TPTP Problem File: SWW613_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW613_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Largest prime factor-T-WP parameter largest prime factor
% 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 : largest_prime_factor-T-WP_parameter_largest_prime_factor [Fil14]
% Status : Theorem
% Rating : 0.00 v8.2.0, 0.12 v7.5.0, 0.20 v7.4.0, 0.25 v7.3.0, 0.00 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.29 v6.2.0, 0.38 v6.1.0
% Syntax : Number of formulae : 151 ( 39 unt; 38 typ; 0 def)
% Number of atoms : 304 ( 76 equ)
% Maximal formula atoms : 65 ( 2 avg)
% Number of connectives : 216 ( 25 ~; 9 |; 59 &)
% ( 9 <=>; 114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number arithmetic : 440 ( 89 atm; 68 fun; 105 num; 178 var)
% Number of types : 7 ( 5 usr; 1 ari)
% Number of type conns : 44 ( 25 >; 19 *; 0 +; 0 <<)
% Number of predicates : 9 ( 6 usr; 0 prp; 1-2 aty)
% Number of functors : 37 ( 27 usr; 12 con; 0-5 aty)
% Number of variables : 240 ( 237 !; 3 ?; 240 :)
% 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(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(divides,type,
divides: ( $int * $int ) > $o ).
tff(divides_def,axiom,
! [D: $int,N: $int] :
( divides(D,N)
<=> ? [Q: $int] : N = $product(Q,D) ) ).
tff(divides_refl,axiom,
! [N: $int] : divides(N,N) ).
tff(divides_1_n,axiom,
! [N: $int] : divides(1,N) ).
tff(divides_0,axiom,
! [N: $int] : divides(N,0) ).
tff(divides_left,axiom,
! [A: $int,B: $int,C: $int] :
( divides(A,B)
=> divides($product(C,A),$product(C,B)) ) ).
tff(divides_right,axiom,
! [A: $int,B: $int,C: $int] :
( divides(A,B)
=> divides($product(A,C),$product(B,C)) ) ).
tff(divides_oppr,axiom,
! [A: $int,B: $int] :
( divides(A,B)
=> divides(A,$uminus(B)) ) ).
tff(divides_oppl,axiom,
! [A: $int,B: $int] :
( divides(A,B)
=> divides($uminus(A),B) ) ).
tff(divides_oppr_rev,axiom,
! [A: $int,B: $int] :
( divides($uminus(A),B)
=> divides(A,B) ) ).
tff(divides_oppl_rev,axiom,
! [A: $int,B: $int] :
( divides(A,$uminus(B))
=> divides(A,B) ) ).
tff(divides_plusr,axiom,
! [A: $int,B: $int,C: $int] :
( divides(A,B)
=> ( divides(A,C)
=> divides(A,$sum(B,C)) ) ) ).
tff(divides_minusr,axiom,
! [A: $int,B: $int,C: $int] :
( divides(A,B)
=> ( divides(A,C)
=> divides(A,$difference(B,C)) ) ) ).
tff(divides_multl,axiom,
! [A: $int,B: $int,C: $int] :
( divides(A,B)
=> divides(A,$product(C,B)) ) ).
tff(divides_multr,axiom,
! [A: $int,B: $int,C: $int] :
( divides(A,B)
=> divides(A,$product(B,C)) ) ).
tff(divides_factorl,axiom,
! [A: $int,B: $int] : divides(A,$product(B,A)) ).
tff(divides_factorr,axiom,
! [A: $int,B: $int] : divides(A,$product(A,B)) ).
tff(divides_n_1,axiom,
! [N: $int] :
( divides(N,1)
=> ( ( N = 1 )
| ( N = $uminus(1) ) ) ) ).
tff(divides_antisym,axiom,
! [A: $int,B: $int] :
( divides(A,B)
=> ( divides(B,A)
=> ( ( A = B )
| ( A = $uminus(B) ) ) ) ) ).
tff(divides_trans,axiom,
! [A: $int,B: $int,C: $int] :
( divides(A,B)
=> ( divides(B,C)
=> divides(A,C) ) ) ).
tff(divides_bounds,axiom,
! [A: $int,B: $int] :
( divides(A,B)
=> ( ( B != 0 )
=> $lesseq(abs(A),abs(B)) ) ) ).
