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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U42#_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,1)) x3 + (2,22)
            tt_A() = (5,11)
            mark#_A(x1) = ((1,0),(0,0)) x1 + (6,15)
            s_A(x1) = x1 + (9,0)
            plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (1,1)
            a__plus#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (6,11)
            mark_A(x1) = ((1,0),(1,1)) x1 + (0,13)
            a__U41#_A(x1,x2,x3) = x2 + ((1,0),(0,0)) x3 + (8,10)
            a__isNat_A(x1) = ((0,0),(1,0)) x1 + (5,12)
            U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (5,1)
            |0|_A() = (11,11)
            a__U31#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,1)
            U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (5,1)
            U31_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,1)
            U21_A(x1) = x1 + (0,1)
            U12_A(x1) = ((1,0),(0,0)) x1 + (0,1)
            U11_A(x1,x2) = ((1,0),(0,0)) x1 + (0,12)
            a__U11_A(x1,x2) = ((1,0),(0,0)) x1 + (0,13)
            a__U12_A(x1) = ((1,0),(0,0)) x1 + (0,12)
            a__U21_A(x1) = x1 + (0,1)
            a__U31_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,7)
            a__U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (5,4)
            a__U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (5,3)
            a__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (1,2)
            isNat_A(x1) = ((0,0),(1,0)) x1 + (5,1)
    
      precedence:
      
        mark > a__plus > a__isNat = a__U31 > a__U11 > |0| = U11 = a__U12 = a__U41 > a__U42 > tt = s = plus = a__plus# = a__U41# = U42 = a__U31# = U41 = U21 = U12 = a__U21 = isNat > a__U42# > U31 > mark#
      
      partial status:
    
        pi(a__U42#) = [1, 3]
        pi(tt) = []
        pi(mark#) = []
        pi(s) = [1]
        pi(plus) = []
        pi(a__plus#) = [2]
        pi(mark) = []
        pi(a__U41#) = [2]
        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) = [1]
        pi(a__U31) = [2]
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__plus) = []
        pi(isNat) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U42#_A(x1,x2,x3) = ((1,0),(1,1)) x1 + (1,1)
            tt_A() = (15,23)
            mark#_A(x1) = (12,21)
            s_A(x1) = (3,1)
            plus_A(x1,x2) = (9,0)
            a__plus#_A(x1,x2) = (14,22)
            mark_A(x1) = (10,9)
            a__U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + (15,15)
            a__isNat_A(x1) = (5,8)
            U42_A(x1,x2,x3) = (1,1)
            |0|_A() = (6,7)
            a__U31#_A(x1,x2) = (13,21)
            U41_A(x1,x2,x3) = (3,10)
            U31_A(x1,x2) = (0,0)
            U21_A(x1) = (1,1)
            U12_A(x1) = (1,1)
            U11_A(x1,x2) = (13,0)
            a__U11_A(x1,x2) = (3,7)
            a__U12_A(x1) = (2,24)
            a__U21_A(x1) = ((0,0),(1,0)) x1 + (2,2)
            a__U31_A(x1,x2) = x2 + (7,8)
            a__U41_A(x1,x2,x3) = (4,7)
            a__U42_A(x1,x2,x3) = (7,24)
            a__plus_A(x1,x2) = (8,25)
            isNat_A(x1) = (1,1)
    
      precedence:
      
        a__U12 > U42 > U21 > a__U41 = a__U42 > tt > mark > a__isNat = a__U11 > U12 > mark# = s = a__plus > a__U31 > isNat > a__U41# > a__U42# > plus = a__plus# > |0| > a__U21 > a__U31# = U41 = U31 = U11
      
      partial status:
    
        pi(a__U42#) = [1]
        pi(tt) = []
        pi(mark#) = []
        pi(s) = []
        pi(plus) = []
        pi(a__plus#) = []
        pi(mark) = []
        pi(a__U41#) = [2]
        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) = [2]
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__plus) = []
        pi(isNat) = []
    

The next rules are strictly ordered:

  p1, p5, p6, p7, p8, p11, p15, p17

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

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

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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            s_A(x1) = x1 + (3,1)
            U11_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (3,1)
            U12_A(x1) = ((1,0),(1,1)) x1 + (3,3)
            U21_A(x1) = x1 + (1,3)
            U31_A(x1,x2) = x1 + x2 + (3,0)
            U41_A(x1,x2,x3) = x1 + x2 + x3
            U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x3 + (3,1)
            plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (3,1)
    
      precedence:
      
        U41 > mark# = U12 = plus > s > U11 = U21 = U31 = U42
      
      partial status:
    
        pi(mark#) = [1]
        pi(s) = [1]
        pi(U11) = [2]
        pi(U12) = []
        pi(U21) = [1]
        pi(U31) = [2]
        pi(U41) = []
        pi(U42) = [3]
        pi(plus) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = (0,0)
            s_A(x1) = x1 + (0,1)
            U11_A(x1,x2) = x2
            U12_A(x1) = (0,0)
            U21_A(x1) = (0,0)
            U31_A(x1,x2) = x2
            U41_A(x1,x2,x3) = (0,0)
            U42_A(x1,x2,x3) = x3
            plus_A(x1,x2) = (0,1)
    
      precedence:
      
