def_empty,axiom,
! [C: map_int_int,I: $int,J: $int] :
( $lesseq(J,I)
=> ( sum(C,I,J) = 0 ) ) ).
tff(t2tb1,type,
t2tb1: map_int_int > uni ).
tff(t2tb_sort1,axiom,
! [X: map_int_int] : sort(map(int,int),t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > map_int_int ).
tff(bridgeL1,axiom,
! [I: map_int_int] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] : t2tb1(tb2t1(J)) = J ).
tff(sum_def_non_empty,axiom,
! [C: map_int_int,I: $int,J: $int] :
( $less(I,J)
=> ( sum(C,I,J) = $sum(tb2t(get(int,int,t2tb1(C),t2tb(I))),sum(C,$sum(I,1),J)) ) ) ).
tff(sum_right_extension,axiom,
! [C: map_int_int,I: $int,J: $int] :
( $less(I,J)
=> ( sum(C,I,J) = $sum(sum(C,I,$difference(J,1)),tb2t(get(int,int,t2tb1(C),t2tb($difference(J,1))))) ) ) ).
tff(sum_transitivity,axiom,
! [C: map_int_int,I: $int,K: $int,J: $int] :
( ( $lesseq(I,K)
& $lesseq(K,J) )
=> ( sum(C,I,J) = $sum(sum(C,I,K),sum(C,K,J)) ) ) ).
tff(sum_eq,axiom,
! [C1: map_int_int,C2: map_int_int,I: $int,J: $int] :
( ! [K: $int] :
( ( $lesseq(I,K)
& $less(K,J) )
=> ( tb2t(get(int,int,t2tb1(C1),t2tb(K))) = tb2t(get(int,int,t2tb1(C2),t2tb(K))) ) )
=> ( sum(C1,I,J) = sum(C2,I,J) ) ) ).
tff(array_int,type,
array_int: $tType ).
tff(sum1,type,
sum1: ( array_int * $int * $int ) > $int ).
tff(t2tb2,type,
t2tb2: array_int > uni ).
tff(t2tb_sort2,axiom,
! [X: array_int] : sort(array(int),t2tb2(X)) ).
tff(tb2t2,type,
tb2t2: uni > array_int ).
tff(bridgeL2,axiom,
! [I: array_int] : tb2t2(t2tb2(I)) = I ).
tff(bridgeR2,axiom,
! [J: uni] : t2tb2(tb2t2(J)) = J ).
tff(sum_def,axiom,
! [A: array_int,L: $int,H: $int] : sum1(A,L,H) = sum(tb2t1(elts(int,t2tb2(A))),L,H) ).
tff(maxsublo,type,
maxsublo: ( array_int * $int * $int ) > $o ).
tff(maxsublo_def,axiom,
! [A: array_int,Maxlo: $int,S: $int] :
( maxsublo(A,Maxlo,S)
<=> ! [L: $int,H: $int] :
( ( $lesseq(0,L)
& $less(L,Maxlo) )
=> ( ( $lesseq(L,H)
& $lesseq(H,length(int,t2tb2(A))) )
=> $lesseq(sum1(A,L,H),S) ) ) ) ).
tff(maxsub,type,
maxsub: ( array_int * $int ) > $o ).
tff(maxsub_def,axiom,
! [A: array_int,S: $int] :
( maxsub(A,S)
<=> ! [L: $int,H: $int] :
( ( $lesseq(0,L)
& $lesseq(L,H)
& $lesseq(H,length(int,t2tb2(A))) )
=> $lesseq(sum1(A,L,H),S) ) ) ).
tff(wP_parameter_maximum_subarray_rec,conjecture,
! [A: $int,A1: map_int_int,L: $int,H: $int] :
( ( $lesseq(0,A)
& $lesseq(0,L)
& $lesseq(L,H)
& $lesseq(H,A) )
=> ( ( H != L )
=> ! [Lo: $int] :
( ( Lo = $sum(L,div($difference(H,L),2)) )
=> ! [Hi: $int] :
( ( Hi = $sum(L,div($difference(H,L),2)) )
=> ( $lesseq(L,$difference($sum(L,div($difference(H,L),2)),1))
=> ! [S: $int,Ms: $int,Lo1: $int] :
( ( $lesseq(L,Lo1)
& $lesseq(Lo1,$sum(L,div($difference(H,L),2)))
& ( $sum(L,div($difference(H,L),2)) = Hi )
& ( Ms = sum(A1,Lo1,Hi) )
& ! [Lqt: $int] :
( ( $less($difference(L,1),Lqt)
& $lesseq(Lqt,$sum(L,div($difference(H,L),2))) )
=> $lesseq(sum(A1,Lqt,$sum(L,div($difference(H,L),2))),Ms) )
& ( S = sum(A1,$sum($difference(L,1),1),$sum(L,div($difference(H,L),2))) ) )
=> ( ! [Lqt: $int] :
( ( $lesseq(L,Lqt)
& $lesseq(Lqt,$sum(L,div($difference(H,L),2))) )
=> $lesseq(sum(A1,Lqt,$sum(L,div($difference(H,L),2))),sum(A1,Lo1,$sum(L,div($difference(H,L),2)))) )
=> ! [S1: $int] :
( ( S1 = Ms )
=> ( $lesseq($sum(L,div($difference(H,L),2)),$difference(H,1))
=> ! [Lqt: $int,Hqt: $int] :
( ( $lesseq(L,Lqt)
& $lesseq(Lqt,$sum(L,div($difference(H,L),2)))
& $lesseq($sum(L,div($difference(H,L),2)),Hqt)
& $lesseq(Hqt,$sum(L,div($difference(H,L),2))) )
=> $lesseq(sum(A1,Lqt,Hqt),Ms) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------