## TPTP Problem File: SWW652=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW652=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Vacid 0 build maze-T-Ineq1
% 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    : vacid_0_build_maze-T-Ineq1 [Fil14]

% Status   : Theorem
% Rating   : 0.62 v7.5.0, 0.80 v7.4.0, 0.62 v7.3.0, 0.67 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.86 v6.2.0, 0.88 v6.1.0
% Syntax   : Number of formulae    :   56 (   9 unt;  32 typ;   0 def)
%            Number of atoms       :   68 (  21 equ)
%            Maximal formula atoms :   13 (   1 avg)
%            Number of connectives :   45 (   1   ~;   3   |;  17   &)
%                                         (   4 <=>;  20  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number arithmetic     :   66 (  19 atm;   5 fun;  10 num;  32 var)
%            Number of types       :    9 (   7 usr;   1 ari)
%            Number of type conns  :   27 (  14   >;  13   *;   0   +;   0  <<)
%            Number of predicates  :    8 (   5 usr;   0 prp; 2-3 aty)
%            Number of functors    :   24 (  20 usr;  13 con; 0-4 aty)
%            Number of variables   :   68 (  58   !;  10   ?;  68   :)
% 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(uf_pure,type,
uf_pure1: \$tType ).

tff(uf_pure1,type,
uf_pure: ty ).

tff(repr,type,
repr1: ( uf_pure1 * \$int * \$int ) > \$o ).

tff(size,type,
size1: uf_pure1 > \$int ).

tff(num,type,
num1: uf_pure1 > \$int ).

tff(repr_function_1,axiom,
! [U: uf_pure1,X: \$int] :
( ( \$lesseq(0,X)
& \$less(X,size1(U)) )
=> ? [Y: \$int] :
( \$lesseq(0,Y)
& \$less(Y,size1(U))
& repr1(U,X,Y) ) ) ).

tff(repr_function_2,axiom,
! [U: uf_pure1,X: \$int,Y: \$int,Z: \$int] :
( ( \$lesseq(0,X)
& \$less(X,size1(U)) )
=> ( repr1(U,X,Y)
=> ( repr1(U,X,Z)
=> ( Y = Z ) ) ) ) ).

tff(same,type,
same1: ( uf_pure1 * \$int * \$int ) > \$o ).

tff(same_def,axiom,
! [U: uf_pure1,X: \$int,Y: \$int] :
( same1(U,X,Y)
<=> ! [R: \$int] :
( repr1(U,X,R)
<=> repr1(U,Y,R) ) ) ).

tff(same_reprs,type,
same_reprs1: ( uf_pure1 * uf_pure1 ) > \$o ).

tff(same_reprs_def,axiom,
! [U1: uf_pure1,U2: uf_pure1] :
( same_reprs1(U1,U2)
<=> ! [X: \$int,R: \$int] :
( repr1(U1,X,R)
<=> repr1(U2,X,R) ) ) ).

tff(oneClass,axiom,
! [U: uf_pure1] :
( ( num1(U) = 1 )
=> ! [X: \$int,Y: \$int] :
( ( \$lesseq(0,X)
& \$less(X,size1(U)) )
=> ( ( \$lesseq(0,Y)
& \$less(Y,size1(U)) )
=> same1(U,X,Y) ) ) ) ).

tff(uf,type,
uf1: \$tType ).

tff(uf1,type,
uf: ty ).

tff(mk_uf,type,
mk_uf1: uf_pure1 > uf1 ).

tff(state,type,
state1: uf1 > uf_pure1 ).

tff(state_def1,axiom,
! [U: uf_pure1] : ( state1(mk_uf1(U)) = U ) ).

tff(uf_inversion1,axiom,
! [U: uf1] : ( U = mk_uf1(state1(U)) ) ).

tff(graph,type,
graph1: \$tType ).

tff(graph1,type,
graph: ty ).

tff(path,type,
path1: ( graph1 * \$int * \$int ) > \$o ).

tff(path_refl,axiom,
! [G: graph1,X: \$int] : path1(G,X,X) ).

tff(path_sym,axiom,
! [G: graph1,X: \$int,Y: \$int] :
( path1(G,X,Y)
=> path1(G,Y,X) ) ).

tff(path_trans,axiom,
! [G: graph1,X: \$int,Y: \$int,Z: \$int] :
( path1(G,X,Y)
=> ( path1(G,Y,Z)
=> path1(G,X,Z) ) ) ).

tff(path_inversion,axiom,
! [Z: graph1,Z1: \$int,Z2: \$int] :
( path1(Z,Z1,Z2)
=> ( ? [G: graph1,X: \$int] :
( ( Z = G )
& ( Z1 = X )
& ( Z2 = X ) )
| ? [G: graph1,X: \$int,Y: \$int] :
( path1(G,X,Y)
& ( Z = G )
& ( Z1 = Y )
& ( Z2 = X ) )
| ? [G: graph1,X: \$int,Y: \$int,Z3: \$int] :
( path1(G,X,Y)
& path1(G,Y,Z3)
& ( Z = G )
& ( Z1 = X )
& ( Z2 = Z3 ) ) ) ) ).

tff(ineq1,conjecture,
! [N: \$int,X: \$int,Y: \$int] :
( \$lesseq(0,N)
=> ( ( \$lesseq(0,X)
& \$less(X,N) )
=> ( ( \$lesseq(0,Y)
& \$less(Y,N) )
=> \$less(\$sum(\$product(X,N),Y),\$product(N,N)) ) ) ) ).

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