TPTP Problem File: SWW578_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW578_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Binary search-T-WP parameter binary search
% 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 : binary_search-T-WP_parameter_binary_search [Fil14]
% Status : Theorem
% Rating : 0.62 v7.5.0, 0.60 v7.4.0, 0.50 v7.0.0, 0.86 v6.4.0, 0.33 v6.3.0, 0.86 v6.2.0, 0.62 v6.1.0
% Syntax : Number of formulae : 113 ( 33 unt; 47 typ; 0 def)
% Number of atoms : 204 ( 78 equ)
% Maximal formula atoms : 86 ( 1 avg)
% Number of connectives : 152 ( 14 ~; 1 |; 64 &)
% ( 5 <=>; 68 =>; 0 <=; 0 <~>)
% Maximal formula depth : 66 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number arithmetic : 232 ( 84 atm; 30 fun; 63 num; 55 var)
% Number of types : 9 ( 7 usr; 1 ari)
% Number of type conns : 61 ( 31 >; 30 *; 0 +; 0 <<)
% Number of predicates : 5 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 47 ( 38 usr; 14 con; 0-5 aty)
% Number of variables : 173 ( 173 !; 0 ?; 173 :)
% 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(int32,type,
int321: $tType ).
tff(int321,type,
int32: ty ).
tff(to_int,type,
to_int1: int321 > $int ).
tff(in_bounds,type,
in_bounds1: $int > $o ).
tff(in_bounds_def,axiom,
! [N: $int] :
( in_bounds1(N)
<=> ( $lesseq($uminus(2147483648),N)
& $lesseq(N,2147483647) ) ) ).
tff(to_int_in_bounds,axiom,
! [N: int321] : in_bounds1(to_int1(N)) ).
tff(extensionality,axiom,
! [X: int321,Y: int321] :
( ( to_int1(X) = to_int1(Y) )
=> ( X = Y ) ) ).
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(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(map,type,
map: ( ty * ty ) > ty ).
tff(get,type,
get: ( ty * ty * uni * uni ) > uni ).
tff(get_sort3,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort1(B,get(B,A,X,X1)) ).
tff(map_int_int32,type,
map_int_int32: $tType ).
tff(get1,type,
get3: ( map_int_int32 * $int ) > int321 ).
tff(set,type,
set: ( ty * ty * uni * uni * uni ) > uni ).
tff(set_sort2,axiom,
! [A: ty,B: ty,X: uni,X1: uni,X2: uni] : sort1(map(A,B),set(B,A,X,X1,X2)) ).
tff(t2tb,type,
t2tb: map_int_int32 > uni ).
tff(t2tb_sort,axiom,
! [X: map_int_int32] : sort1(map(int,int32),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > map_int_int32 ).
tff(bridgeL,axiom,
! [I: map_int_int32] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] :
( sort1(map(int,int32),J)
=> ( t2tb(tb2t(J)) = J ) ) ).
tff(t2tb1,type,
t2tb1: int321 > uni ).
tff(t2tb_sort1,axiom,
! [X: int321] : sort1(int32,t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > int321 ).
tff(bridgeL1,axiom,
! [I: int321] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] :
( sort1(int32,J)
=> ( t2tb1(tb2t1(J)) = J ) ) ).
tff(t2tb2,type,
t2tb2: $int > uni ).
tff(t2tb_sort2,axiom,
! [X: $int] : sort1(int,t2tb2(X)) ).
tff(tb2t2,type,
tb2t2: uni > $int ).
tff(bridgeL2,axiom,
! [I: $int] : tb2t2(t2tb2(I)) = I ).
tff(bridgeR2,axiom,
! [J: uni] : t2tb2(tb2t2(J)) = J ).
tff(select_eq,axiom,
! [M: map_int_int32,A1: $int,A2: $int,B: int321] :
( ( A1 = A2 )
=> ( get3(tb2t(set(int32,int,t2tb(M),t2tb2(A1),t2tb1(B))),A2) = B ) ) ).
tff(select_eq1,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni,B1: uni] :
( sort1(B,B1)
=> ( ( A1 = A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = B1 ) ) ) ).
tff(select_neq,axiom,
! [M: map_int_int32,A1: $int,A2: $int,B: int321] :
( ( A1 != A2 )
=> ( get3(tb2t(set(int32,int,t2tb(M),t2tb2(A1),t2tb1(B))),A2) = get3(M,A2) ) ) ).
