YES
We show the termination of the TRS R:

  eq(|0|(),|0|()) -> true()
  eq(|0|(),s(x)) -> false()
  eq(s(x),|0|()) -> false()
  eq(s(x),s(y)) -> eq(x,y)
  le(|0|(),y) -> true()
  le(s(x),|0|()) -> false()
  le(s(x),s(y)) -> le(x,y)
  app(nil(),y) -> y
  app(add(n,x),y) -> add(n,app(x,y))
  min(add(n,nil())) -> n
  min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
  if_min(true(),add(n,add(m,x))) -> min(add(n,x))
  if_min(false(),add(n,add(m,x))) -> min(add(m,x))
  rm(n,nil()) -> nil()
  rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
  if_rm(true(),n,add(m,x)) -> rm(n,x)
  if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
  minsort(nil(),nil()) -> nil()
  minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
  if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
  if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))


-- SCC decomposition.

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

p1: eq#(s(x),s(y)) -> eq#(x,y)
p2: le#(s(x),s(y)) -> le#(x,y)
p3: app#(add(n,x),y) -> app#(x,y)
p4: min#(add(n,add(m,x))) -> if_min#(le(n,m),add(n,add(m,x)))
p5: min#(add(n,add(m,x))) -> le#(n,m)
p6: if_min#(true(),add(n,add(m,x))) -> min#(add(n,x))
p7: if_min#(false(),add(n,add(m,x))) -> min#(add(m,x))
p8: rm#(n,add(m,x)) -> if_rm#(eq(n,m),n,add(m,x))
p9: rm#(n,add(m,x)) -> eq#(n,m)
p10: if_rm#(true(),n,add(m,x)) -> rm#(n,x)
p11: if_rm#(false(),n,add(m,x)) -> rm#(n,x)
p12: minsort#(add(n,x),y) -> if_minsort#(eq(n,min(add(n,x))),add(n,x),y)
p13: minsort#(add(n,x),y) -> eq#(n,min(add(n,x)))
p14: minsort#(add(n,x),y) -> min#(add(n,x))
p15: if_minsort#(true(),add(n,x),y) -> minsort#(app(rm(n,x),y),nil())
p16: if_minsort#(true(),add(n,x),y) -> app#(rm(n,x),y)
p17: if_minsort#(true(),add(n,x),y) -> rm#(n,x)
p18: if_minsort#(false(),add(n,x),y) -> minsort#(x,add(n,y))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The estimated dependency graph contains the following SCCs:

  {p12, p15, p18}
  {p8, p10, p11}
  {p1}
  {p4, p6, p7}
  {p2}
  {p3}


-- Reduction pair.

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

p1: if_minsort#(false(),add(n,x),y) -> minsort#(x,add(n,y))
p2: minsort#(add(n,x),y) -> if_minsort#(eq(n,min(add(n,x))),add(n,x),y)
p3: if_minsort#(true(),add(n,x),y) -> minsort#(app(rm(n,x),y),nil())

