TPTP Problem File: SWW647_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW647_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Sum of digits-T-WP parameter f
% 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 : sum_of_digits-T-WP_parameter_f [Fil14]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.75 v7.5.0, 0.80 v7.4.0, 0.62 v7.3.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.67 v6.3.0, 0.57 v6.2.0, 0.75 v6.1.0
% Syntax : Number of formulae : 110 ( 24 unt; 37 typ; 0 def)
% Number of atoms : 185 ( 61 equ)
% Maximal formula atoms : 13 ( 1 avg)
% Number of connectives : 125 ( 13 ~; 1 |; 32 &)
% ( 2 <=>; 77 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 8 ( 1 avg)
% Number arithmetic : 415 ( 91 atm; 84 fun; 118 num; 122 var)
% Number of types : 7 ( 5 usr; 1 ari)
% Number of type conns : 56 ( 24 >; 32 *; 0 +; 0 <<)
% Number of predicates : 5 ( 2 usr; 0 prp; 2-2 aty)
% Number of functors : 41 ( 30 usr; 13 con; 0-6 aty)
% Number of variables : 202 ( 202 !; 0 ?; 202 :)
% 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(bool,type,
bool1: $tType ).
tff(bool1,type,
bool: ty ).
tff(true,type,
true1: bool1 ).
tff(false,type,
false1: bool1 ).
tff(match_bool,type,
match_bool1: ( ty * bool1 * uni * uni ) > uni ).
tff(match_bool_sort1,axiom,
! [A: ty,X: bool1,X1: uni,X2: uni] : sort1(A,match_bool1(A,X,X1,X2)) ).
tff(match_bool_True,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort1(A,Z)
=> ( match_bool1(A,true1,Z,Z1) = Z ) ) ).
tff(match_bool_False,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort1(A,Z1)
=> ( match_bool1(A,false1,Z,Z1) = Z1 ) ) ).
tff(true_False,axiom,
true1 != false1 ).
tff(bool_inversion,axiom,
! [U: bool1] :
( ( U = true1 )
| ( U = false1 ) ) ).
tff(tuple0,type,
tuple02: $tType ).
tff(tuple01,type,
tuple0: ty ).
tff(tuple02,type,
tuple03: tuple02 ).
tff(tuple0_inversion,axiom,
! [U: tuple02] : U = tuple03 ).
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_sort1,axiom,
! [A: ty,X: uni] : sort1(ref(A),mk_ref(A,X)) ).
tff(contents,type,
contents: ( ty * uni ) > uni ).
tff(contents_sort1,axiom,
! [A: ty,X: uni] : sort1(A,contents(A,X)) ).
tff(contents_def1,axiom,
! [A: ty,U: uni] :
( sort1(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).
tff(ref_inversion1,axiom,
! [A: ty,U: uni] :
( sort1(ref(A),U)
=> ( U = mk_ref(A,contents(A,U)) ) ) ).
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_mult,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_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( $less(0,X)
=> ( $remainder_e($sum($product(X,Y),Z),X) = $remainder_e(Z,X) ) ) ).
tff(power,type,
power1: ( $int * $int ) > $int ).
tff(power_0,axiom,
! [X: $int] : power1(X,0) = 1 ).
tff(power_s,axiom,
! [X: $int,N: $int] :
( $lesseq(0,N)
=> ( power1(X,$sum(N,1)) = $product(X,power1(X,N)) ) ) ).
tff(power_s_alt,axiom,
! [X: $int,N: $int] :
( $less(0,N)
=> ( power1(X,N) = $product(X,power1(X,$difference(N,1))) ) ) ).
tff(power_1,axiom,
! [X: $int] : power1(X,1) = X ).
tff(power_sum,axiom,
! [X: $int,N: $int,M: $int] :
( $lesseq(0,N)
=> ( $lesseq(0,M)
=> ( power1(X,$sum(N,M)) = $product(power1(X,N),power1(X,M)) ) ) ) ).
tff(power_mult,axiom,
! [X: $int,N: $int,M: $int] :
( $lesseq(0,N)
=> ( $lesseq(0,M)
=> ( power1(X,$product(N,M)) = power1(power1(X,N),M) ) ) ) ).
