YES

We show the termination of the TRS R:

  a__U11(tt(),V2) -> a__U12(a__isNat(V2))
  a__U12(tt()) -> tt()
  a__U21(tt()) -> tt()
  a__U31(tt(),N) -> mark(N)
  a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
  a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
  a__isNat(|0|()) -> tt()
  a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
  a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
  a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
  mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
  mark(U12(X)) -> a__U12(mark(X))
  mark(isNat(X)) -> a__isNat(X)
  mark(U21(X)) -> a__U21(mark(X))
  mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
  mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
  mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
  mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
  mark(tt()) -> tt()
  mark(s(X)) -> s(mark(X))
  mark(|0|()) -> |0|()
  a__U11(X1,X2) -> U11(X1,X2)
  a__U12(X) -> U12(X)
  a__isNat(X) -> isNat(X)
  a__U21(X) -> U21(X)
  a__U31(X1,X2) -> U31(X1,X2)
  a__U41(X1,X2,X3) -> U41(X1,X2,X3)
  a__U42(X1,X2,X3) -> U42(X1,X2,X3)
  a__plus(X1,X2) -> plus(X1,X2)

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: a__U11#(tt(),V2) -> a__U12#(a__isNat(V2))
p2: a__U11#(tt(),V2) -> a__isNat#(V2)
p3: a__U31#(tt(),N) -> mark#(N)
p4: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N)
p5: a__U41#(tt(),M,N) -> a__isNat#(N)
p6: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p7: a__U42#(tt(),M,N) -> mark#(N)
p8: a__U42#(tt(),M,N) -> mark#(M)
p9: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2)
p10: a__isNat#(plus(V1,V2)) -> a__isNat#(V1)
p11: a__isNat#(s(V1)) -> a__U21#(a__isNat(V1))
p12: a__isNat#(s(V1)) -> a__isNat#(V1)
p13: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p14: a__plus#(N,|0|()) -> a__isNat#(N)
p15: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N)
p16: a__plus#(N,s(M)) -> a__isNat#(M)
p17: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)
p18: mark#(U11(X1,X2)) -> mark#(X1)
p19: mark#(U12(X)) -> a__U12#(mark(X))
p20: mark#(U12(X)) -> mark#(X)
p21: mark#(isNat(X)) -> a__isNat#(X)
p22: mark#(U21(X)) -> a__U21#(mark(X))
p23: mark#(U21(X)) -> mark#(X)
p24: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p27: mark#(U41(X1,X2,X3)) -> mark#(X1)
p28: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p29: mark#(U42(X1,X2,X3)) -> mark#(X1)
p30: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p31: mark#(plus(X1,X2)) -> mark#(X1)
p32: mark#(plus(X1,X2)) -> mark#(X2)
p33: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p3, p4, p6, p7, p8, p13, p15, p18, p20, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33}
  {p2, p9, p10, p12}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: a__U42#(tt(),M,N) -> mark#(M)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p6: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N)
p7: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N)
p8: a__U42#(tt(),M,N) -> mark#(N)
p9: mark#(U42(X1,X2,X3)) -> mark#(X1)
p10: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p11: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p12: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p13: a__U31#(tt(),N) -> mark#(N)
p14: mark#(U41(X1,X2,X3)) -> mark#(X1)
p15: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p16: mark#(U31(X1,X2)) -> mark#(X1)
p17: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p18: mark#(U21(X)) -> mark#(X)
p19: mark#(U12(X)) -> mark#(X)
p20: mark#(U11(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__U42#_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (5,1)
          tt_A() = (1,12)
          mark#_A(x1) = ((0,1),(0,1)) x1 + (3,1)
          s_A(x1) = ((0,0),(0,1)) x1 + (25,23)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (17,11)
          a__plus#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (4,1)
          mark_A(x1) = x1 + (7,0)
          a__U41#_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (13,0)
          a__isNat_A(x1) = (26,12)
          U42_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (6,22)
          |0|_A() = (12,10)
          a__U31#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + (1,1)
          U41_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (13,22)
          U31_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (4,9)
          U21_A(x1) = ((0,0),(0,1)) x1 + (23,0)
          U12_A(x1) = ((0,0),(0,1)) x1 + (19,0)
          U11_A(x1,x2) = ((0,0),(0,1)) x1 + (20,0)
          a__U11_A(x1,x2) = ((0,0),(0,1)) x1 + (26,0)
          a__U12_A(x1) = ((0,0),(0,1)) x1 + (26,0)
          a__U21_A(x1) = ((0,0),(0,1)) x1 + (24,0)
          a__U31_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (11,9)
          a__U41_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (13,22)
          a__U42_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (13,22)
          a__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (17,11)
          isNat_A(x1) = (20,12)
  