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_minsort#_A(x1,x2,x3) = x2 + x3 + 1
          false_A() = 1
          add_A(x1,x2) = x1 + x2 + 7
          minsort#_A(x1,x2) = x1 + x2 + 1
          eq_A(x1,x2) = 4
          min_A(x1) = x1 + 5
          true_A() = 3
          app_A(x1,x2) = x1 + x2 + 2
          rm_A(x1,x2) = x1 + x2 + 2
          nil_A() = 1
          |0|_A() = 4
          s_A(x1) = 5
          le_A(x1,x2) = 4
          if_min_A(x1,x2) = x1 + x2
          if_rm_A(x1,x2,x3) = x2 + x3 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_minsort#_A(x1,x2,x3) = max{0, x1 - 116, x2 - 96, x3 - 96}
          false_A = 3
          add_A(x1,x2) = max{x1 + 50, x2}
          minsort#_A(x1,x2) = max{0, x1 - 96, x2 - 96}
          eq_A(x1,x2) = max{71, x1 + 70, x2 + 69}
          min_A(x1) = max{2, x1 - 49}
          true_A = 63
          app_A(x1,x2) = max{88, x1, x2}
          rm_A(x1,x2) = max{88, x1 - 56, x2}
          nil_A = 95
          |0|_A = 0
          s_A(x1) = max{2, x1}
          le_A(x1,x2) = max{141, x1}
          if_min_A(x1,x2) = max{2, x2 - 49}
          if_rm_A(x1,x2,x3) = max{88, x1 - 126, x2 - 56, x3}
          if_minsort#!_A(x1,x2,x3) = max{x1 - 107, x2 + 57, x3 - 67}
          false!_A = 1
          add!_A(x1,x2) = max{27, x2 + 7}
          minsort#!_A(x1,x2) = max{x1 + 57, x2 - 67}
          eq!_A(x1,x2) = x1 + 170
          min!_A(x1) = x1 + 87
          true!_A = 3
          app!_A(x1,x2) = max{x1 + 55, x2 + 96}
          rm!_A(x1,x2) = max{147, x1 - 47, x2 + 118}
          nil!_A = 0
          |0|!_A = 4
          s!_A(x1) = max{28, x1 - 2}
          le!_A(x1,x2) = 27
          if_min!_A(x1,x2) = max{6, x1 - 4}
          if_rm!_A(x1,x2,x3) = max{x1 - 119, x2 - 48, x3 + 96}
  
    partial status:
  
      pi(if_minsort#) = [2]
      pi(false) = []
      pi(add) = [1, 2]
      pi(minsort#) = [1, 2]
      pi(eq) = []
      pi(min) = [1]
      pi(true) = []
      pi(app) = [1]
      pi(rm) = [2]
      pi(nil) = []
      pi(|0|) = []
      pi(s) = []
      pi(le) = []
      pi(if_min) = [1, 2]
      pi(if_rm) = [3]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: if_rm#(false(),n,add(m,x)) -> rm#(n,x)
p2: rm#(n,add(m,x)) -> if_rm#(eq(n,m),n,add(m,x))
p3: if_rm#(true(),n,add(m,x)) -> rm#(n,x)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The set of usable rules consists of

  r1, r2, r3, r5, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_rm#_A(x1,x2,x3) = x3 + 1
          false_A() = 6
          add_A(x1,x2) = x2 + 10
          rm#_A(x1,x2) = x2 + 5
          eq_A(x1,x2) = x1 + 20
          true_A() = 8
          |0|_A() = 9
          s_A(x1) = x1 + 21
          le_A(x1,x2) = x1 + 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_rm#_A(x1,x2,x3) = max{14, x2 - 1, x3 + 5}
          false_A = 0
          add_A(x1,x2) = max{8, x1 - 5, x2}
          rm#_A(x1,x2) = max{14, x1 - 1, x2 + 5}
          eq_A(x1,x2) = max{0, x1 - 1, x2 - 1}
          true_A = 0
          |0|_A = 0
          s_A(x1) = max{0, x1 - 6}
          le_A(x1,x2) = max{x1 - 1, x2 + 5}
          if_rm#!_A(x1,x2,x3) = 7
          false!_A = 24
          add!_A(x1,x2) = max{x1 + 1, x2 + 6}
          rm#!_A(x1,x2) = max{21, x1 + 8, x2 + 13}
          eq!_A(x1,x2) = max{22, x1 - 1, x2 - 34}
          true!_A = 22
          |0|!_A = 25
          s!_A(x1) = max{25, x1 - 35}
          le!_A(x1,x2) = 14
  
    partial status:
  
      pi(if_rm#) = [3]
      pi(false) = []
      pi(add) = [2]
      pi(rm#) = [2]
      pi(eq) = [1]
      pi(true) = []
      pi(|0|) = []
      pi(s) = [1]
      pi(le) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: eq#(s(x),s(y)) -> eq#(x,y)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The set of usable rules consists of

