Certification Problem

Input (TPDB TRS_Standard/EEG_IJCAR_12/rybalchenko-nonloop-popl08)

The rewrite relation of the following TRS is considered.

notZero(pos(s(x)))	→	true	(1)
notZero(neg(s(x)))	→	true	(2)
notZero(neg(0))	→	false	(3)
notZero(pos(0))	→	false	(4)
greaterZero(pos(s(x)))	→	true	(5)
greaterZero(pos(0))	→	false	(6)
greaterZero(neg(x))	→	false	(7)
and(false,false)	→	false	(8)
and(false,true)	→	false	(9)
and(true,false)	→	false	(10)
and(true,true)	→	true	(11)
minusT(0,y)	→	neg(y)	(12)
minusT(x,0)	→	pos(x)	(13)
minusT(s(x),s(y))	→	minusT(x,y)	(14)
plusNat(0,y)	→	y	(15)
plusNat(s(x),y)	→	plusNat(x,s(y))	(16)
negate(pos(x))	→	neg(x)	(17)
negate(neg(x))	→	pos(x)	(18)
minus(pos(x),pos(y))	→	minusT(x,y)	(19)
minus(neg(x),neg(y))	→	negate(minusT(x,y))	(20)
minus(pos(x),neg(y))	→	pos(plusNat(x,y))	(21)
minus(neg(x),pos(y))	→	neg(plusNat(x,y))	(22)
while(true,i,y)	→	while(and(notZero(y),greaterZero(i)),minus(i,y),y)	(23)

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

minusT^#(s(x),s(y))	→	minusT^#(x,y)	(24)
plusNat^#(s(x),y)	→	plusNat^#(x,s(y))	(25)
minus^#(pos(x),pos(y))	→	minusT^#(x,y)	(26)
minus^#(neg(x),neg(y))	→	negate^#(minusT(x,y))	(27)
minus^#(neg(x),neg(y))	→	minusT^#(x,y)	(28)
minus^#(pos(x),neg(y))	→	plusNat^#(x,y)	(29)
minus^#(neg(x),pos(y))	→	plusNat^#(x,y)	(30)
while^#(true,i,y)	→	while^#(and(notZero(y),greaterZero(i)),minus(i,y),y)	(31)
while^#(true,i,y)	→	and^#(notZero(y),greaterZero(i))	(32)
while^#(true,i,y)	→	notZero^#(y)	(33)
while^#(true,i,y)	→	greaterZero^#(i)	(34)
while^#(true,i,y)	→	minus^#(i,y)	(35)

It remains to prove infiniteness of the resulting DP problem.

1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

minusT^#(s(x),s(y))	→	minusT^#(x,y)	(24)
plusNat^#(s(x),y)	→	plusNat^#(x,s(y))	(25)
minus^#(pos(x),pos(y))	→	minusT^#(x,y)	(26)
minus^#(neg(x),neg(y))	→	negate^#(minusT(x,y))	(27)
minus^#(neg(x),neg(y))	→	minusT^#(x,y)	(28)
minus^#(pos(x),neg(y))	→	plusNat^#(x,y)	(29)
minus^#(neg(x),pos(y))	→	plusNat^#(x,y)	(30)
while^#(true,i,y)	→	and^#(notZero(y),greaterZero(i))	(32)
while^#(true,i,y)	→	notZero^#(y)	(33)
while^#(true,i,y)	→	greaterZero^#(i)	(34)
while^#(true,i,y)	→	minus^#(i,y)	(35)

and the following rules have been deleted.

1.1.1 Non-Loop

An infinite (possibly non-looping) derivation has been detected due to the following pattern rule.

while^#(true,pos(s(s(zr1))),neg(s(0))) {zr1/0} ⁿ {zr1/s(zr1)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(0))) {zr1/0} ⁿ {zr1/s(zr1)}
The pattern rule is equivalent to the following pattern rule