    precedence:
    
      mark = a__isNat = |0| = a__U11 = a__U21 = a__U31 = a__plus > plus > U31 > tt = mark# = a__plus# = a__U31# = U12 = a__U12 > U21 > a__U42# = s = a__U41# = U42 = U41 = U11 = a__U41 = a__U42 = isNat
    
    partial status:
  
      pi(a__U42#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(plus) = [1]
      pi(a__plus#) = []
      pi(mark) = []
      pi(a__U41#) = []
      pi(a__isNat) = []
      pi(U42) = []
      pi(|0|) = []
      pi(a__U31#) = []
      pi(U41) = []
      pi(U31) = []
      pi(U21) = []
      pi(U12) = []
      pi(U11) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U21) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__U42) = []
      pi(a__plus) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p15

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: a__U42#(tt(),M,N) -> mark#(M)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p6: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N)
p7: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N)
p8: a__U42#(tt(),M,N) -> mark#(N)
p9: mark#(U42(X1,X2,X3)) -> mark#(X1)
p10: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p11: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p12: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p13: a__U31#(tt(),N) -> mark#(N)
p14: mark#(U41(X1,X2,X3)) -> mark#(X1)
p15: mark#(U31(X1,X2)) -> mark#(X1)
p16: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p17: mark#(U21(X)) -> mark#(X)
p18: mark#(U12(X)) -> mark#(X)
p19: mark#(U11(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: a__U42#(tt(),M,N) -> mark#(M)
p2: mark#(U11(X1,X2)) -> mark#(X1)
p3: mark#(U12(X)) -> mark#(X)
p4: mark#(U21(X)) -> mark#(X)
p5: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p6: a__U31#(tt(),N) -> mark#(N)
p7: mark#(U31(X1,X2)) -> mark#(X1)
p8: mark#(U41(X1,X2,X3)) -> mark#(X1)
p9: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p11: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p12: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N)
p13: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N)
p14: a__U42#(tt(),M,N) -> mark#(N)
p15: mark#(U42(X1,X2,X3)) -> mark#(X1)
p16: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p17: mark#(plus(X1,X2)) -> mark#(X1)
p18: mark#(plus(X1,X2)) -> mark#(X2)
p19: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__U42#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (3,11)
          tt_A() = (14,10)
          mark#_A(x1) = ((0,1),(0,0)) x1 + (1,11)
          U11_A(x1,x2) = ((0,1),(0,1)) x1 + (6,0)
          U12_A(x1) = ((0,0),(0,1)) x1 + (15,0)
          U21_A(x1) = ((0,1),(0,1)) x1 + (4,0)
          U31_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (28,28)
          a__U31#_A(x1,x2) = ((0,1),(0,0)) x2 + (29,11)
          mark_A(x1) = x1
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(1,1)) x2 + x3 + (25,11)
          U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(1,1)) x2 + x3 + (24,11)
          a__plus#_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (13,11)
          |0|_A() = (15,11)
          a__isNat_A(x1) = ((0,1),(0,0)) x1 + (5,10)
          s_A(x1) = x1 + (6,9)
          a__U41#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (4,11)
          plus_A(x1,x2) = x1 + ((0,1),(1,1)) x2 + (17,12)
          a__U11_A(x1,x2) = ((0,1),(0,1)) x1 + (6,0)
          a__U12_A(x1) = ((0,0),(0,1)) x1 + (15,0)
          a__U21_A(x1) = ((0,1),(0,1)) x1 + (4,0)
          a__U31_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (28,28)
          a__U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(1,1)) x2 + x3 + (25,11)
          a__U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(1,1)) x2 + x3 + (24,11)
          a__plus_A(x1,x2) = x1 + ((0,1),(1,1)) x2 + (17,12)
          isNat_A(x1) = ((0,1),(0,0)) x1 + (5,10)
  
    precedence:
    
      a__isNat = isNat > U21 = a__U21 > U11 = a__U11 > U12 = U31 = mark = U41 = a__U12 = a__U31 = a__U41 = a__U42 = a__plus > tt > plus > s > a__U42# = mark# = a__U31# = a__plus# = a__U41# > |0| > U42
    