tff(power_mult2,axiom,
! [X: $int,Y: $int,N: $int] :
( $lesseq(0,N)
=> ( power1($product(X,Y),N) = $product(power1(X,N),power1(Y,N)) ) ) ).
tff(sum_digits,type,
sum_digits1: $int > $int ).
tff(sum_digits_def,axiom,
! [N: $int] :
( ( $lesseq(N,0)
=> ( sum_digits1(N) = 0 ) )
& ( ~ $lesseq(N,0)
=> ( sum_digits1(N) = $sum(sum_digits1($quotient_e(N,10)),$remainder_e(N,10)) ) ) ) ).
tff(tuple3,type,
tuple3: ( ty * ty * ty ) > ty ).
tff(tuple31,type,
tuple31: ( ty * ty * ty * uni * uni * uni ) > uni ).
tff(tuple3_sort1,axiom,
! [A: ty,A1: ty,A2: ty,X: uni,X1: uni,X2: uni] : sort1(tuple3(A2,A1,A),tuple31(A2,A1,A,X,X1,X2)) ).
tff(tuple3_proj_1,type,
tuple3_proj_11: ( ty * ty * ty * uni ) > uni ).
tff(tuple3_proj_1_sort1,axiom,
! [A: ty,A1: ty,A2: ty,X: uni] : sort1(A2,tuple3_proj_11(A2,A1,A,X)) ).
tff(tuple3_proj_1_def1,axiom,
! [A: ty,A1: ty,A2: ty,U: uni,U1: uni,U2: uni] :
( sort1(A2,U)
=> ( tuple3_proj_11(A2,A1,A,tuple31(A2,A1,A,U,U1,U2)) = U ) ) ).
tff(tuple3_proj_2,type,
tuple3_proj_21: ( ty * ty * ty * uni ) > uni ).
tff(tuple3_proj_2_sort1,axiom,
! [A: ty,A1: ty,A2: ty,X: uni] : sort1(A1,tuple3_proj_21(A2,A1,A,X)) ).
tff(tuple3_proj_2_def1,axiom,
! [A: ty,A1: ty,A2: ty,U: uni,U1: uni,U2: uni] :
( sort1(A1,U1)
=> ( tuple3_proj_21(A2,A1,A,tuple31(A2,A1,A,U,U1,U2)) = U1 ) ) ).
tff(tuple3_proj_3,type,
tuple3_proj_31: ( ty * ty * ty * uni ) > uni ).
tff(tuple3_proj_3_sort1,axiom,
! [A: ty,A1: ty,A2: ty,X: uni] : sort1(A,tuple3_proj_31(A2,A1,A,X)) ).
tff(tuple3_proj_3_def1,axiom,
! [A: ty,A1: ty,A2: ty,U: uni,U1: uni,U2: uni] :
( sort1(A,U2)
=> ( tuple3_proj_31(A2,A1,A,tuple31(A2,A1,A,U,U1,U2)) = U2 ) ) ).
tff(tuple3_inversion1,axiom,
! [A: ty,A1: ty,A2: ty,U: uni] :
( sort1(tuple3(A2,A1,A),U)
=> ( U = tuple31(A2,A1,A,tuple3_proj_11(A2,A1,A,U),tuple3_proj_21(A2,A1,A,U),tuple3_proj_31(A2,A1,A,U)) ) ) ).
tff(lpintcm_intcm_intrp,type,
lpintcm_intcm_intrp: $tType ).
tff(p,type,
p1: ( lpintcm_intcm_intrp * $int ) > $o ).
tff(t2tb,type,
t2tb: lpintcm_intcm_intrp > uni ).
tff(t2tb_sort,axiom,
! [X: lpintcm_intcm_intrp] : sort1(tuple3(int,int,int),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > lpintcm_intcm_intrp ).