tff(select_neq1,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni] :
( sort1(A,A1)
=> ( sort1(A,A2)
=> ! [B1: uni] :
( ( A1 != A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = get(B,A,M,A2) ) ) ) ) ).
tff(const2,type,
const: ( ty * ty * uni ) > uni ).
tff(const_sort1,axiom,
! [A: ty,B: ty,X: uni] : sort1(map(A,B),const(B,A,X)) ).
tff(const,axiom,
! [B: int321,A: $int] : get3(tb2t(const(int32,int,t2tb1(B))),A) = B ).
tff(const1,axiom,
! [A: ty,B: ty,B1: uni,A1: uni] :
( sort1(B,B1)
=> ( get(B,A,const(B,A,B1),A1) = B1 ) ) ).
tff(array,type,
array: ty > ty ).
tff(mk_array,type,
mk_array1: ( ty * int321 * uni ) > uni ).
tff(mk_array_sort1,axiom,
! [A: ty,X: int321,X1: uni] : sort1(array(A),mk_array1(A,X,X1)) ).
tff(length,type,
length1: ( ty * uni ) > int321 ).
tff(length_def1,axiom,
! [A: ty,U: int321,U1: uni] : length1(A,mk_array1(A,U,U1)) = U ).
tff(elts,type,
elts: ( ty * uni ) > uni ).
tff(elts_sort1,axiom,
! [A: ty,X: uni] : sort1(map(int,A),elts(A,X)) ).
tff(elts_def1,axiom,
! [A: ty,U: int321,U1: uni] :
( sort1(map(int,A),U1)
=> ( elts(A,mk_array1(A,U,U1)) = U1 ) ) ).
tff(array_inversion1,axiom,
! [A: ty,U: uni] :
( sort1(array(A),U)
=> ( U = mk_array1(A,length1(A,U),elts(A,U)) ) ) ).
tff(get2,type,
get4: ( ty * uni * $int ) > uni ).
tff(get_sort4,axiom,
! [A: ty,X: uni,X1: $int] : sort1(A,get4(A,X,X1)) ).
tff(array_int32,type,
array_int32: $tType ).
tff(t2tb3,type,
t2tb3: array_int32 > uni ).
tff(t2tb_sort3,axiom,
! [X: array_int32] : sort1(array(int32),t2tb3(X)) ).
tff(tb2t3,type,
tb2t3: uni > array_int32 ).
tff(bridgeL3,axiom,
! [I: array_int32] : tb2t3(t2tb3(I)) = I ).
tff(bridgeR3,axiom,
! [J: uni] :
( sort1(array(int32),J)
=> ( t2tb3(tb2t3(J)) = J ) ) ).
tff(get_def,axiom,
! [A: array_int32,I: $int] : tb2t1(get4(int32,t2tb3(A),I)) = get3(tb2t(elts(int32,t2tb3(A))),I) ).
tff(get_def1,axiom,
! [A: ty,A1: uni,I: $int] : get4(A,A1,I) = get(A,int,elts(A,A1),t2tb2(I)) ).
tff(set1,type,
set2: ( ty * uni * $int * uni ) > uni ).
tff(set_sort3,axiom,
! [A: ty,X: uni,X1: $int,X2: uni] : sort1(array(A),set2(A,X,X1,X2)) ).
tff(set_def,axiom,
! [A: ty,A1: uni,I: $int,V: uni] : set2(A,A1,I,V) = mk_array1(A,length1(A,A1),set(A,int,elts(A,A1),t2tb2(I),V)) ).
tff(make,type,
make1: ( ty * int321 * uni ) > uni ).
tff(make_sort1,axiom,
! [A: ty,X: int321,X1: uni] : sort1(array(A),make1(A,X,X1)) ).
tff(make_def,axiom,
! [A: ty,N: int321,V: uni] : make1(A,N,V) = mk_array1(A,N,const(A,int,V)) ).