        U42 > s = plus > U11 > mark# = U12 = U21 = U31 = U41
      
      partial status:
    
        pi(mark#) = []
        pi(s) = []
        pi(U11) = [2]
        pi(U12) = []
        pi(U21) = []
        pi(U31) = [2]
        pi(U41) = []
        pi(U42) = [3]
        pi(plus) = []
    

The next rules are strictly ordered:

  p1, p2, p4, p7, p8

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U12(X)) -> mark#(X)
p2: mark#(U31(X1,X2)) -> mark#(X1)
p3: mark#(U41(X1,X2,X3)) -> mark#(X1)
p4: mark#(plus(X1,X2)) -> mark#(X2)

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#(U12(X)) -> mark#(X)
p2: mark#(plus(X1,X2)) -> mark#(X2)
p3: mark#(U41(X1,X2,X3)) -> mark#(X1)
p4: mark#(U31(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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = ((0,0),(1,0)) x1 + (1,0)
            U12_A(x1) = x1 + (2,1)
            plus_A(x1,x2) = x2 + (0,1)
            U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,1)
            U31_A(x1,x2) = ((1,0),(0,0)) x1 + (2,1)
    
      precedence:
      
        mark# = U12 = plus > U41 = U31
      
      partial status:
    
        pi(mark#) = []
        pi(U12) = []
        pi(plus) = [2]
        pi(U41) = []
        pi(U31) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = (0,0)
            U12_A(x1) = (0,0)
            plus_A(x1,x2) = ((1,0),(0,0)) x2 + (1,0)
            U41_A(x1,x2,x3) = (0,0)
            U31_A(x1,x2) = (0,0)
    
      precedence:
      
        mark# = plus = U41 > U12 = U31
      
      partial status:
    
        pi(mark#) = []
        pi(U12) = []
        pi(plus) = []
        pi(U41) = []
        pi(U31) = []
    

The next rules are strictly ordered:

  p1, p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(plus(X1,X2)) -> mark#(X2)
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#(plus(X1,X2)) -> mark#(X2)
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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = ((1,0),(0,0)) x1
            plus_A(x1,x2) = ((1,0),(1,1)) x2 + (1,1)
            U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + x3 + (1,0)
    
      precedence:
      
        mark# = plus = U41
      
      partial status:
    
        pi(mark#) = []
        pi(plus) = []
        pi(U41) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = (0,0)
            plus_A(x1,x2) = (1,1)
            U41_A(x1,x2,x3) = x2 + ((1,0),(1,1)) x3 + (1,1)
    
      precedence:
      
        plus > mark# = U41
      
      partial status:
    
        pi(mark#) = []
        pi(plus) = []
        pi(U41) = [2, 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#(plus(X1,X2)) -> mark#(X2)

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#(plus(X1,X2)) -> mark#(X2)

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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = x1 + (1,2)
            plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,1)
    
      precedence:
      
        mark# > plus
      
      partial status:
    
        pi(mark#) = [1]
        pi(plus) = [1]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = (0,0)
            plus_A(x1,x2) = (1,1)
    
      precedence:
      
        mark# > plus
      
      partial status:
    
        pi(mark#) = []
        pi(plus) = []
    

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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U11#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (4,1)
            tt_A() = (3,4)
            a__isNat#_A(x1) = ((1,0),(1,1)) x1 + (2,5)
            s_A(x1) = ((1,0),(1,1)) x1 + (7,6)
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (9,6)
            a__isNat_A(x1) = ((1,0),(1,1)) x1 + (6,1)
            a__U12_A(x1) = (4,5)
            U12_A(x1) = (0,0)
            a__U11_A(x1,x2) = ((1,0),(1,1)) x1 + (2,2)
            a__U21_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            U11_A(x1,x2) = (1,1)
            U21_A(x1) = (1,1)
            |0|_A() = (1,1)
            isNat_A(x1) = (0,0)
    
      precedence:
      
        plus > U12 > a__isNat = a__U11 > a__U12 > a__U11# = tt = a__isNat# = s = U11 > U21 = |0| = isNat > a__U21
      
      partial status:
    
        pi(a__U11#) = [1]
        pi(tt) = []
        pi(a__isNat#) = []
        pi(s) = [1]
        pi(plus) = [1, 2]
        pi(a__isNat) = []
        pi(a__U12) = []
        pi(U12) = []
        pi(a__U11) = []
        pi(a__U21) = [1]
        pi(U11) = []
        pi(U21) = []
        pi(|0|) = []
        pi(isNat) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U11#_A(x1,x2) = (1,0)
            tt_A() = (0,0)
            a__isNat#_A(x1) = (0,0)
            s_A(x1) = ((1,0),(1,1)) x1
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,0)
            a__isNat_A(x1) = (2,2)
            a__U12_A(x1) = (0,0)
            U12_A(x1) = (0,0)
            a__U11_A(x1,x2) = (1,1)
            a__U21_A(x1) = ((1,0),(1,1)) x1 + (1,1)
            U11_A(x1,x2) = (0,0)
            U21_A(x1) = (0,0)
            |0|_A() = (0,0)
            isNat_A(x1) = (0,0)
    
      precedence:
      
        a__U11 > a__U21 > U21 > U11 > a__isNat# = plus > a__isNat > isNat > |0| > a__U11# > a__U12 > tt > s > U12
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__isNat#) = []
        pi(s) = [1]
        pi(plus) = [1, 2]
        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:

  p1, p2, p3, p4

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