TPTP Problem File: SWW600_2.p
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%------------------------------------------------------------------------------
% File : SWW600_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Gcd bezout-T-WP parameter gcd
% 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 : gcd_bezout-T-WP_parameter_gcd [Fil14]
% Status : Theorem
% Rating : 0.38 v8.2.0, 0.50 v7.5.0, 0.60 v7.4.0, 0.75 v7.3.0, 0.67 v7.1.0, 0.33 v7.0.0, 0.43 v6.4.0, 1.00 v6.3.0, 0.57 v6.2.0, 0.62 v6.1.0
% Syntax : Number of formulae : 108 ( 23 unt; 25 typ; 0 def)
% Number of atoms : 184 ( 56 equ)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 119 ( 18 ~; 4 |; 20 &)
% ( 6 <=>; 71 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number arithmetic : 343 ( 49 atm; 68 fun; 76 num; 150 var)
% Number of types : 6 ( 4 usr; 1 ari)
% Number of type conns : 23 ( 13 >; 10 *; 0 +; 0 <<)
% Number of predicates : 7 ( 4 usr; 0 prp; 1-2 aty)
% Number of functors : 26 ( 17 usr; 11 con; 0-4 aty)
% Number of variables : 171 ( 166 !; 5 ?; 171 :)
% 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(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(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(wP_parameter_gcd,conjecture,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $lesseq(0,Y) )
=> ! [D: $int,C: $int,B: $int,A: $int,Y1: $int,X1: $int] :
( ( $lesseq(0,X1)
& $lesseq(0,Y1)
& ( gcd(X1,Y1) = gcd(X,Y) )
& ( $sum($product(A,X),$product(B,Y)) = X1 )
& ( $sum($product(C,X),$product(D,Y)) = Y1 ) )
=> ( ~ $less(0,Y1)
=> ? [A1: $int,B1: $int] : $sum($product(A1,X),$product(B1,Y)) = X1 ) ) ) ).
%------------------------------------------------------------------------------