## TPTP Problem File: SWW664=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW664=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Verifythis PrefixSumRec-T-WP parameter downsweep
% 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    : verifythis_PrefixSumRec-T-WP_parameter_downsweep [Fil14]

% Status   : Theorem
% Rating   : 0.62 v7.5.0, 0.70 v7.4.0, 0.38 v7.3.0, 0.33 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.57 v6.2.0, 0.75 v6.1.0
% Syntax   : Number of formulae    :  121 (  34 unt;  45 typ;   0 def)
%            Number of atoms       :  187 (  74 equ)
%            Maximal formula atoms :   14 (   1 avg)
%            Number of connectives :  119 (   8   ~;   2   |;  50   &)
%                                         (   3 <=>;  56  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   5 avg)
%            Maximal term depth    :    6 (   1 avg)
%            Number arithmetic     :  339 (  77 atm;  68 fun;  89 num; 105 var)
%            Number of types       :    8 (   6 usr;   1 ari)
%            Number of type conns  :   71 (  31   >;  40   *;   0   +;   0  <<)
%            Number of predicates  :    7 (   4 usr;   0 prp; 1-4 aty)
%            Number of functors    :   42 (  35 usr;  11 con; 0-5 aty)
%            Number of variables   :  204 ( 196   !;   8   ?; 204   :)
% SPC      : TF0_THM_EQU_ARI

%------------------------------------------------------------------------------
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(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(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(map,type,
map: ( ty * ty ) > ty ).

tff(get,type,
get: ( ty * ty * uni * uni ) > uni ).

tff(get_sort2,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort1(B,get(B,A,X,X1)) ).

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(select_eq,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,
! [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(const1,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,
! [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 * \$int * uni ) > uni ).

tff(mk_array_sort1,axiom,
! [A: ty,X: \$int,X1: uni] : sort1(array(A),mk_array1(A,X,X1)) ).

tff(length,type,
length1: ( ty * uni ) > \$int ).

tff(length_def1,axiom,
! [A: ty,U: \$int,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: \$int,U1: uni] :
( sort1(map(int,A),U1)
=> ( elts(A,mk_array1(A,U,U1)) = U1 ) ) ).

tff(array_inversion1,axiom,
! [A: ty,U: uni] : ( U = mk_array1(A,length1(A,U),elts(A,U)) ) ).

tff(get1,type,
get2: ( ty * uni * \$int ) > uni ).

tff(get_sort3,axiom,
! [A: ty,X: uni,X1: \$int] : sort1(A,get2(A,X,X1)) ).

tff(t2tb,type,
t2tb: \$int > uni ).

tff(t2tb_sort,axiom,
! [X: \$int] : sort1(int,t2tb(X)) ).

tff(tb2t,type,
tb2t: uni > \$int ).

tff(bridgeL,axiom,
! [I: \$int] : ( tb2t(t2tb(I)) = I ) ).

tff(bridgeR,axiom,
! [J: uni] : ( t2tb(tb2t(J)) = J ) ).

tff(get_def,axiom,
! [A: ty,A1: uni,I: \$int] : ( get2(A,A1,I) = get(A,int,elts(A,A1),t2tb(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),t2tb(I),V)) ) ).

tff(make,type,
make1: ( ty * \$int * uni ) > uni ).

tff(make_sort1,axiom,
! [A: ty,X: \$int,X1: uni] : sort1(array(A),make1(A,X,X1)) ).

tff(make_def,axiom,
! [A: ty,N: \$int,V: uni] : ( make1(A,N,V) = mk_array1(A,N,const(A,int,V)) ) ).

tff(map_int_int,type,
map_int_int: \$tType ).

tff(sum,type,
sum2: ( map_int_int * \$int * \$int ) > \$int ).

tff(sum_def_empty,axiom,
! [C: map_int_int,I: \$int,J: \$int] :
( \$lesseq(J,I)
=> ( sum2(C,I,J) = 0 ) ) ).

tff(t2tb1,type,
t2tb1: map_int_int > uni ).

tff(t2tb_sort1,axiom,
! [X: map_int_int] : sort1(map(int,int),t2tb1(X)) ).

tff(tb2t1,type,
tb2t1: uni > map_int_int ).

tff(bridgeL1,axiom,
! [I: map_int_int] : ( tb2t1(t2tb1(I)) = I ) ).

tff(bridgeR1,axiom,
! [J: uni] : ( t2tb1(tb2t1(J)) = J ) ).