tff(div_mult1,axiom,
! [X: $int,Y: $int,Z: $int] :
( $less(0,X)
=> ( $quotient_e($sum($product(X,Y),Z),X) = $sum(Y,$quotient_e(Z,X)) ) ) ).
tff(mod_mult1,axiom,
! [X: $int,Y: $int,Z: $int] :
( $less(0,X)
=> ( $remainder_e($sum($product(X,Y),Z),X) = $remainder_e(Z,X) ) ) ).
tff(mod_divides_euclidean,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( ( $remainder_e(A,B) = 0 )
=> divides(B,A) ) ) ).
tff(divides_mod_euclidean,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( divides(B,A)
=> ( $remainder_e(A,B) = 0 ) ) ) ).
tff(mod_divides_computer,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( ( mod(A,B) = 0 )
=> divides(B,A) ) ) ).
tff(divides_mod_computer,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( divides(B,A)
=> ( mod(A,B) = 0 ) ) ) ).
tff(even,type,
even: $int > $o ).
tff(even_def,axiom,
! [N: $int] :
( even(N)
<=> ? [K: $int] : N = $product(2,K) ) ).
tff(odd,type,
odd: $int > $o ).
tff(odd_def,axiom,
! [N: $int] :
( odd(N)
<=> ? [K: $int] : N = $sum($product(2,K),1) ) ).
tff(even_or_odd,axiom,
! [N: $int] :
( even(N)
| odd(N) ) ).
tff(even_not_odd,axiom,
! [N: $int] :
( even(N)
=> ~ odd(N) ) ).
tff(odd_not_even,axiom,
! [N: $int] :
( odd(N)
=> ~ even(N) ) ).
tff(even_odd,axiom,
! [N: $int] :
( even(N)
=> odd($sum(N,1)) ) ).
tff(odd_even,axiom,
! [N: $int] :
( odd(N)
=> even($sum(N,1)) ) ).
tff(even_even,axiom,
! [N: $int] :
( even(N)
=> even($sum(N,2)) ) ).
tff(odd_odd,axiom,
! [N: $int] :
( odd(N)
=> odd($sum(N,2)) ) ).
tff(even_2k,axiom,
! [K: $int] : even($product(2,K)) ).
tff(odd_2k1,axiom,
! [K: $int] : odd($sum($product(2,K),1)) ).
tff(even_divides,axiom,
! [A: $int] :
( even(A)
<=> divides(2,A) ) ).
tff(odd_divides,axiom,
! [A: $int] :
( odd(A)
<=> ~ divides(2,A) ) ).
tff(prime,type,
prime: $int > $o ).
tff(prime_def,axiom,
! [P: $int] :
( prime(P)
<=> ( $lesseq(2,P)
& ! [N: $int] :
( ( $less(1,N)
& $less(N,P) )
=> ~ divides(N,P) ) ) ) ).
tff(not_prime_1,axiom,
~ prime(1) ).
tff(prime_2,axiom,
prime(2) ).
tff(prime_3,axiom,
prime(3) ).
tff(prime_divisors,axiom,
! [P: $int] :
( prime(P)
=> ! [D: $int] :
( divides(D,P)
=> ( ( D = 1 )
| ( D = $uminus(1) )
| ( D = P )
| ( D = $uminus(P) ) ) ) ) ).
tff(small_divisors,axiom,
! [P: $int] :
( $lesseq(2,P)
=> ( ! [D: $int] :
( $lesseq(2,D)
=> ( prime(D)
=> ( ( $less(1,$product(D,D))
& $lesseq($product(D,D),P) )
=> ~ divides(D,P) ) ) )
=> prime(P) ) ) ).
tff(even_prime,axiom,
! [P: $int] :
( prime(P)
=> ( even(P)
=> ( P = 2 ) ) ) ).
tff(odd_prime,axiom,
! [P: $int] :
( prime(P)
=> ( $lesseq(3,P)
=> odd(P) ) ) ).
tff(gcd,type,
gcd: ( $int * $int ) > $int ).
tff(gcd_nonneg,axiom,
! [A: $int,B: $int] : $lesseq(0,gcd(A,B)) ).
tff(gcd_def1,axiom,
! [A: $int,B: $int] : divides(gcd(A,B),A) ).
tff(gcd_def2,axiom,
! [A: $int,B: $int] : divides(gcd(A,B),B) ).