tff(bridgeL,axiom,
! [I: lpintcm_intcm_intrp] : 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] : sort1(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(p_def,axiom,
! [N: $int,A: $int,B: $int,C: $int] :
( p1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(C))),N)
<=> ( $lesseq(0,$remainder_e(N,10))
& $less($remainder_e(N,10),C)
& ( sum_digits1(N) = $sum(sum_digits1($sum($product(137,N),A)),B) ) ) ) ).
tff(num_of,type,
num_of1: ( lpintcm_intcm_intrp * $int * $int ) > $int ).
tff(num_of_empty,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( $lesseq(B,A)
=> ( num_of1(P,A,B) = 0 ) ) ).
tff(num_of_right_no_add,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( ~ p1(P,$difference(B,1))
=> ( num_of1(P,A,B) = num_of1(P,A,$difference(B,1)) ) ) ) ).
tff(num_of_right_add,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( p1(P,$difference(B,1))
=> ( num_of1(P,A,B) = $sum(1,num_of1(P,A,$difference(B,1))) ) ) ) ).
tff(num_of_bounds,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( $lesseq(0,num_of1(P,A,B))
& $lesseq(num_of1(P,A,B),$difference(B,A)) ) ) ).
tff(num_of_append,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int,C: $int] :
( ( $lesseq(A,B)
& $lesseq(B,C) )
=> ( num_of1(P,A,C) = $sum(num_of1(P,A,B),num_of1(P,B,C)) ) ) ).
tff(num_of_left_no_add,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( ~ p1(P,A)
=> ( num_of1(P,A,B) = num_of1(P,$sum(A,1),B) ) ) ) ).
tff(num_of_left_add,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( p1(P,A)
=> ( num_of1(P,A,B) = $sum(1,num_of1(P,$sum(A,1),B)) ) ) ) ).
tff(empty,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( ! [N: $int] :
( ( $lesseq(A,N)
& $less(N,B) )
=> ~ p1(P,N) )
=> ( num_of1(P,A,B) = 0 ) ) ).
tff(full,axiom,
! [P: lpintcm_intcm_intrp,A: $int,B: $int] :
( $lesseq(A,B)
=> ( ! [N: $int] :
( ( $lesseq(A,N)
& $less(N,B) )
=> p1(P,N) )
=> ( num_of1(P,A,B) = $difference(B,A) ) ) ) ).
tff(num_of_increasing,axiom,
! [P: lpintcm_intcm_intrp,I: $int,J: $int,K: $int] :
( ( $lesseq(I,J)
& $lesseq(J,K) )
=> $lesseq(num_of1(P,I,J),num_of1(P,I,K)) ) ).
tff(num_of_strictly_increasing,axiom,
! [P: lpintcm_intcm_intrp,I: $int,J: $int,K: $int,L: $int] :
( ( $lesseq(I,J)
& $lesseq(J,K)
& $less(K,L) )
=> ( p1(P,K)
=> $less(num_of1(P,I,J),num_of1(P,I,L)) ) ) ).
tff(num_of_change_any,axiom,
! [P1: lpintcm_intcm_intrp,P2: lpintcm_intcm_intrp,A: $int,B: $int] :
( ! [J: $int] :
( ( $lesseq(A,J)
& $less(J,B) )
=> ( p1(P1,J)
=> p1(P2,J) ) )
=> $lesseq(num_of1(P1,A,B),num_of1(P2,A,B)) ) ).
tff(num_of_change_some,axiom,
! [P1: lpintcm_intcm_intrp,P2: lpintcm_intcm_intrp,A: $int,B: $int,I: $int] :
( ( $lesseq(A,I)
& $less(I,B) )
=> ( ! [J: $int] :
( ( $lesseq(A,J)
& $less(J,B) )
=> ( p1(P1,J)
=> p1(P2,J) ) )
=> ( ~ p1(P1,I)
=> ( p1(P2,I)
=> $less(num_of1(P1,A,B),num_of1(P2,A,B)) ) ) ) ) ).
tff(solution,type,
solution1: ( $int * $int * $int ) > $int ).
tff(solution_def,axiom,
! [A: $int,B: $int,M: $int] : solution1(A,B,M) = num_of1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(10))),0,power1(10,M)) ).