  r1, r2, r3, r5, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          eq#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 8
          eq_A(x1,x2) = 3
          |0|_A() = 3
          true_A() = 2
          false_A() = 2
          le_A(x1,x2) = x2 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          eq#_A(x1,x2) = max{0, x1 - 2, x2 - 2}
          s_A(x1) = x1 + 1
          eq_A(x1,x2) = max{45, x1 - 2, x2 - 2}
          |0|_A = 0
          true_A = 0
          false_A = 0
          le_A(x1,x2) = max{2, x1 - 2, x2 - 2}
          eq#!_A(x1,x2) = max{1, x1 - 2, x2 - 2}
          s!_A(x1) = 0
          eq!_A(x1,x2) = max{28, x2 - 159}
          |0|!_A = 28
          true!_A = 3
          false!_A = 27
          le!_A(x1,x2) = max{75, x1 - 2, x2 - 17}
  
    partial status:
  
      pi(eq#) = [2]
      pi(s) = [1]
      pi(eq) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(le) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: if_min#(false(),add(n,add(m,x))) -> min#(add(m,x))
p2: min#(add(n,add(m,x))) -> if_min#(le(n,m),add(n,add(m,x)))
p3: if_min#(true(),add(n,add(m,x))) -> min#(add(n,x))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The set of usable rules consists of

  r5, r6, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_min#_A(x1,x2) = 4
          false_A() = 7
          add_A(x1,x2) = x1 + x2 + 5
          min#_A(x1) = 4
          le_A(x1,x2) = x1 + x2 + 3
          true_A() = 7
          |0|_A() = 8
          s_A(x1) = x1 + 8
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          if_min#_A(x1,x2) = max{0, x1 - 44, x2 - 21}
          false_A = 37
          add_A(x1,x2) = max{15, x1 - 4, x2 + 4}
          min#_A(x1) = max{0, x1 - 21}
          le_A(x1,x2) = max{43, x1 - 33, x2 - 30}
          true_A = 0
          |0|_A = 2
          s_A(x1) = max{30, x1 + 29}
          if_min#!_A(x1,x2) = max{16, x2 + 2}
          false!_A = 18
          add!_A(x1,x2) = max{14, x2 + 11}
          min#!_A(x1) = x1 + 2
          le!_A(x1,x2) = max{25, x2 - 3}
          true!_A = 13
          |0|!_A = 19
          s!_A(x1) = max{0, x1 - 70}
  
    partial status:
  
      pi(if_min#) = []
      pi(false) = []
      pi(add) = [1, 2]
      pi(min#) = []
      pi(le) = []
      pi(true) = []
      pi(|0|) = []
      pi(s) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: min#(add(n,add(m,x))) -> if_min#(le(n,m),add(n,add(m,x)))
p2: if_min#(true(),add(n,add(m,x))) -> min#(add(n,x))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: min#(add(n,add(m,x))) -> if_min#(le(n,m),add(n,add(m,x)))
p2: if_min#(true(),add(n,add(m,x))) -> min#(add(n,x))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          min#_A(x1) = 1
          add_A(x1,x2) = x1 + 3
          if_min#_A(x1,x2) = 1
          le_A(x1,x2) = x1 + x2 + 6
          true_A() = 2
          eq_A(x1,x2) = x1 + 1
          |0|_A() = 5
          s_A(x1) = x1 + 5
          false_A() = 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          min#_A(x1) = max{1, x1 - 83}
          add_A(x1,x2) = x2 + 41
          if_min#_A(x1,x2) = max{1, x1 - 53, x2 - 83}
          le_A(x1,x2) = 49
          true_A = 1
          eq_A(x1,x2) = max{x1 - 2, x2 + 75}
          |0|_A = 1
          s_A(x1) = max{0, x1 - 76}
          false_A = 1
          min#!_A(x1) = x1 + 7
          add!_A(x1,x2) = max{x1 + 2, x2 + 57}
          if_min#!_A(x1,x2) = max{x1 + 39, x2 - 26}
          le!_A(x1,x2) = max{0, x1 - 40}
          true!_A = 4
          eq!_A(x1,x2) = max{5, x1 - 2}
          |0|!_A = 5
          s!_A(x1) = max{2, x1 - 41}
          false!_A = 1
  
    partial status:
  