    partial status:
  
      pi(a__U42#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(U21) = []
      pi(U31) = []
      pi(a__U31#) = []
      pi(mark) = [1]
      pi(U41) = [3]
      pi(U42) = []
      pi(a__plus#) = []
      pi(|0|) = []
      pi(a__isNat) = []
      pi(s) = [1]
      pi(a__U41#) = []
      pi(plus) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U21) = []
      pi(a__U31) = [2]
      pi(a__U41) = [3]
      pi(a__U42) = []
      pi(a__plus) = [1]
      pi(isNat) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p5: a__U31#(tt(),N) -> mark#(N)
p6: mark#(U31(X1,X2)) -> mark#(X1)
p7: mark#(U41(X1,X2,X3)) -> mark#(X1)
p8: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p9: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p10: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p11: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N)
p12: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N)
p13: a__U42#(tt(),M,N) -> mark#(N)
p14: mark#(U42(X1,X2,X3)) -> mark#(X1)
p15: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p16: mark#(plus(X1,X2)) -> mark#(X1)
p17: mark#(plus(X1,X2)) -> mark#(X2)
p18: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p6: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N)
p7: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N)
p8: a__U42#(tt(),M,N) -> mark#(N)
p9: mark#(U42(X1,X2,X3)) -> mark#(X1)
p10: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p11: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p12: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p13: a__U31#(tt(),N) -> mark#(N)
p14: mark#(U41(X1,X2,X3)) -> mark#(X1)
p15: mark#(U31(X1,X2)) -> mark#(X1)
p16: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p17: mark#(U21(X)) -> mark#(X)
p18: mark#(U12(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(1,0)) x1 + (6,7)
          U11_A(x1,x2) = x1
          s_A(x1) = x1 + (12,12)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (10,10)
          a__plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (12,1)
          mark_A(x1) = ((0,1),(1,0)) x1
          a__U41#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (36,12)
          a__isNat_A(x1) = (11,11)
          tt_A() = (11,11)
          a__U42#_A(x1,x2,x3) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,12)
          U42_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (5,5)
          |0|_A() = (10,10)
          a__U31#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + (8,7)
          U41_A(x1,x2,x3) = x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (17,17)
          U31_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (7,7)
          U21_A(x1) = x1
          U12_A(x1) = x1
          a__U11_A(x1,x2) = x1
          a__U12_A(x1) = x1
          a__U21_A(x1) = x1
          a__U31_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (7,7)
          a__U41_A(x1,x2,x3) = x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (17,17)
          a__U42_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (5,5)
          a__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (10,10)
          isNat_A(x1) = (11,11)
  
    precedence:
    
      mark# > a__plus# = mark = a__U41# = a__isNat = tt = a__U11 = a__U12 = a__U21 = a__U31 = a__U41 = a__U42 = a__plus > U11 = a__U42# = |0| > a__U31# > plus = U41 > U31 > U12 > s > U42 = U21 > isNat
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(s) = [1]
      pi(plus) = []
      pi(a__plus#) = [2]
      pi(mark) = []
      pi(a__U41#) = [2]
      pi(a__isNat) = []
      pi(tt) = []
      pi(a__U42#) = []
      pi(U42) = []
      pi(|0|) = []
      pi(a__U31#) = []
      pi(U41) = []
      pi(U31) = []
      pi(U21) = []
      pi(U12) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U21) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__U42) = []
      pi(a__plus) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p6: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N)
p7: a__U42#(tt(),M,N) -> mark#(N)
p8: mark#(U42(X1,X2,X3)) -> mark#(X1)
p9: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p11: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p12: a__U31#(tt(),N) -> mark#(N)
p13: mark#(U41(X1,X2,X3)) -> mark#(X1)
p14: mark#(U31(X1,X2)) -> mark#(X1)
p15: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p16: mark#(U21(X)) -> mark#(X)
p17: mark#(U12(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p5: a__U31#(tt(),N) -> mark#(N)
p6: mark#(U31(X1,X2)) -> mark#(X1)
p7: mark#(U41(X1,X2,X3)) -> mark#(X1)
p8: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p9: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p10: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p11: a__U42#(tt(),M,N) -> mark#(N)
p12: mark#(U42(X1,X2,X3)) -> mark#(X1)
p13: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p14: mark#(plus(X1,X2)) -> mark#(X1)
p15: mark#(plus(X1,X2)) -> mark#(X2)
p16: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (2,5)
          U11_A(x1,x2) = x1
          U12_A(x1) = x1
          U21_A(x1) = x1
          U31_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (7,7)
          a__U31#_A(x1,x2) = ((1,1),(1,0)) x2 + (3,5)
          mark_A(x1) = ((0,1),(1,0)) x1
          tt_A() = (0,0)
          U41_A(x1,x2,x3) = x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (5,5)
          U42_A(x1,x2,x3) = x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (5,5)
          a__U42#_A(x1,x2,x3) = ((0,1),(1,0)) x2 + ((1,1),(1,1)) x3 + (6,6)
          a__plus#_A(x1,x2) = ((1,1),(1,0)) x1 + (4,5)
          |0|_A() = (1,1)
          a__isNat_A(x1) = (0,0)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (5,5)
          s_A(x1) = x1
          a__U11_A(x1,x2) = x1
          a__U12_A(x1) = x1
          a__U21_A(x1) = x1
          a__U31_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (7,7)
          a__U41_A(x1,x2,x3) = x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (5,5)
          a__U42_A(x1,x2,x3) = x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (5,5)
          a__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (5,5)
          isNat_A(x1) = (0,0)
  