tff(sum_def_non_empty,axiom,
! [C: map_int_int,I: \$int,J: \$int] :
( \$less(I,J)
=> ( sum2(C,I,J) = \$sum(tb2t(get(int,int,t2tb1(C),t2tb(I))),sum2(C,\$sum(I,1),J)) ) ) ).

tff(sum_right_extension,axiom,
! [C: map_int_int,I: \$int,J: \$int] :
( \$less(I,J)
=> ( sum2(C,I,J) = \$sum(sum2(C,I,\$difference(J,1)),tb2t(get(int,int,t2tb1(C),t2tb(\$difference(J,1))))) ) ) ).

tff(sum_transitivity,axiom,
! [C: map_int_int,I: \$int,K: \$int,J: \$int] :
( ( \$lesseq(I,K)
& \$lesseq(K,J) )
=> ( sum2(C,I,J) = \$sum(sum2(C,I,K),sum2(C,K,J)) ) ) ).

tff(sum_eq,axiom,
! [C1: map_int_int,C2: map_int_int,I: \$int,J: \$int] :
( ! [K: \$int] :
( ( \$lesseq(I,K)
& \$less(K,J) )
=> ( tb2t(get(int,int,t2tb1(C1),t2tb(K))) = tb2t(get(int,int,t2tb1(C2),t2tb(K))) ) )
=> ( sum2(C1,I,J) = sum2(C2,I,J) ) ) ).

tff(array_int,type,
array_int: \$tType ).

tff(sum1,type,
sum3: ( array_int * \$int * \$int ) > \$int ).

tff(t2tb2,type,
t2tb2: array_int > uni ).

tff(t2tb_sort2,axiom,
! [X: array_int] : sort1(array(int),t2tb2(X)) ).

tff(tb2t2,type,
tb2t2: uni > array_int ).

tff(bridgeL2,axiom,
! [I: array_int] : ( tb2t2(t2tb2(I)) = I ) ).

tff(bridgeR2,axiom,
! [J: uni] : ( t2tb2(tb2t2(J)) = J ) ).

tff(sum_def,axiom,
! [A: array_int,L: \$int,H: \$int] : ( sum3(A,L,H) = sum2(tb2t1(elts(int,t2tb2(A))),L,H) ) ).

tff(div_mod_2,axiom,
! [X: \$int] :
( \$lesseq(0,X)
=> ( \$lesseq(\$product(2,div1(X,2)),X)
& \$lesseq(\$difference(X,1),\$product(2,div1(X,2))) ) ) ).

tff(is_power_of_2,type,
is_power_of_21: \$int > \$o ).

tff(is_power_of_2_def,axiom,
! [X: \$int] :
( is_power_of_21(X)
<=> ? [K: \$int] :
( \$lesseq(0,K)
& ( X = power1(2,K) ) ) ) ).

tff(is_power_of_2_1,axiom,
! [X: \$int] :
( is_power_of_21(X)
=> ( \$less(1,X)
=> ( \$product(2,div1(X,2)) = X ) ) ) ).

tff(go_left,type,
go_left1: ( \$int * \$int ) > \$int ).

tff(go_left_def,axiom,
! [Left: \$int,Right: \$int] : ( go_left1(Left,Right) = \$difference(Left,div1(\$difference(Right,Left),2)) ) ).

tff(go_right,type,
go_right1: ( \$int * \$int ) > \$int ).

tff(go_right_def,axiom,
! [Left: \$int,Right: \$int] : ( go_right1(Left,Right) = \$difference(Right,div1(\$difference(Right,Left),2)) ) ).

tff(phase1,type,
phase11: ( \$int * \$int * array_int * array_int ) > \$o ).

tff(leaf,axiom,
! [Left: \$int,Right: \$int,A0: array_int,A: array_int] :
( ( Right = \$sum(Left,1) )
=> ( ( tb2t(get2(int,t2tb2(A),Left)) = tb2t(get2(int,t2tb2(A0),Left)) )
=> phase11(Left,Right,A0,A) ) ) ).

tff(node,axiom,
! [Left: \$int,Right: \$int,A0: array_int,A: array_int] :
( \$less(\$sum(Left,1),Right)
=> ( phase11(go_left1(Left,Right),Left,A0,A)
=> ( phase11(go_right1(Left,Right),Right,A0,A)
=> ( ( tb2t(get2(int,t2tb2(A),Left)) = sum3(A0,\$sum(\$difference(Left,\$difference(Right,Left)),1),\$sum(Left,1)) )
=> phase11(Left,Right,A0,A) ) ) ) ) ).

