TPTP Problem File: SWW590_2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWW590_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Euler001-T-Closed formula n 15
% 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 : euler001-T-Closed_formula_n_15 [Fil14]
% Status : Theorem
% Rating : 0.75 v7.5.0, 0.80 v7.4.0, 0.75 v7.3.0, 0.67 v7.0.0, 0.71 v6.4.0, 0.33 v6.3.0, 0.57 v6.2.0, 0.75 v6.1.0
% Syntax : Number of formulae : 60 ( 13 unt; 13 typ; 0 def)
% Number of atoms : 130 ( 62 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 97 ( 14 ~; 2 |; 32 &)
% ( 3 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 354 ( 57 atm; 67 fun; 154 num; 76 var)
% Number of types : 4 ( 2 usr; 1 ari)
% Number of type conns : 12 ( 9 >; 3 *; 0 +; 0 <<)
% Number of predicates : 5 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 19 ( 9 usr; 8 con; 0-2 aty)
% Number of variables : 77 ( 76 !; 1 ?; 77 :)
% SPC : TF0_THM_EQU_ARI
% Comments :
%------------------------------------------------------------------------------
tff(uni,type,
uni: $tType ).
tff(ty,type,
ty: $tType ).
tff(sort,type,
sort1: ( ty * uni ) > $o ).
tff(witness,type,
witness1: ty > uni ).
tff(witness_sort1,axiom,
! [A: ty] : sort1(A,witness1(A)) ).
tff(int,type,
int: ty ).
tff(real,type,
real: ty ).
tff(compatOrderMult,axiom,
! [X: $int,Y: $int,Z: $int] :
( $lesseq(X,Y)
=> ( $lesseq(0,Z)
=> $lesseq($product(X,Z),$product(Y,Z)) ) ) ).
tff(abs,type,
abs1: $int > $int ).
tff(abs_def,axiom,
! [X: $int] :
( ( $lesseq(0,X)
=> ( abs1(X) = X ) )
& ( ~ $lesseq(0,X)
=> ( abs1(X) = $uminus(X) ) ) ) ).
tff(abs_le,axiom,
! [X: $int,Y: $int] :
( $lesseq(abs1(X),Y)
<=> ( $lesseq($uminus(Y),X)
& $lesseq(X,Y) ) ) ).
tff(abs_pos,axiom,
! [X: $int] : $lesseq(0,abs1(X)) ).
tff(div,type,
div1: ( $int * $int ) > $int ).
tff(mod,type,
mod1: ( $int * $int ) > $int ).
tff(div_mod,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( X = $sum($product(Y,div1(X,Y)),mod1(X,Y)) ) ) ).
tff(div_bound,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( $lesseq(0,div1(X,Y))
& $lesseq(div1(X,Y),X) ) ) ).
tff(mod_bound,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( $less($uminus(abs1(Y)),mod1(X,Y))
& $less(mod1(X,Y),abs1(Y)) ) ) ).
tff(div_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> $lesseq(0,div1(X,Y)) ) ).
tff(div_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& $less(0,Y) )
=> $lesseq(div1(X,Y),0) ) ).
tff(mod_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& ( Y != 0 ) )
=> $lesseq(0,mod1(X,Y)) ) ).
tff(mod_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& ( Y != 0 ) )
=> $lesseq(mod1(X,Y),0) ) ).
tff(rounds_toward_zero,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> $lesseq(abs1($product(div1(X,Y),Y)),abs1(X)) ) ).
tff(div_1,axiom,
! [X: $int] : div1(X,1) = X ).
tff(mod_1,axiom,
! [X: $int] : mod1(X,1) = 0 ).
tff(div_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( div1(X,Y) = 0 ) ) ).
tff(mod_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( mod1(X,Y) = X ) ) ).
tff(div_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( div1($sum($product(X,Y),Z),X) = $sum(Y,div1(Z,X)) ) ) ).
tff(mod_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( mod1($sum($product(X,Y),Z),X) = mod1(Z,X) ) ) ).
tff(sum_multiple_3_5_lt,type,
sum_multiple_3_5_lt1: $int > $int ).
tff(sumEmpty,axiom,
sum_multiple_3_5_lt1(0) = 0 ).
tff(sumNo,axiom,
! [N: $int] :
( $lesseq(0,N)
=> ( ( ( mod1(N,3) != 0 )
& ( mod1(N,5) != 0 ) )
=> ( sum_multiple_3_5_lt1($sum(N,1)) = sum_multiple_3_5_lt1(N) ) ) ) ).
tff(sumYes,axiom,
! [N: $int] :
( $lesseq(0,N)
=> ( ( ( mod1(N,3) = 0 )
| ( mod1(N,5) = 0 ) )
=> ( sum_multiple_3_5_lt1($sum(N,1)) = $sum(sum_multiple_3_5_lt1(N),N) ) ) ) ).
tff(div2,axiom,
! [X: $int] :
? [Y: $int] :
( ( X = $product(2,Y) )
| ( X = $sum($product(2,Y),1) ) ) ).