while^#(true,pos(s(s(zr1))),neg(s(0))) {zr1/0} ⁿ {zr1/s(zr1)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(zl1))) {zl1/0, zr1/0} ⁿ {zr1/s(zr1)}
The pattern rule is equivalent to the following pattern rule
- while^#(true,pos(s(s(zr1))),neg(s(zl1))) {zr1/0, zl1/0} ⁿ {zr1/s(zr1)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(zl1))) {zl1/0, zr1/0} ⁿ {zr1/s(zr1)}
  The pattern rule is equivalent to the following pattern rule
  - while^#(true,pos(s(s(zr1))),neg(s(zl1))) {zr1/0, zl1/0} ⁿ {zr1/s(zr1)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(zr2))) {zr2/0, zr1/0} ⁿ {zr1/s(zr1)}
    The pattern rule is equivalent to the following pattern rule
    - while^#(true,pos(s(s(zl0))),neg(s(zl1))) {zl0/0, zl1/0} ⁿ {zl0/s(zl0)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(zr2))) {zr2/0, zr1/0} ⁿ {zr1/s(zr1)}
      The pattern rule is equivalent to the following pattern rule
      - while^#(true,pos(s(s(zl0))),neg(s(zl1))) {zl0/0, zl1/0} ⁿ {zl0/s(zl0)} →⁺while^#(true,pos(s(s(s(zr0)))),neg(s(x0))) {x0/0, zr0/0} ⁿ {zr0/s(zr0)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zr0))),neg(s(x0))) {zr0/0, x0/0} ⁿ {zr0/s(zr0)} →⁺while^#(true,pos(s(s(s(zr0)))),neg(s(x0))) {x0/0, zr0/0} ⁿ {zr0/s(zr0)}
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(s(zr0))),neg(s(x0))) {zr0/0, x0/zr0} ⁿ {zr0/s(zr0)} →⁺while^#(true,pos(s(s(s(zr0)))),neg(s(x0))) {x0/zr0} ⁿ {zr0/s(zr0)}
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(s(zr0))),neg(s(x0))) {zr0/0} ⁿ {zr0/s(zr0)} →⁺while^#(true,pos(s(s(s(zr0)))),neg(s(x0))) {zr0/x0} ⁿ {zr0/s(zr0)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl0))),neg(s(x0))) {zl0/0} ⁿ {zl0/s(zl0)} →⁺while^#(true,pos(s(s(s(zr0)))),neg(s(x0))) {zr0/x0} ⁿ {zr0/s(zr0)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl0))),neg(s(x0))) {zl0/0} ⁿ {zl0/s(zl0)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(x0))) {zr1/x0} ⁿ {zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zl1/0} ⁿ {zl1/s(zl1)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(x0))) {zr1/x0} ⁿ {zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zl1/0} ⁿ {zl1/s(zl1)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(x0))) {zr1/x0, zl1/0} ⁿ {zr1/s(zr1), zl1/s(zl1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zr1/x0, zl1/0} ⁿ {zr1/s(zr1), zl1/s(zl1)} →⁺while^#(true,pos(s(s(s(zr1)))),neg(s(x0))) {zr1/x0, zl1/0} ⁿ {zr1/s(zr1), zl1/s(zl1)}
        The pattern rule is obtained by narrowing the following two pattern rules.
        
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zr1/x0, zl1/0} ⁿ {zr1/s(zr1), zl1/s(zl1)} →⁺while^#(true,pos(plusNat(0,s(s(s(zr1))))),neg(s(x0))) {zl1/0, zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zr1/x0, zl1/0} ⁿ {zr1/s(zr1), zl1/s(zl1)} →⁺while^#(true,pos(plusNat(0,s(s(s(zr1))))),neg(s(x0))) {zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zl1/0} ⁿ {zr1/s(zr1), zl1/s(zl1)} →⁺while^#(true,pos(plusNat(0,s(s(s(zr1))))),neg(s(x0))) {zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zl1/0} ⁿ {zr1/s(zr1), zl1/s(zl1)} →⁺while^#(true,pos(plusNat(0,s(s(s(zr1))))),neg(s(x0))) {zr1/x0} ⁿ {zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zl1/0} ⁿ {zl1/s(zl1)} →⁺while^#(true,pos(plusNat(0,s(s(s(zr1))))),neg(s(x0))) {zr1/x0} ⁿ {zr1/s(zr1)}
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zl1/x1} ⁿ {zl1/s(zl1)} →⁺while^#(true,pos(plusNat(x1,s(s(s(zr1))))),neg(s(x0))) {zr1/x0} ⁿ {zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl1))),neg(s(x0))) {zl1/x1} ⁿ {zl1/s(zl1)} →⁺while^#(true,pos(plusNat(x1,s(s(s(zr0))))),neg(s(x0))) {zr0/x0} ⁿ {zr0/s(zr0)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl0))),neg(s(x0))) {zl0/x1} ⁿ {zl0/s(zl0)} →⁺while^#(true,pos(plusNat(x1,s(s(s(zr0))))),neg(s(x0))) {zr0/x0} ⁿ {zr0/s(zr0)}
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(s(zl0))),neg(s(x0))) {zl0/y1} ⁿ {zl0/s(zl0)} →⁺while^#(true,pos(plusNat(y1,s(s(s(zr0))))),neg(s(x0))) {zr0/x0} ⁿ {zr0/s(zr0)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zl0))),neg(s(x0))) {zl0/y1} ⁿ {zl0/s(zl0)} →⁺while^#(true,pos(plusNat(y1,s(s(s(zt1))))),neg(s(x0))) {zt1/x0} ⁿ {zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zs1))),neg(s(x0))) {zs1/y1} ⁿ {zs1/s(zs1)} →⁺while^#(true,pos(plusNat(y1,s(s(s(zt1))))),neg(s(x0))) {zt1/x0} ⁿ {zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zs1))),neg(s(x0))) {zs1/y1} ⁿ {zs1/s(zs1)} →⁺while^#(true,pos(plusNat(y1,s(s(s(zt1))))),neg(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zs1))),neg(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺while^#(true,pos(plusNat(y1,s(s(s(zt1))))),neg(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)}
        The pattern rule is obtained by narrowing the following two pattern rules.
        