    precedence:
    
      mark# = a__U31# > a__U42# > mark = |0| > a__plus# > plus = a__plus > U11 = U12 = U31 = tt = U41 = U42 = a__isNat = a__U11 = a__U12 = a__U21 = a__U31 = a__U41 = a__U42 = isNat > U21 = s
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(U21) = []
      pi(U31) = [1]
      pi(a__U31#) = []
      pi(mark) = []
      pi(tt) = []
      pi(U41) = [1]
      pi(U42) = []
      pi(a__U42#) = []
      pi(a__plus#) = []
      pi(|0|) = []
      pi(a__isNat) = []
      pi(plus) = []
      pi(s) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U21) = []
      pi(a__U31) = [1]
      pi(a__U41) = [1]
      pi(a__U42) = [1]
      pi(a__plus) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p13

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p5: a__U31#(tt(),N) -> mark#(N)
p6: mark#(U31(X1,X2)) -> mark#(X1)
p7: mark#(U41(X1,X2,X3)) -> mark#(X1)
p8: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p9: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p10: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p11: a__U42#(tt(),M,N) -> mark#(N)
p12: mark#(U42(X1,X2,X3)) -> mark#(X1)
p13: mark#(plus(X1,X2)) -> mark#(X1)
p14: mark#(plus(X1,X2)) -> mark#(X2)
p15: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> mark#(X1)
p6: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p7: a__U42#(tt(),M,N) -> mark#(N)
p8: mark#(U41(X1,X2,X3)) -> mark#(X1)
p9: mark#(U31(X1,X2)) -> mark#(X1)
p10: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p11: a__U31#(tt(),N) -> mark#(N)
p12: mark#(U21(X)) -> mark#(X)
p13: mark#(U12(X)) -> mark#(X)
p14: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p15: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (10,4)
          U11_A(x1,x2) = x1 + (0,5)
          s_A(x1) = ((1,0),(0,0)) x1 + (1,3)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,0),(0,0)) x2 + (12,5)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (2,4)
          a__U42#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,1),(1,1)) x3 + (10,5)
          mark_A(x1) = x1
          tt_A() = (11,9)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (2,5)
          U31_A(x1,x2) = ((1,1),(0,0)) x1 + x2 + (1,5)
          a__U31#_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,0),(1,0)) x2 + (2,0)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,9)
          U12_A(x1) = ((1,0),(0,0)) x1 + (0,9)
          a__plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,0),(1,0)) x2 + (9,0)
          |0|_A() = (18,5)
          a__isNat_A(x1) = ((0,0),(0,1)) x1 + (11,6)
          a__U11_A(x1,x2) = x1 + (0,5)
          a__U12_A(x1) = ((1,0),(0,0)) x1 + (0,9)
          a__U21_A(x1) = ((1,0),(0,0)) x1 + (0,9)
          a__U31_A(x1,x2) = ((1,1),(0,0)) x1 + x2 + (1,5)
          a__U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (2,5)
          a__U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (2,4)
          a__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,0),(0,0)) x2 + (12,5)
          isNat_A(x1) = ((0,0),(0,1)) x1 + (11,6)
  