tff(num_of_modc,type,
num_of_modc1: ( lpintcm_intcm_intrp * $int * $int ) > $int ).
tff(num_of_modc_def,axiom,
! [X: $int,Y: $int,A: $int,B: $int,C: $int] : num_of_modc1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(C))),X,Y) = $difference(num_of1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1($sum(C,1)))),X,Y),num_of1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(C))),X,Y)) ).
tff(base,axiom,
! [A: $int,B: $int] :
( $lesseq(0,A)
=> ( ( $sum(sum_digits1(A),B) = 0 )
=> p1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(10))),0) ) ) ).
tff(empty1,axiom,
! [A: $int,B: $int,X: $int,Y: $int] : num_of1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(0))),X,Y) = 0 ).
tff(induc,axiom,
! [A: $int,B: $int,C: $int,M: $int] :
( $lesseq(0,A)
=> ( ( $lesseq(0,C)
& $less(C,10) )
=> ( $less(0,M)
=> ( solution1($quotient_e($sum($product(137,C),A),10),$difference($sum($remainder_e($sum($product(137,C),A),10),B),C),$difference(M,1)) = num_of_modc1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(C))),0,power1(10,M)) ) ) ) ) ).
tff(div,type,
div2: ( $int * $int ) > $int ).
tff(mod,type,
mod2: ( $int * $int ) > $int ).
tff(div_mod1,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( X = $sum($product(Y,div2(X,Y)),mod2(X,Y)) ) ) ).
tff(div_bound1,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( $lesseq(0,div2(X,Y))
& $lesseq(div2(X,Y),X) ) ) ).
tff(mod_bound1,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( $less($uminus(abs1(Y)),mod2(X,Y))
& $less(mod2(X,Y),abs1(Y)) ) ) ).
tff(div_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> $lesseq(0,div2(X,Y)) ) ).
tff(div_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& $less(0,Y) )
=> $lesseq(div2(X,Y),0) ) ).
tff(mod_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& ( Y != 0 ) )
=> $lesseq(0,mod2(X,Y)) ) ).
tff(mod_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& ( Y != 0 ) )
=> $lesseq(mod2(X,Y),0) ) ).
tff(rounds_toward_zero,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> $lesseq(abs1($product(div2(X,Y),Y)),abs1(X)) ) ).
tff(div_11,axiom,
! [X: $int] : div2(X,1) = X ).
tff(mod_11,axiom,
! [X: $int] : mod2(X,1) = 0 ).
tff(div_inf1,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( div2(X,Y) = 0 ) ) ).
tff(mod_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( mod2(X,Y) = X ) ) ).
tff(div_mult1,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( div2($sum($product(X,Y),Z),X) = $sum(Y,div2(Z,X)) ) ) ).
tff(mod_mult1,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( mod2($sum($product(X,Y),Z),X) = mod2(Z,X) ) ) ).
tff(wP_parameter_f,conjecture,
! [M: $int,A: $int,B: $int] :
( ( $lesseq(0,M)
& $lesseq(0,A) )
=> ( ( M != 0 )
=> ( $lesseq(0,9)
=> ! [Sum: $int,C: $int] :
( ( $lesseq(0,C)
& $lesseq(C,9) )
=> ( ( Sum = num_of1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(C))),0,power1(10,M)) )
=> ( ( $lesseq(0,$difference(M,1))
& $lesseq(0,div2($sum($product(137,C),A),10)) )
=> ! [Sum1: $int] :
( ( Sum1 = $sum(Sum,solution1(div2($sum($product(137,C),A),10),$difference($sum(mod2($sum($product(137,C),A),10),B),C),$difference(M,1))) )
=> ( ( div2($sum($product(137,C),A),10) = $quotient_e($sum($product(137,C),A),10) )
=> ( ( mod2($sum($product(137,C),A),10) = $remainder_e($sum($product(137,C),A),10) )
=> ( $difference(Sum1,Sum) = num_of_modc1(tb2t(tuple31(int,int,int,t2tb1(A),t2tb1(B),t2tb1(C))),0,power1(10,M)) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------