tff(phase1_inversion,axiom,
! [Z: \$int,Z1: \$int,Z2: array_int,Z3: array_int] :
( phase11(Z,Z1,Z2,Z3)
=> ( ? [Left: \$int,A0: array_int,A: array_int] :
( ( tb2t(get2(int,t2tb2(A),Left)) = tb2t(get2(int,t2tb2(A0),Left)) )
& ( Z = Left )
& ( Z1 = \$sum(Left,1) )
& ( Z2 = A0 )
& ( Z3 = A ) )
| ? [Left: \$int,Right: \$int,A0: array_int,A: array_int] :
( \$less(\$sum(Left,1),Right)
& phase11(go_left1(Left,Right),Left,A0,A)
& phase11(go_right1(Left,Right),Right,A0,A)
& ( tb2t(get2(int,t2tb2(A),Left)) = sum3(A0,\$sum(\$difference(Left,\$difference(Right,Left)),1),\$sum(Left,1)) )
& ( Z = Left )
& ( Z1 = Right )
& ( Z2 = A0 )
& ( Z3 = A ) ) ) ) ).

tff(phase1_frame,axiom,
! [Left: \$int,Right: \$int,A0: array_int,A: array_int,Aqt: array_int] :
( ( \$lesseq(0,length1(int,t2tb2(A0)))
& \$lesseq(0,length1(int,t2tb2(A)))
& \$lesseq(0,length1(int,t2tb2(Aqt)))
& ! [I: \$int] :
( ( \$less(\$difference(Left,\$difference(Right,Left)),I)
& \$less(I,Right) )
=> ( tb2t(get2(int,t2tb2(A),I)) = tb2t(get2(int,t2tb2(Aqt),I)) ) )
& phase11(Left,Right,A0,A) )
=> phase11(Left,Right,A0,Aqt) ) ).

tff(phase1_frame2,axiom,
! [Left: \$int,Right: \$int,A0: array_int,A0qt: array_int,A: array_int] :
( ( \$lesseq(0,length1(int,t2tb2(A0)))
& \$lesseq(0,length1(int,t2tb2(A0qt)))
& \$lesseq(0,length1(int,t2tb2(A)))
& ! [I: \$int] :
( ( \$less(\$difference(Left,\$difference(Right,Left)),I)
& \$less(I,Right) )
=> ( tb2t(get2(int,t2tb2(A0),I)) = tb2t(get2(int,t2tb2(A0qt),I)) ) )
& phase11(Left,Right,A0,A) )
=> phase11(Left,Right,A0qt,A) ) ).

tff(partial_sum,type,
partial_sum1: ( \$int * \$int * array_int * array_int ) > \$o ).

tff(partial_sum_def,axiom,
! [Left: \$int,Right: \$int,A0: array_int,A: array_int] :
( partial_sum1(Left,Right,A0,A)
<=> ! [I: \$int] :
( ( \$less(\$difference(Left,\$difference(Right,Left)),I)
& \$lesseq(I,Right) )
=> ( tb2t(get2(int,t2tb2(A),I)) = sum3(A0,0,I) ) ) ) ).

tff(wP_parameter_downsweep,conjecture,
! [Left: \$int,Right: \$int,A0: \$int,A01: map_int_int,A: \$int,A1: map_int_int] :
( ( \$lesseq(0,A0)
& \$lesseq(0,A)
& \$lesseq(0,Left)
& \$less(Left,Right)
& \$less(Right,A)
& \$lesseq(\$uminus(1),\$difference(Left,\$difference(Right,Left)))
& is_power_of_21(\$difference(Right,Left))
& ( tb2t(get(int,int,t2tb1(A1),t2tb(Right))) = sum2(A01,0,\$sum(\$difference(Left,\$difference(Right,Left)),1)) )
& phase11(Left,Right,tb2t2(mk_array1(int,A0,t2tb1(A01))),tb2t2(mk_array1(int,A,t2tb1(A1)))) )
=> ( ( \$lesseq(0,Right)
& \$less(Right,A) )
=> ( ( tb2t(get(int,int,t2tb1(A1),t2tb(Right))) = sum2(A01,0,\$sum(\$difference(Left,\$difference(Right,Left)),1)) )
=> ( tb2t(get(int,int,t2tb1(A1),t2tb(Left))) = sum2(A01,\$sum(\$difference(Left,\$difference(Right,Left)),1),\$sum(Left,1)) ) ) ) ) ).

%------------------------------------------------------------------------------
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