    precedence:
    
      mark# = a__U42# > a__plus# > a__U31# > U11 = s = plus = U42 = mark = tt = U41 = U31 = U21 = U12 = |0| = a__isNat = a__U11 = a__U12 = a__U21 = a__U31 = a__U41 = a__U42 = a__plus = isNat
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(s) = []
      pi(plus) = []
      pi(U42) = []
      pi(a__U42#) = [3]
      pi(mark) = []
      pi(tt) = []
      pi(U41) = []
      pi(U31) = []
      pi(a__U31#) = []
      pi(U21) = []
      pi(U12) = []
      pi(a__plus#) = [1]
      pi(|0|) = []
      pi(a__isNat) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U21) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__U42) = []
      pi(a__plus) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> mark#(X1)
p6: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p7: a__U42#(tt(),M,N) -> mark#(N)
p8: mark#(U41(X1,X2,X3)) -> mark#(X1)
p9: mark#(U31(X1,X2)) -> mark#(X1)
p10: a__U31#(tt(),N) -> mark#(N)
p11: mark#(U21(X)) -> mark#(X)
p12: mark#(U12(X)) -> mark#(X)
p13: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p14: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U31(X1,X2)) -> mark#(X1)
p5: mark#(U41(X1,X2,X3)) -> mark#(X1)
p6: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p7: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p8: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p9: a__U31#(tt(),N) -> mark#(N)
p10: mark#(U42(X1,X2,X3)) -> mark#(X1)
p11: mark#(plus(X1,X2)) -> mark#(X1)
p12: mark#(plus(X1,X2)) -> mark#(X2)
p13: mark#(s(X)) -> mark#(X)
p14: a__U42#(tt(),M,N) -> mark#(N)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (5,7)
          U11_A(x1,x2) = ((1,0),(1,0)) x1 + (0,2)
          U12_A(x1) = ((1,0),(0,0)) x1 + (0,6)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,5)
          U31_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (6,2)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + x3 + (4,2)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + x3 + (4,2)
          a__U42#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (5,7)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,2)
          tt_A() = (8,6)
          a__plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,7)
          |0|_A() = (7,1)
          a__U31#_A(x1,x2) = ((1,0),(1,0)) x2 + (6,7)
          a__isNat_A(x1) = ((0,0),(0,1)) x1 + (8,12)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (9,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (3,6)
          a__U11_A(x1,x2) = ((1,0),(1,0)) x1 + (0,4)
          a__U12_A(x1) = ((1,0),(0,0)) x1 + (0,6)
          a__U21_A(x1) = ((1,0),(0,0)) x1 + (0,7)
          a__U31_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (6,3)
          a__U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + x3 + (4,7)
          a__U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + x3 + (4,7)
          a__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (9,4)
          isNat_A(x1) = ((0,0),(0,1)) x1 + (8,2)
  
    precedence:
    
      mark# = U31 = a__U42# = mark = tt = a__isNat = a__U11 = a__U12 = a__U21 = a__U31 = a__plus > U21 = U42 = a__U41 = a__U42 > U11 = U12 = U41 = |0| = plus = s = isNat > a__plus# = a__U31#
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(U42) = []
      pi(a__U42#) = []
      pi(mark) = []
      pi(tt) = []
      pi(a__plus#) = []
      pi(|0|) = []
      pi(a__U31#) = []
      pi(a__isNat) = []
      pi(plus) = []
      pi(s) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U21) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__U42) = []
      pi(a__plus) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p9