        while^#(true,pos(s(s(zs1))),neg(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺while^#(true,pos(plusNat(s(zs1),s(s(x0)))),neg(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zs1))),neg(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺while^#(true,pos(plusNat(s(zs1),s(s(x0)))),neg(s(x0))) {zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zs1))),neg(s(x0))) {zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺while^#(true,pos(plusNat(s(zs1),s(s(x0)))),neg(s(x0))) {zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)}
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(s(zs1))),neg(s(x0)))∅ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺while^#(true,pos(plusNat(s(zs1),s(s(x0)))),neg(s(x0)))∅ⁿ {zt1/s(zt1), zs1/s(zs1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zs1))),neg(s(x0)))∅ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺while^#(true,pos(plusNat(s(zs1),s(s(x0)))),neg(s(x0)))∅ⁿ {zs1/s(zs1)}
        The pattern rule is equivalent to the following pattern rule
        while^#(true,pos(s(s(zs1))),neg(s(x0)))∅ⁿ {zs1/s(zs1)} →⁺while^#(true,pos(plusNat(s(zs1),s(s(x0)))),neg(s(x0)))∅ⁿ {zs1/s(zs1)}
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(s(zs1))),neg(s(x0)))∅ⁿ∅→⁺while^#(true,pos(plusNat(s(zs1),s(s(x0)))),neg(s(x0)))∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(x1)),neg(s(x0)))∅ⁿ∅→⁺while^#(true,pos(plusNat(x1,s(s(x0)))),neg(s(x0)))∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,pos(s(y0)),neg(s(x1)))∅ⁿ∅→⁺while^#(true,pos(plusNat(y0,s(s(x1)))),neg(s(x1)))∅ⁿ∅
        The pattern rule is obtained by rewriting the following pattern rule.
        while^#(true,pos(s(y0)),neg(s(x1)))∅ⁿ∅→⁺while^#(and(true,true),minus(pos(s(y0)),neg(s(x1))),neg(s(x1)))∅ⁿ∅
        The pattern rule is obtained by narrowing the following two pattern rules.
        
        while^#(true,pos(s(y0)),neg(s(x1)))∅ⁿ∅→⁺while^#(and(true,greaterZero(pos(s(y0)))),minus(pos(s(y0)),neg(s(x1))),neg(s(x1)))∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,x0,neg(s(x1)))∅ⁿ∅→⁺while^#(and(true,greaterZero(x0)),minus(x0,neg(s(x1))),neg(s(x1)))∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,x0,neg(s(y0)))∅ⁿ∅→⁺while^#(and(true,greaterZero(x0)),minus(x0,neg(s(y0))),neg(s(y0)))∅ⁿ∅
        The pattern rule is obtained by narrowing the following two pattern rules.
        
        while^#(true,x0,neg(s(y0)))∅ⁿ∅→⁺while^#(and(notZero(neg(s(y0))),greaterZero(x0)),minus(x0,neg(s(y0))),neg(s(y0)))∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,x0,x1)∅ⁿ∅→⁺while^#(and(notZero(x1),greaterZero(x0)),minus(x0,x1),x1)∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        while^#(true,i,y)∅ⁿ∅→⁺while^#(and(notZero(y),greaterZero(i)),minus(i,y),y)∅ⁿ∅
        The pattern rule is obtained from the original rule
        
        notZero(neg(s(y0)))∅ⁿ∅→⁺true∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        notZero(neg(s(x)))∅ⁿ∅→⁺true∅ⁿ∅
        The pattern rule is obtained from the original rule
        