tff(gcd_def3,axiom,
! [A: $int,B: $int,X: $int] :
( divides(X,A)
=> ( divides(X,B)
=> divides(X,gcd(A,B)) ) ) ).
tff(gcd_unique,axiom,
! [A: $int,B: $int,D: $int] :
( $lesseq(0,D)
=> ( divides(D,A)
=> ( divides(D,B)
=> ( ! [X: $int] :
( divides(X,A)
=> ( divides(X,B)
=> divides(X,D) ) )
=> ( D = gcd(A,B) ) ) ) ) ) ).
tff(assoc,axiom,
! [X: $int,Y: $int,Z: $int] : gcd(gcd(X,Y),Z) = gcd(X,gcd(Y,Z)) ).
tff(comm,axiom,
! [X: $int,Y: $int] : gcd(X,Y) = gcd(Y,X) ).
tff(gcd_0_pos,axiom,
! [A: $int] :
( $lesseq(0,A)
=> ( gcd(A,0) = A ) ) ).
tff(gcd_0_neg,axiom,
! [A: $int] :
( $less(A,0)
=> ( gcd(A,0) = $uminus(A) ) ) ).
tff(gcd_opp,axiom,
! [A: $int,B: $int] : gcd(A,B) = gcd($uminus(A),B) ).
tff(gcd_euclid,axiom,
! [A: $int,B: $int,Q: $int] : gcd(A,B) = gcd(A,$difference(B,$product(Q,A))) ).
tff(gcd_computer_mod,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( gcd(B,mod(A,B)) = gcd(A,B) ) ) ).
tff(gcd_euclidean_mod,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( gcd(B,$remainder_e(A,B)) = gcd(A,B) ) ) ).
tff(gcd_mult,axiom,
! [A: $int,B: $int,C: $int] :
( $lesseq(0,C)
=> ( gcd($product(C,A),$product(C,B)) = $product(C,gcd(A,B)) ) ) ).
tff(coprime,type,
coprime: ( $int * $int ) > $o ).
tff(coprime_def,axiom,
! [A: $int,B: $int] :
( coprime(A,B)
<=> ( gcd(A,B) = 1 ) ) ).
tff(prime_coprime,axiom,
! [P: $int] :
( prime(P)
<=> ( $lesseq(2,P)
& ! [N: $int] :
( ( $lesseq(1,N)
& $less(N,P) )
=> coprime(N,P) ) ) ) ).
tff(gauss,axiom,
! [A: $int,B: $int,C: $int] :
( ( divides(A,$product(B,C))
& coprime(A,B) )
=> divides(A,C) ) ).
tff(euclid,axiom,
! [P: $int,A: $int,B: $int] :
( ( prime(P)
& divides(P,$product(A,B)) )
=> ( divides(P,A)
| divides(P,B) ) ) ).
tff(gcd_coprime,axiom,
! [A: $int,B: $int,C: $int] :
( coprime(A,B)
=> ( gcd(A,$product(B,C)) = gcd(A,C) ) ) ).
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(list,type,
list: ty > ty ).
tff(nil,type,
nil: ty > uni ).
tff(nil_sort,axiom,
! [A: ty] : sort(list(A),nil(A)) ).
tff(cons,type,
cons: ( ty * uni * uni ) > uni ).
tff(cons_sort,axiom,
! [A: ty,X: uni,X1: uni] : sort(list(A),cons(A,X,X1)) ).
tff(match_list,type,
match_list: ( ty * ty * uni * uni * uni ) > uni ).
tff(match_list_sort,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort(A1,match_list(A1,A,X,X1,X2)) ).
tff(match_list_Nil,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni] :
( sort(A1,Z)
=> ( match_list(A1,A,nil(A),Z,Z1) = Z ) ) ).
tff(match_list_Cons,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni,U1: uni] :
( sort(A1,Z1)
=> ( match_list(A1,A,cons(A,U,U1),Z,Z1) = Z1 ) ) ).
tff(nil_Cons,axiom,
! [A: ty,V: uni,V1: uni] : nil(A) != cons(A,V,V1) ).
tff(cons_proj_1,type,
cons_proj_1: ( ty * uni ) > uni ).
tff(cons_proj_1_sort,axiom,
! [A: ty,X: uni] : sort(A,cons_proj_1(A,X)) ).