      pi(min#) = []
      pi(add) = [1]
      pi(if_min#) = []
      pi(le) = []
      pi(true) = []
      pi(eq) = [1]
      pi(|0|) = []
      pi(s) = [1]
      pi(false) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

p1: le#(s(x),s(y)) -> le#(x,y)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The set of usable rules consists of

  r1, r2, r3, r5, r6, r14, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          le#_A(x1,x2) = x1 + x2 + 9
          s_A(x1) = x1 + 8
          eq_A(x1,x2) = 4
          |0|_A() = 4
          true_A() = 3
          false_A() = 3
          le_A(x1,x2) = x1 + x2 + 7
          rm_A(x1,x2) = x1 + x2 + 13
          nil_A() = 2
          minsort_A(x1,x2) = x2 + 5
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          le#_A(x1,x2) = max{0, x1 - 6, x2 - 6}
          s_A(x1) = x1 + 5
          eq_A(x1,x2) = max{49, x1 - 8, x2 - 8}
          |0|_A = 7
          true_A = 0
          false_A = 30
          le_A(x1,x2) = max{39, x1 - 8, x2 - 8}
          rm_A(x1,x2) = max{x1 + 7, x2 + 7}
          nil_A = 0
          minsort_A(x1,x2) = max{7, x1 - 39, x2 - 8}
          le#!_A(x1,x2) = max{4, x1 - 6, x2 - 6}
          s!_A(x1) = max{3, x1 - 1}
          eq!_A(x1,x2) = max{6, x1 - 62, x2 - 149}
          |0|!_A = 0
          true!_A = 1
          false!_A = 4
          le!_A(x1,x2) = max{2, x2 - 2}
          rm!_A(x1,x2) = max{5, x1 + 1, x2}
          nil!_A = 0
          minsort!_A(x1,x2) = max{7, x1 - 68, x2 - 24}
  
    partial status:
  
      pi(le#) = [2]
      pi(s) = [1]
      pi(eq) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(le) = []
      pi(rm) = [2]
      pi(nil) = []
      pi(minsort) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: app#(add(n,x),y) -> app#(x,y)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: le(|0|(),y) -> true()
r6: le(s(x),|0|()) -> false()
r7: le(s(x),s(y)) -> le(x,y)
r8: app(nil(),y) -> y
r9: app(add(n,x),y) -> add(n,app(x,y))
r10: min(add(n,nil())) -> n
r11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
r12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
r13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
r14: rm(n,nil()) -> nil()
r15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
r16: if_rm(true(),n,add(m,x)) -> rm(n,x)
r17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
r18: minsort(nil(),nil()) -> nil()
r19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
r20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
r21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))

The set of usable rules consists of

  r1, r2, r3, r5, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1
          add_A(x1,x2) = x1 + x2 + 6
          eq_A(x1,x2) = x1 + 5
          |0|_A() = 5
          true_A() = 4
          s_A(x1) = x1 + 6
          false_A() = 3
          le_A(x1,x2) = x1 + x2 + 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{0, x1 - 1}
          add_A(x1,x2) = max{32, x1 - 22, x2}
          eq_A(x1,x2) = max{42, x1 - 10, x2 - 10}
          |0|_A = 0
          true_A = 0
          s_A(x1) = max{9, x1 + 1}
          false_A = 0
          le_A(x1,x2) = max{2, x1 - 10, x2 - 10}
          app#!_A(x1,x2) = max{27, x1 - 33, x2 - 1}
          add!_A(x1,x2) = max{30, x1 - 56, x2 + 28}
          eq!_A(x1,x2) = max{6, x2 - 2}
          |0|!_A = 3
          true!_A = 2
          s!_A(x1) = 4
          false!_A = 4
          le!_A(x1,x2) = max{5, x1 - 3, x2 - 10}
  
    partial status:
  
      pi(app#) = [1]
      pi(add) = [1]
      pi(eq) = []
      pi(|0|) = []
      pi(true) = []
      pi(s) = [1]
      pi(false) = []
      pi(le) = []

The following rules are removed:

  p1

Then no dependency pair remains.