## TPTP Axioms File: SWV003+0.ax

```%------------------------------------------------------------------------------
% File     : SWV003+0 : TPTP v7.5.0. Bugfixed v3.3.0.
% Domain   : Software Verification
% Axioms   : NASA software certification axioms
% Version  : [DFS04] axioms : Especial.
% English  :

% Refs     : [Fis04] Fischer (2004), Email to G. Sutcliffe
%          : [DFS04] Denney et al. (2004), Using Automated Theorem Provers
% Source   : [Fis04]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :   52 (  23 unit)
%            Number of atoms       :  190 (  54 equality)
%            Maximal formula depth :   18 (   5 average)
%            Number of connectives :  143 (   5 ~  ;   2  |;  86  &)
%                                         (   5 <=>;  45 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   28 (  10 constant; 0-4 arity)
%            Number of variables   :  162 (   0 singleton; 162 !;   0 ?)
%            Maximal term depth    :    9 (   1 average)
% SPC      :

% Bugfixes : v3.3.0 - Fixed symmetry axioms
%------------------------------------------------------------------------------
%----Axioms for gt
fof(totality,axiom,(
! [X,Y] :
( gt(X,Y)
| gt(Y,X)
| X = Y ) )).

fof(transitivity_gt,axiom,(
! [X,Y,Z] :
( ( gt(X,Y)
& gt(Y,Z) )
=> gt(X,Z) ) )).

fof(irreflexivity_gt,axiom,(
! [X] : ~ gt(X,X) )).

%----Axioms for leq
fof(reflexivity_leq,axiom,(
! [X] : leq(X,X) )).

fof(transitivity_leq,axiom,(
! [X,Y,Z] :
( ( leq(X,Y)
& leq(Y,Z) )
=> leq(X,Z) ) )).

%----Axioms for lt/geq
fof(lt_gt,axiom,(
! [X,Y] :
( lt(X,Y)
<=> gt(Y,X) ) )).

fof(leq_geq,axiom,(
! [X,Y] :
( geq(X,Y)
<=> leq(Y,X) ) )).

%----Axioms for combinations of gt and leq
fof(leq_gt1,axiom,(
! [X,Y] :
( gt(Y,X)
=> leq(X,Y) ) )).

fof(leq_gt2,axiom,(
! [X,Y] :
( ( leq(X,Y)
& X != Y )
=> gt(Y,X) ) )).

%----leq/gt and pred/succ
fof(leq_gt_pred,axiom,(
! [X,Y] :
( leq(X,pred(Y))
<=> gt(Y,X) ) )).

fof(gt_succ,axiom,(
! [X] : gt(succ(X),X) )).

fof(leq_succ,axiom,(
! [X,Y] :
( leq(X,Y)
=> leq(X,succ(Y)) ) )).

fof(leq_succ_gt_equiv,axiom,(
! [X,Y] :
( leq(X,Y)
<=> gt(succ(Y),X) ) )).

%----uniform_int_rand
%----Restriction:  LB of uniform_int_rnd is 0
fof(uniform_int_rand_ranges_hi,axiom,(
! [X,C] :
( leq(n0,X)
=> leq(uniform_int_rnd(C,X),X) ) )).

fof(uniform_int_rand_ranges_lo,axiom,(
! [X,C] :
( leq(n0,X)
=> leq(n0,uniform_int_rnd(C,X)) ) )).

%----Axioms for constant arrays
fof(const_array1_select,axiom,(
! [I,L,U,Val] :
( ( leq(L,I)
& leq(I,U) )
=> a_select2(tptp_const_array1(dim(L,U),Val),I) = Val ) )).

fof(const_array2_select,axiom,(
! [I,L1,U1,J,L2,U2,Val] :
( ( leq(L1,I)
& leq(I,U1)
& leq(L2,J)
& leq(J,U2) )
=> a_select3(tptp_const_array2(dim(L1,U1),dim(L2,U2),Val),I,J) = Val ) )).

%----Symmetry axioms for matrix operations
fof(matrix_symm_trans,axiom,(
! [A,N] :
( ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(A,I,J) = a_select3(A,J,I) )
=> ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(trans(A),I,J) = a_select3(trans(A),J,I) ) ) )).

fof(matrix_symm_inv,axiom,(
! [A,N] :
( ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(A,I,J) = a_select3(A,J,I) )
=> ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(inv(A),I,J) = a_select3(inv(A),J,I) ) ) )).

fof(matrix_symm_update_diagonal,axiom,(
! [A,N] :
( ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(A,I,J) = a_select3(A,J,I) )
=> ! [I,J,K,VAL] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N)
& leq(n0,K)
& leq(K,N) )
=> a_select3(tptp_update3(A,K,K,VAL),I,J) = a_select3(tptp_update3(A,K,K,VAL),J,I) ) ) )).

! [A,B,N] :
( ( ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(A,I,J) = a_select3(A,J,I) )
& ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(B,I,J) = a_select3(B,J,I) ) )
=> ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )

fof(matrix_symm_sub,axiom,(
! [A,B,N] :
( ( ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(A,I,J) = a_select3(A,J,I) )
& ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(B,I,J) = a_select3(B,J,I) ) )
=> ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(tptp_msub(A,B),I,J) = a_select3(tptp_msub(A,B),J,I) ) ) )).