tff(cons_proj_1_def,axiom,
! [A: ty,U: uni,U1: uni] :
( sort(A,U)
=> ( cons_proj_1(A,cons(A,U,U1)) = U ) ) ).
tff(cons_proj_2,type,
cons_proj_2: ( ty * uni ) > uni ).
tff(cons_proj_2_sort,axiom,
! [A: ty,X: uni] : sort(list(A),cons_proj_2(A,X)) ).
tff(cons_proj_2_def,axiom,
! [A: ty,U: uni,U1: uni] : cons_proj_2(A,cons(A,U,U1)) = U1 ).
tff(list_inversion,axiom,
! [A: ty,U: uni] :
( ( U = nil(A) )
| ( U = cons(A,cons_proj_1(A,U),cons_proj_2(A,U)) ) ) ).
tff(list_int,type,
list_int: $tType ).
tff(t2tb,type,
t2tb: list_int > uni ).
tff(t2tb_sort,axiom,
! [X: list_int] : sort(list(int),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > list_int ).
tff(bridgeL,axiom,
! [I: list_int] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] : t2tb(tb2t(J)) = J ).
tff(t2tb1,type,
t2tb1: $int > uni ).
tff(t2tb_sort1,axiom,
! [X: $int] : sort(int,t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > $int ).
tff(bridgeL1,axiom,
! [I: $int] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] : t2tb1(tb2t1(J)) = J ).
tff(wP_parameter_largest_prime_factor,conjecture,
! [N: $int] :
( $lesseq(2,N)
=> ( ( $lesseq(2,N)
& $lesseq(2,2)
& $lesseq(2,N)
& ! [I: $int] :
( ( $lesseq(2,I)
& $less(I,2) )
=> ~ divides(I,N) ) )
=> ! [D: $int] :
( ( $lesseq(2,D)
& $lesseq(D,N)
& divides(D,N)
& ! [I: $int] :
( ( $lesseq(2,I)
& $less(I,D) )
=> ~ divides(I,N) ) )
=> ! [Factors: list_int] :
( ( Factors = tb2t(cons(int,t2tb1(D),nil(int))) )
=> ( ( ( $product(div(N,D),D) = N )
& divides(div(N,D),N) )
=> ( ! [I: $int] :
( ( prime(I)
& divides(I,N)
& $less(D,I) )
=> ( coprime(D,I)
& divides(I,div(N,D)) ) )
=> ! [Target: $int,Factor: $int,Factors1: list_int] :
( ( $lesseq(1,Target)
& $lesseq(Target,N)
& $lesseq(2,Factor)
& $lesseq(Factor,N)
& divides(Factor,N)
& prime(Factor)
& ! [I: $int] :
( ( divides(I,Target)
& $lesseq(2,I) )
=> ( $lesseq(Factor,I)
& divides(I,N) ) )
& ! [I: $int] :
( ( prime(I)
& divides(I,N)
& $less(Factor,I) )
=> divides(I,Target) ) )
=> ( $lesseq(2,Target)
=> ( ( divides(Target,Target)
& $lesseq(2,Target)
& $lesseq(Factor,Target) )
=> ( ( $lesseq(2,Target)
& $lesseq(2,Factor)
& $lesseq(Factor,Target)
& ! [I: $int] :
( ( $lesseq(2,I)
& $less(I,Factor) )
=> ~ divides(I,Target) ) )
=> ! [D1: $int] :
( ( $lesseq(Factor,D1)
& $lesseq(D1,Target)
& divides(D1,Target)
& ! [I: $int] :
( ( $lesseq(2,I)
& $less(I,D1) )
=> ~ divides(I,Target) ) )
=> ( prime(D1)
=> ! [Factor1: $int] :
( ( Factor1 = D1 )
=> ! [Factors2: list_int] :
( ( Factors2 = tb2t(cons(int,t2tb1(D1),t2tb(Factors1))) )
=> ! [Target1: $int] :
( ( Target1 = div(Target,D1) )
=> ( ( ( $product(Target1,D1) = Target )
& divides(Target1,Target) )
=> ! [I: $int] :
( ( prime(I)
& divides(I,N)
& $less(D1,I) )
=> ( $less(Factor,I)
=> ( divides(I,Target)
=> ( ( $lesseq(1,D1)
& $less(D1,I) )
=> coprime(D1,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------