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U31(X1,X2)) -> mark#(X1)
p5: mark#(U41(X1,X2,X3)) -> mark#(X1)
p6: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p7: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p8: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N)
p9: mark#(U42(X1,X2,X3)) -> mark#(X1)
p10: mark#(plus(X1,X2)) -> mark#(X1)
p11: mark#(plus(X1,X2)) -> mark#(X2)
p12: mark#(s(X)) -> mark#(X)
p13: a__U42#(tt(),M,N) -> mark#(N)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p9, p10, p11, p12, p13}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> mark#(X1)
p6: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p7: a__U42#(tt(),M,N) -> mark#(N)
p8: mark#(U41(X1,X2,X3)) -> mark#(X1)
p9: mark#(U31(X1,X2)) -> mark#(X1)
p10: mark#(U21(X)) -> mark#(X)
p11: mark#(U12(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (0,8)
          U11_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,1),(0,0)) x2
          s_A(x1) = ((0,0),(0,1)) x1 + (5,6)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (0,12)
          U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (5,7)
          a__U42#_A(x1,x2,x3) = ((0,1),(0,0)) x3 + (6,8)
          mark_A(x1) = ((1,1),(0,1)) x1 + (8,0)
          tt_A() = (8,11)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (1,7)
          U31_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (8,8)
          U21_A(x1) = ((0,1),(0,1)) x1
          U12_A(x1) = ((0,0),(0,1)) x1
          a__U11_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,1),(0,0)) x2 + (8,0)
          a__U12_A(x1) = ((0,0),(0,1)) x1 + (8,0)
          a__isNat_A(x1) = ((1,0),(0,0)) x1 + (8,11)
          a__U21_A(x1) = ((0,1),(0,1)) x1 + (1,0)
          a__U31_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(0,1)) x2 + (24,8)
          a__U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (16,7)
          a__U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (6,7)
          a__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (11,12)
          |0|_A() = (1,14)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (1,11)
  
    precedence:
    
      |0| > plus = mark = U41 = U31 = a__U11 = a__isNat = a__U21 = a__U31 = a__U41 = a__U42 = a__plus > s > U12 = a__U12 > tt > isNat > mark# = U11 = U42 = a__U42# = U21
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(s) = []
      pi(plus) = []
      pi(U42) = []
      pi(a__U42#) = []
      pi(mark) = []
      pi(tt) = []
      pi(U41) = []
      pi(U31) = []
      pi(U21) = []
      pi(U12) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__isNat) = []
      pi(a__U21) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__U42) = []
      pi(a__plus) = []
      pi(|0|) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p9

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> mark#(X1)
p6: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p7: a__U42#(tt(),M,N) -> mark#(N)
p8: mark#(U41(X1,X2,X3)) -> mark#(X1)
p9: mark#(U21(X)) -> mark#(X)
p10: mark#(U12(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2,X3)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p6: a__U42#(tt(),M,N) -> mark#(N)
p7: mark#(U42(X1,X2,X3)) -> mark#(X1)
p8: mark#(plus(X1,X2)) -> mark#(X1)
p9: mark#(plus(X1,X2)) -> mark#(X2)
p10: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (0,2)
          U11_A(x1,x2) = ((0,0),(0,1)) x1 + (9,0)
          U12_A(x1) = ((0,0),(0,1)) x1 + (9,0)
          U21_A(x1) = ((0,0),(0,1)) x1
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (3,1)
          U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (1,1)
          a__U42#_A(x1,x2,x3) = ((0,1),(0,0)) x3 + (0,2)
          mark_A(x1) = x1 + (4,0)
          tt_A() = (0,5)
          plus_A(x1,x2) = x1 + ((0,0),(0,1)) x2 + (6,1)
          s_A(x1) = ((0,0),(0,1)) x1 + (1,5)
          a__U11_A(x1,x2) = ((0,0),(0,1)) x1 + (13,0)
          a__U12_A(x1) = ((0,0),(0,1)) x1 + (13,0)
          a__isNat_A(x1) = ((1,0),(0,0)) x1 + (7,5)
          a__U21_A(x1) = ((0,0),(0,1)) x1
          a__U31_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (0,1)
          a__U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (3,1)
          a__U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (2,1)
          a__plus_A(x1,x2) = x1 + ((0,0),(0,1)) x2 + (6,1)
          |0|_A() = (0,0)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (3,5)
          U31_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (0,1)
  
    precedence:
    
      |0| > a__U42# > mark# > a__U31 = U31 > mark = tt = a__U11 = a__U12 = a__isNat = a__U21 = a__plus = isNat > plus > U41 = a__U41 > U42 = s = a__U42 > U11 > U12 = U21
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(U21) = []
      pi(U41) = []
      pi(U42) = []
      pi(a__U42#) = []
      pi(mark) = [1]
      pi(tt) = []
      pi(plus) = []
      pi(s) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__isNat) = []
      pi(a__U21) = []
      pi(a__U31) = [2]
      pi(a__U41) = []
      pi(a__U42) = []
      pi(a__plus) = [1]
      pi(|0|) = []
      pi(isNat) = []
      pi(U31) = [2]

