## TPTP Problem File: SWW588=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW588=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Division-T-WP parameter division
% 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    : division-T-WP_parameter_division [Fil14]

% Status   : Theorem
% Rating   : 0.25 v7.5.0, 0.30 v7.4.0, 0.25 v7.3.0, 0.17 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.57 v6.2.0, 0.62 v6.1.0
% Syntax   : Number of formulae    :   31 (   6 unt;  18 typ;   0 def)
%            Number of atoms       :   36 (  14 equ)
%            Maximal formula atoms :   17 (   1 avg)
%            Number of connectives :   25 (   2   ~;   1   |;  10   &)
%                                         (   0 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number arithmetic     :   46 (  14 atm;  12 fun;  10 num;  10 var)
%            Number of types       :    6 (   4 usr;   1 ari)
%            Number of type conns  :   12 (   6   >;   6   *;   0   +;   0  <<)
%            Number of predicates  :    4 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :   18 (  13 usr;  10 con; 0-4 aty)
%            Number of variables   :   31 (  30   !;   1   ?;  31   :)
% 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(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(wP_parameter_division,conjecture,
! [A: \$int,B: \$int] :
( ( \$lesseq(0,A)
& \$less(0,B) )
=> ( ( \$sum(\$product(0,B),A) = A )
& \$lesseq(0,A)
& ! [R: \$int,Q: \$int] :
( ( ( \$sum(\$product(Q,B),R) = A )
& \$lesseq(0,R) )
=> ( ( \$lesseq(B,R)
=> ! [Q1: \$int] :
( ( Q1 = \$sum(Q,1) )
=> ! [R1: \$int] :
( ( R1 = \$difference(R,B) )
=> ( ( \$sum(\$product(Q1,B),R1) = A )
& \$lesseq(0,R1)
& \$lesseq(0,R)
& \$less(R1,R) ) ) ) )
& ( ~ \$lesseq(B,R)
=> ? [R1: \$int] :
( ( \$sum(\$product(Q,B),R1) = A )
& \$lesseq(0,R1)
& \$less(R1,B) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------
```