        greaterZero(pos(s(y0)))∅ⁿ∅→⁺true∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        greaterZero(pos(s(x)))∅ⁿ∅→⁺true∅ⁿ∅
        The pattern rule is obtained from the original rule
        
        plusNat(s(zs1),s(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺plusNat(y1,s(s(s(zt1)))) {zs1/y1, zt1/x0} ⁿ {zs1/s(zs1), zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs1),s(s(x0))) {zt1/x0, zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺plusNat(y1,s(s(s(zt1)))) {zt1/x0} ⁿ {zs1/s(zs1), zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs1),s(s(x0))) {zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺plusNat(y1,s(s(s(zt1)))) {zt1/x0} ⁿ {zs1/s(zs1), zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs1),s(s(x0))) {zs1/y1} ⁿ {zt1/s(zt1), zs1/s(zs1)} →⁺plusNat(y1,s(s(s(zt1)))) {zt1/x0} ⁿ {zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs1),s(s(x0))) {zs1/y1} ⁿ {zs1/s(zs1)} →⁺plusNat(y1,s(s(s(zt1)))) {zt1/x0} ⁿ {zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs1),s(s(x0))) {zs1/y1} ⁿ {zs1/s(zs1)} →⁺plusNat(y1,s(zt1)) {zt1/s(s(x0))} ⁿ {zt1/s(zt1)}
        The pattern rule is obtained by instantiating the following pattern rule.
        plusNat(s(zs1),y0) {zs1/y1} ⁿ {zs1/s(zs1)} →⁺plusNat(y1,s(zt1)) {zt1/y0} ⁿ {zt1/s(zt1)}
        The pattern rule is obtained by instantiating the following pattern rule.
        plusNat(s(zs1),y1) {zs1/y0} ⁿ {zs1/s(zs1)} →⁺plusNat(y0,s(zt1)) {zt1/y1} ⁿ {zt1/s(zt1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs1),y1) {zs1/y0} ⁿ {zs1/s(zs1)} →⁺plusNat(y0,s(zt0)) {zt0/y1} ⁿ {zt0/s(zt0)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs0),y1) {zs0/y0} ⁿ {zs0/s(zs0)} →⁺plusNat(y0,s(zt0)) {zt0/y1} ⁿ {zt0/s(zt0)}
        The pattern rule is obtained by instantiating the following pattern rule.
        plusNat(s(zs0),y) {zs0/x} ⁿ {zs0/s(zs0)} →⁺plusNat(x,s(zt0)) {zt0/y} ⁿ {zt0/s(zt0)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(zs0),y) {zs0/x} ⁿ {zs0/s(zs0)} →⁺plusNat(x,s(y))∅ⁿ {y/s(y)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(s(x),y)∅ⁿ {x/s(x)} →⁺plusNat(x,s(y))∅ⁿ {y/s(y)}
        The pattern rule is obtained by adding an initial pumping substitution from
        plusNat(s(x),y)∅ⁿ∅→⁺plusNat(x,s(y))∅ⁿ∅
        The pattern rule is obtained from the original rule
        
        plusNat(0,s(s(s(zr1)))) {zl1/0, zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)} →⁺s(s(s(zr1))) {zl1/0, zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(0,s(s(s(zr1)))) {zl1/0, zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)} →⁺s(s(s(zr1))) {zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(0,s(s(s(zr1)))) {zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)} →⁺s(s(s(zr1))) {zr1/x0} ⁿ {zl1/s(zl1), zr1/s(zr1)}
        The pattern rule is obtained by instantiating the following pattern rule.
        plusNat(0,s(s(s(zr1))))∅ⁿ {zl1/s(zl1), zr1/s(zr1)} →⁺s(s(s(zr1)))∅ⁿ {zl1/s(zl1), zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(0,s(s(s(zr1))))∅ⁿ {zl1/s(zl1), zr1/s(zr1)} →⁺s(s(s(zr1)))∅ⁿ {zr1/s(zr1)}
        The pattern rule is equivalent to the following pattern rule
        plusNat(0,s(s(s(zr1))))∅ⁿ {zr1/s(zr1)} →⁺s(s(s(zr1)))∅ⁿ {zr1/s(zr1)}
        The pattern rule is obtained by instantiating the following pattern rule.
        plusNat(0,s(s(s(zr1))))∅ⁿ∅→⁺s(s(s(zr1)))∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        plusNat(0,y0)∅ⁿ∅→⁺y0∅ⁿ∅
        The pattern rule is obtained by instantiating the following pattern rule.
        plusNat(0,y)∅ⁿ∅→⁺y∅ⁿ∅
        The pattern rule is obtained from the original rule