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2,X3)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p6: mark#(U42(X1,X2,X3)) -> mark#(X1)
p7: mark#(plus(X1,X2)) -> mark#(X1)
p8: mark#(plus(X1,X2)) -> mark#(X2)
p9: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p6, p7, p8, p9}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> mark#(X1)
p6: mark#(U41(X1,X2,X3)) -> mark#(X1)
p7: mark#(U21(X)) -> mark#(X)
p8: mark#(U12(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (0,2)
          U11_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
          s_A(x1) = ((1,1),(1,1)) x1 + (1,2)
          plus_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (0,2)
          U42_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
          U21_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          U12_A(x1) = ((0,0),(0,1)) x1 + (1,1)
  
    precedence:
    
      mark# = U11 = s > plus = U41 = U21 = U12 > U42
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(s) = []
      pi(plus) = [2]
      pi(U42) = []
      pi(U41) = [3]
      pi(U21) = [1]
      pi(U12) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U42(X1,X2,X3)) -> mark#(X1)
p5: mark#(U41(X1,X2,X3)) -> mark#(X1)
p6: mark#(U21(X)) -> mark#(X)
p7: mark#(U12(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2,X3)) -> mark#(X1)
p5: mark#(U42(X1,X2,X3)) -> mark#(X1)
p6: mark#(plus(X1,X2)) -> mark#(X1)
p7: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          U11_A(x1,x2) = ((0,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (3,2)
          U12_A(x1) = ((1,1),(1,1)) x1 + (3,2)
          U21_A(x1) = x1 + (1,1)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
          U42_A(x1,x2,x3) = x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
          plus_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (3,1)
          s_A(x1) = ((0,0),(0,1)) x1 + (1,1)
  
    precedence:
    
      plus > U41 = s > U21 = U42 > mark# = U11 = U12
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = [1]
      pi(U21) = [1]
      pi(U41) = []
      pi(U42) = []
      pi(plus) = []
      pi(s) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U41(X1,X2,X3)) -> mark#(X1)
p4: mark#(U42(X1,X2,X3)) -> mark#(X1)
p5: mark#(plus(X1,X2)) -> mark#(X1)
p6: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U42(X1,X2,X3)) -> mark#(X1)
p5: mark#(U41(X1,X2,X3)) -> mark#(X1)
p6: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          U11_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (3,1)
          s_A(x1) = ((1,0),(1,1)) x1 + (3,2)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,1),(1,1)) x2 + (1,1)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
          U41_A(x1,x2,x3) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
          U21_A(x1) = ((1,0),(1,0)) x1 + (3,1)
  
    precedence:
    
      mark# = U42 > U11 = U21 > s = plus = U41
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = [2]
      pi(s) = [1]
      pi(plus) = [2]
      pi(U42) = []
      pi(U41) = []
      pi(U21) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U41(X1,X2,X3)) -> mark#(X1)
p5: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U41(X1,X2,X3)) -> mark#(X1)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,1)) x1 + (2,2)
          U11_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
          U21_A(x1) = ((1,1),(1,1)) x1 + (3,2)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
          plus_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (3,1)
          s_A(x1) = ((0,1),(0,1)) x1 + (3,2)
  
    precedence:
    
      s > U41 = plus > mark# = U11 > U21
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(U21) = []
      pi(U41) = [2]
      pi(plus) = []
      pi(s) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U41(X1,X2,X3)) -> mark#(X1)
p4: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(U41(X1,X2,X3)) -> mark#(X1)
p4: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          U11_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + (1,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          U41_A(x1,x2,x3) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (0,1)
          U21_A(x1) = ((1,0),(1,0)) x1 + (1,1)
  
    precedence:
    
      s > mark# = U41 = U21 > U11
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(s) = []
      pi(U41) = [2]
      pi(U21) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U41(X1,X2,X3)) -> mark#(X1)
p3: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U41(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,1)) x1 + (2,2)
          U11_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (1,1)
          U21_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
  
    precedence:
    
      U21 > mark# = U11 = U41
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = [1, 2]
      pi(U21) = []
      pi(U41) = [3]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U41(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U41(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          U11_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (1,1)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
  
    precedence:
    
      mark# = U41 > U11
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = [1]
      pi(U41) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U41(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U41(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((1,1),(1,1)) x1 + (1,1)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (2,1)
  
    precedence:
    
      mark# = U41
    
    partial status:
  
      pi(mark#) = [1]
      pi(U41) = [1, 3]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: a__U11#(tt(),V2) -> a__isNat#(V2)
p2: a__isNat#(s(V1)) -> a__isNat#(V1)
p3: a__isNat#(plus(V1,V2)) -> a__isNat#(V1)
p4: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r7, r8, r9, r23, r24, r25, r26