tff(mod_div_unique,axiom,
! [X: $int,Y: $int,Q: $int,R: $int] :
( ( $lesseq(0,X)
& $less(0,Y)
& ( X = $sum($product(Q,Y),R) )
& $lesseq(0,R)
& $less(R,Y) )
=> ( ( Q = div1(X,Y) )
& ( R = mod1(X,Y) ) ) ) ).
tff(mod_succ_1,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( ( mod1($sum(X,1),Y) != 0 )
=> ( mod1($sum(X,1),Y) = $sum(mod1(X,Y),1) ) ) ) ).
tff(mod_succ_2,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( ( mod1($sum(X,1),Y) = 0 )
=> ( mod1(X,Y) = $difference(Y,1) ) ) ) ).
tff(div_succ_1,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( ( mod1($sum(X,1),Y) = 0 )
=> ( div1($sum(X,1),Y) = $sum(div1(X,Y),1) ) ) ) ).
tff(div_succ_2,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( ( mod1($sum(X,1),Y) != 0 )
=> ( div1($sum(X,1),Y) = div1(X,Y) ) ) ) ).
tff(mod2_mul2,axiom,
! [X: $int] : mod1($product(2,X),2) = 0 ).
tff(mod2_mul2_aux,axiom,
! [X: $int,Y: $int] : mod1($product(Y,$product(2,X)),2) = 0 ).
tff(mod2_mul2_aux2,axiom,
! [X: $int,Y: $int,Z: $int,T: $int] : mod1($sum($product(Y,$product(2,X)),$product(Z,$product(2,T))),2) = 0 ).
tff(div2_mul2,axiom,
! [X: $int] : div1($product(2,X),2) = X ).
tff(div2_mul2_aux,axiom,
! [X: $int,Y: $int] : div1($product(Y,$product(2,X)),2) = $product(Y,X) ).
tff(div2_add,axiom,
! [X: $int,Y: $int] :
( ( ( mod1(X,2) = 0 )
& ( mod1(Y,2) = 0 ) )
=> ( div1($sum(X,Y),2) = $sum(div1(X,2),div1(Y,2)) ) ) ).
tff(div2_sub,axiom,
! [X: $int,Y: $int] :
( ( ( mod1(X,2) = 0 )
& ( mod1(Y,2) = 0 ) )
=> ( div1($difference(X,Y),2) = $difference(div1(X,2),div1(Y,2)) ) ) ).
tff(tr_mod_2,axiom,
! [N: $int] :
( $lesseq(0,N)
=> ( mod1($product(N,$sum(N,1)),2) = 0 ) ) ).
tff(tr,type,
tr1: $int > $int ).
tff(tr_def,axiom,
! [N: $int] : tr1(N) = div1($product(N,$sum(N,1)),2) ).
tff(tr_repr,axiom,
! [N: $int] :
( $lesseq(0,N)
=> ( $product(N,$sum(N,1)) = $product(2,tr1(N)) ) ) ).
tff(tr_succ,axiom,
! [N: $int] :
( $lesseq(0,N)
=> ( tr1($sum(N,1)) = $sum($sum(tr1(N),N),1) ) ) ).
tff(closed_formula_aux,type,
closed_formula_aux1: $int > $int ).
tff(closed_formula_aux_def,axiom,
! [N: $int] : closed_formula_aux1(N) = $difference($sum($product(3,tr1(div1(N,3))),$product(5,tr1(div1(N,5)))),$product(15,tr1(div1(N,15)))) ).
tff(p,type,
p1: $int > $o ).
tff(p_def,axiom,
! [N: $int] :
( p1(N)
<=> ( sum_multiple_3_5_lt1($sum(N,1)) = closed_formula_aux1(N) ) ) ).
tff(mod_15,axiom,
! [N: $int] :
( ( mod1(N,15) = 0 )
<=> ( ( mod1(N,3) = 0 )
& ( mod1(N,5) = 0 ) ) ) ).
tff(closed_formula_0,axiom,
p1(0) ).
tff(closed_formula_n,axiom,
! [N: $int] :
( $less(0,N)
=> ( p1($difference(N,1))
=> ( ( ( mod1(N,3) != 0 )
& ( mod1(N,5) != 0 ) )
=> p1(N) ) ) ) ).
tff(closed_formula_n_3,axiom,
! [N: $int] :
( $less(0,N)
=> ( p1($difference(N,1))
=> ( ( ( mod1(N,3) = 0 )
& ( mod1(N,5) != 0 ) )
=> p1(N) ) ) ) ).
tff(closed_formula_n_5,axiom,
! [N: $int] :
( $less(0,N)
=> ( p1($difference(N,1))
=> ( ( ( mod1(N,3) != 0 )
& ( mod1(N,5) = 0 ) )
=> p1(N) ) ) ) ).
tff(closed_formula_n_15,conjecture,
! [N: $int] :
( $less(0,N)
=> ( p1($difference(N,1))
=> ( ( ( mod1(N,3) = 0 )
& ( mod1(N,5) = 0 ) )
=> p1(N) ) ) ) ).
%------------------------------------------------------------------------------