fof(matrix_symm_aba1,axiom,(
! [A,B,N] :
( ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(B,I,J) = a_select3(B,J,I) )
=> ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),I,J) = a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),J,I) ) ) )).

%----This is the generalized version where the matrix dimensions
%----can be different for B and the ABA'
fof(matrix_symm_aba2,axiom,(
! [A,B,N,M] :
( ! [I,J] :
( ( leq(n0,I)
& leq(I,M)
& leq(n0,J)
& leq(J,M) )
=> a_select3(B,I,J) = a_select3(B,J,I) )
=> ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),I,J) = a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),J,I) ) ) )).

fof(matrix_symm_joseph_update,axiom,(
! [A,B,C,D,E,F,N,M] :
( ( ! [I,J] :
( ( leq(n0,I)
& leq(I,M)
& leq(n0,J)
& leq(J,M) )
=> a_select3(D,I,J) = a_select3(D,J,I) )
& ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(A,I,J) = a_select3(A,J,I) )
& ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )
=> a_select3(F,I,J) = a_select3(F,J,I) ) )
=> ! [I,J] :
( ( leq(n0,I)
& leq(I,N)
& leq(n0,J)
& leq(J,N) )

%----handling of sums
fof(sum_plus_base,axiom,(
! [Body] : sum(n0,tptp_minus_1,Body) = n0 )).

fof(sum_plus_base_float,axiom,(
! [Body] : tptp_float_0_0 = sum(n0,tptp_minus_1,Body) )).

%----successor/predecessor
fof(succ_tptp_minus_1,axiom,(
succ(tptp_minus_1) = n0 )).

fof(succ_plus_1_r,axiom,(
! [X] : plus(X,n1) = succ(X) )).

fof(succ_plus_1_l,axiom,(
! [X] : plus(n1,X) = succ(X) )).

fof(succ_plus_2_r,axiom,(
! [X] : plus(X,n2) = succ(succ(X)) )).

fof(succ_plus_2_l,axiom,(
! [X] : plus(n2,X) = succ(succ(X)) )).

fof(succ_plus_3_r,axiom,(
! [X] : plus(X,n3) = succ(succ(succ(X))) )).

fof(succ_plus_3_l,axiom,(
! [X] : plus(n3,X) = succ(succ(succ(X))) )).

fof(succ_plus_4_r,axiom,(
! [X] : plus(X,n4) = succ(succ(succ(succ(X)))) )).

fof(succ_plus_4_l,axiom,(
! [X] : plus(n4,X) = succ(succ(succ(succ(X)))) )).

fof(succ_plus_5_r,axiom,(
! [X] : plus(X,n5) = succ(succ(succ(succ(succ(X))))) )).

fof(succ_plus_5_l,axiom,(
! [X] : plus(n5,X) = succ(succ(succ(succ(succ(X))))) )).

fof(pred_minus_1,axiom,(
! [X] : minus(X,n1) = pred(X) )).

fof(pred_succ,axiom,(
! [X] : pred(succ(X)) = X )).

fof(succ_pred,axiom,(
! [X] : succ(pred(X)) = X )).

%----leq/gt and successor
fof(leq_succ_succ,axiom,(
! [X,Y] :
( leq(succ(X),succ(Y))
<=> leq(X,Y) ) )).

fof(leq_succ_gt,axiom,(
! [X,Y] :
( leq(succ(X),Y)
=> gt(Y,X) ) )).

%----leq/gt and plus/minus
fof(leq_minus,axiom,(
! [X,Y] :
( leq(minus(X,Y),X)
=> leq(n0,Y) ) )).

%----select_update
fof(sel3_update_1,axiom,(
! [X,U,V,VAL] : a_select3(tptp_update3(X,U,V,VAL),U,V) = VAL )).

fof(sel3_update_2,axiom,(
! [I,J,U,V,X,VAL,VAL2] :
( ( I != U
& J = V
& a_select3(X,U,V) = VAL )
=> a_select3(tptp_update3(X,I,J,VAL2),U,V) = VAL ) )).

fof(sel3_update_3,axiom,(
! [I,J,U,V,X,VAL] :
( ( ! [I0,J0] :
( ( leq(n0,I0)
& leq(n0,J0)
& leq(I0,U)
& leq(J0,V) )
=> a_select3(X,I0,J0) = VAL )
& leq(n0,I)
& leq(I,U)
& leq(n0,J)
& leq(J,V) )
=> a_select3(tptp_update3(X,U,V,VAL),I,J) = VAL ) )).

fof(sel2_update_1,axiom,(
! [X,U,VAL] : a_select2(tptp_update2(X,U,VAL),U) = VAL )).

fof(sel2_update_2,axiom,(
! [I,U,X,VAL,VAL2] :
( ( I != U
& a_select2(X,U) = VAL )
=> a_select2(tptp_update2(X,I,VAL2),U) = VAL ) )).

fof(sel2_update_3,axiom,(
! [I,U,X,VAL] :
( ( ! [I0] :
( ( leq(n0,I0)
& leq(I0,U) )
=> a_select2(X,I0) = VAL )
& leq(n0,I)
& leq(I,U) )
=> a_select2(tptp_update2(X,U,VAL),I) = VAL ) )).

%----True
fof(ttrue,axiom,(
true )).

%----def and use inequality
fof(defuse,axiom,(
def != use )).
```