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__U11#_A(x1,x2) = ((1,1),(0,1)) x2 + (2,3)
          tt_A() = (13,2)
          a__isNat#_A(x1) = ((1,1),(0,1)) x1 + (1,3)
          s_A(x1) = ((1,1),(0,1)) x1 + (15,3)
          plus_A(x1,x2) = x1 + ((0,0),(1,1)) x2 + (10,0)
          a__isNat_A(x1) = ((1,1),(0,0)) x1 + (12,4)
          a__U12_A(x1) = ((1,1),(0,0)) x1 + (2,3)
          U12_A(x1) = (1,1)
          a__U11_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (17,4)
          a__U21_A(x1) = (14,2)
          U11_A(x1,x2) = (1,1)
          U21_A(x1) = (0,0)
          |0|_A() = (1,1)
          isNat_A(x1) = (1,1)
  
    precedence:
    
      a__isNat = a__U12 = a__U11 > a__U11# = a__isNat# = s > U12 = U11 > tt > plus = a__U21 = |0| = isNat > U21
    
    partial status:
  
      pi(a__U11#) = []
      pi(tt) = []
      pi(a__isNat#) = []
      pi(s) = []
      pi(plus) = []
      pi(a__isNat) = []
      pi(a__U12) = []
      pi(U12) = []
      pi(a__U11) = []
      pi(a__U21) = []
      pi(U11) = []
      pi(U21) = []
      pi(|0|) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: a__U11#(tt(),V2) -> a__isNat#(V2)
p2: a__isNat#(plus(V1,V2)) -> a__isNat#(V1)
p3: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: a__U11#(tt(),V2) -> a__isNat#(V2)
p2: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2)
p3: a__isNat#(plus(V1,V2)) -> a__isNat#(V1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r7, r8, r9, r23, r24, r25, r26

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__U11#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + (0,3)
          tt_A() = (2,2)
          a__isNat#_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (8,4)
          a__isNat_A(x1) = x1 + (3,4)
          a__U12_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          U12_A(x1) = (1,1)
          a__U11_A(x1,x2) = ((1,0),(1,1)) x2 + (6,3)
          a__U21_A(x1) = (3,2)
          U11_A(x1,x2) = ((0,0),(0,1)) x2 + (1,1)
          U21_A(x1) = (1,1)
          |0|_A() = (1,1)
          s_A(x1) = (1,1)
          isNat_A(x1) = (1,1)
  
    precedence:
    
      a__U11# = a__isNat# > a__U11 > a__U12 > tt > plus = a__isNat > U12 > U11 > a__U21 = U21 > |0| > s > isNat
    
    partial status:
  
      pi(a__U11#) = [2]
      pi(tt) = []
      pi(a__isNat#) = [1]
      pi(plus) = [2]
      pi(a__isNat) = [1]
      pi(a__U12) = []
      pi(U12) = []
      pi(a__U11) = []
      pi(a__U21) = []
      pi(U11) = []
      pi(U21) = []
      pi(|0|) = []
      pi(s) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2)
p2: a__isNat#(plus(V1,V2)) -> a__isNat#(V1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: a__isNat#(plus(V1,V2)) -> a__isNat#(V1)

and R consists of:

r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
r2: a__U12(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt(),N) -> mark(N)
r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r7: a__isNat(|0|()) -> tt()
r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N)
r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r13: mark(U12(X)) -> a__U12(mark(X))
r14: mark(isNat(X)) -> a__isNat(X)
r15: mark(U21(X)) -> a__U21(mark(X))
r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r20: mark(tt()) -> tt()
r21: mark(s(X)) -> s(mark(X))
r22: mark(|0|()) -> |0|()
r23: a__U11(X1,X2) -> U11(X1,X2)
r24: a__U12(X) -> U12(X)
r25: a__isNat(X) -> isNat(X)
r26: a__U21(X) -> U21(X)
r27: a__U31(X1,X2) -> U31(X1,X2)
r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r30: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__isNat#_A(x1) = x1 + (1,1)
          plus_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (2,1)
  
    precedence:
    
      plus > a__isNat#
    
    partial status:
  
      pi(a__isNat#) = [1]
      pi(plus) = [1, 2]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.