tff(wP_parameter_binary_search,conjecture,
! [A: int321,A1: map_int_int32,V: int321] :
( ( $lesseq(0,to_int1(A))
& ! [I1: $int,I2: $int] :
( ( $lesseq(0,I1)
& $lesseq(I1,I2)
& $less(I2,to_int1(A)) )
=> $lesseq(to_int1(get3(A1,I1)),to_int1(get3(A1,I2))) ) )
=> ( in_bounds1(0)
& ! [O: int321] :
( ( to_int1(O) = 0 )
=> ( in_bounds1(1)
& ! [O1: int321] :
( ( to_int1(O1) = 1 )
=> ( in_bounds1($difference(to_int1(A),to_int1(O1)))
& ! [O2: int321] :
( ( to_int1(O2) = $difference(to_int1(A),to_int1(O1)) )
=> ( $lesseq(0,to_int1(O))
& $less(to_int1(O2),to_int1(A))
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,to_int1(A)) )
=> ( ( get3(A1,I) = V )
=> ( $lesseq(to_int1(O),I)
& $lesseq(I,to_int1(O2)) ) ) )
& ! [U: int321,L: int321] :
( ( $lesseq(0,to_int1(L))
& $less(to_int1(U),to_int1(A))
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,to_int1(A)) )
=> ( ( get3(A1,I) = V )
=> ( $lesseq(to_int1(L),I)
& $lesseq(I,to_int1(U)) ) ) ) )
=> ! [Result: bool1] :
( ( ( Result = true1 )
<=> $lesseq(to_int1(L),to_int1(U)) )
=> ( ( ( Result = true1 )
=> ( in_bounds1(2)
& ! [O3: int321] :
( ( to_int1(O3) = 2 )
=> ( in_bounds1($difference(to_int1(U),to_int1(L)))
& ! [O4: int321] :
( ( to_int1(O4) = $difference(to_int1(U),to_int1(L)) )
=> ( ( to_int1(O3) != 0 )
& in_bounds1(div1(to_int1(O4),to_int1(O3)))
& ! [O5: int321] :
( ( to_int1(O5) = div1(to_int1(O4),to_int1(O3)) )
=> ( in_bounds1($sum(to_int1(L),to_int1(O5)))
& ! [M: int321] :
( ( to_int1(M) = $sum(to_int1(L),to_int1(O5)) )
=> ( $lesseq(to_int1(L),to_int1(M))
& $lesseq(to_int1(M),to_int1(U))
& $lesseq(0,to_int1(M))
& $less(to_int1(M),to_int1(A))
& ! [Result1: bool1] :
( ( ( Result1 = true1 )
<=> $less(to_int1(get3(A1,to_int1(M))),to_int1(V)) )
=> ( ( ( Result1 = true1 )
=> ( in_bounds1(1)
& ! [O6: int321] :
( ( to_int1(O6) = 1 )
=> ( in_bounds1($sum(to_int1(M),to_int1(O6)))
& ! [O7: int321] :
( ( to_int1(O7) = $sum(to_int1(M),to_int1(O6)) )
=> ! [L1: int321] :
( ( L1 = O7 )
=> ( $lesseq(0,to_int1(L1))
& $less(to_int1(U),to_int1(A))
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,to_int1(A)) )
=> ( ( get3(A1,I) = V )
=> ( $lesseq(to_int1(L1),I)
& $lesseq(I,to_int1(U)) ) ) )
& $lesseq(0,$difference(to_int1(U),to_int1(L)))
& $less($difference(to_int1(U),to_int1(L1)),$difference(to_int1(U),to_int1(L))) ) ) ) ) ) ) )
& ( ( Result1 != true1 )
=> ( $lesseq(0,to_int1(M))
& $less(to_int1(M),to_int1(A))
& ! [Result2: bool1] :
( ( ( Result2 = true1 )
<=> $less(to_int1(V),to_int1(get3(A1,to_int1(M)))) )
=> ( ( ( Result2 = true1 )
=> ( in_bounds1(1)
& ! [O6: int321] :
( ( to_int1(O6) = 1 )
=> ( in_bounds1($difference(to_int1(M),to_int1(O6)))
& ! [O7: int321] :
( ( to_int1(O7) = $difference(to_int1(M),to_int1(O6)) )
=> ! [U1: int321] :
( ( U1 = O7 )
=> ( $lesseq(0,to_int1(L))
& $less(to_int1(U1),to_int1(A))
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,to_int1(A)) )
=> ( ( get3(A1,I) = V )
=> ( $lesseq(to_int1(L),I)
& $lesseq(I,to_int1(U1)) ) ) )
& $lesseq(0,$difference(to_int1(U),to_int1(L)))
& $less($difference(to_int1(U1),to_int1(L)),$difference(to_int1(U),to_int1(L))) ) ) ) ) ) ) )
& ( ( Result2 != true1 )
=> ( $lesseq(0,to_int1(M))
& $less(to_int1(M),to_int1(A))
& ( get3(A1,to_int1(M)) = V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ( Result != true1 )
=> ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,to_int1(A)) )
=> ( get3(A1,I) != V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------