YES

We show the termination of the relative TRS R/S:

  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))

  S:
  rand(x) -> x
  rand(x) -> rand(s(x))

-- SCC decomposition.

Consider the non-minimal 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

Consider the non-minimal 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            if_minsort#_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (1,10)
            false_A() = (3,0)
            add_A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (6,9)
            minsort#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,10)
            eq_A(x1,x2) = ((0,0),(1,1)) x1 + (3,11)
            min_A(x1) = ((1,1),(1,1)) x1 + (2,8)
            true_A() = (2,0)
            app_A(x1,x2) = x1 + x2 + (0,1)
            rm_A(x1,x2) = ((1,1),(0,0)) x1 + x2 + (1,0)
            nil_A() = (0,1)
            |0|_A() = (4,1)
            s_A(x1) = x1
            le_A(x1,x2) = x2 + (33,10)
            if_min_A(x1,x2) = ((1,1),(1,1)) x2 + (1,0)
            if_rm_A(x1,x2,x3) = ((1,1),(0,0)) x2 + x3 + (1,0)
            minsort_A(x1,x2) = ((0,1),(1,1)) x1 + ((0,1),(1,1)) x2 + (5,7)
            if_minsort_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((0,1),(1,1)) x3 + (5,7)
            rand_A(x1) = x1 + (1,0)
    
      precedence:
      
        if_minsort# = false = add = minsort# = eq = min = true = app = rm = nil = |0| = s = le = if_min = if_rm = minsort = if_minsort = rand
      
      partial status:
    
        pi(if_minsort#) = []
        pi(false) = []
        pi(add) = []
        pi(minsort#) = []
        pi(eq) = []
        pi(min) = []
        pi(true) = []
        pi(app) = []
        pi(rm) = []
        pi(nil) = []
        pi(|0|) = []
        pi(s) = []
        pi(le) = []
        pi(if_min) = []
        pi(if_rm) = []
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            if_minsort#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + (1,1)
            false_A() = (30,14)
            add_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (28,12)
            minsort#_A(x1,x2) = ((0,1),(0,1)) x1 + (21,1)
            eq_A(x1,x2) = (32,14)
            min_A(x1) = (27,14)
            true_A() = (31,11)
            app_A(x1,x2) = ((1,0),(1,1)) x1 + x2
            rm_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (0,2)
            nil_A() = (30,0)
            |0|_A() = (0,1)
            s_A(x1) = ((0,1),(0,1)) x1
            le_A(x1,x2) = (57,15)
            if_min_A(x1,x2) = (27,14)
            if_rm_A(x1,x2,x3) = ((0,0),(0,1)) x2 + x3 + (0,2)
            minsort_A(x1,x2) = (29,0)
            if_minsort_A(x1,x2,x3) = (29,0)
            rand_A(x1) = (1,1)
    
      precedence:
      
        rm > if_rm > eq > true > app > minsort = if_minsort > le > minsort# > if_minsort# > false = add = min = nil = |0| = s = if_min = rand
      
      partial status:
    
        pi(if_minsort#) = []
        pi(false) = []
        pi(add) = []
        pi(minsort#) = []
        pi(eq) = []
        pi(min) = []
        pi(true) = []
        pi(app) = [1, 2]
        pi(rm) = [2]
        pi(nil) = []
        pi(|0|) = []
        pi(s) = []
        pi(le) = []
        pi(if_min) = []
        pi(if_rm) = [3]
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p2, p3

We remove them from the problem.

-- SCC decomposition.

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

p1: 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

Consider the non-minimal 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            if_rm#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,1),(0,1)) x3 + (0,1)
            false_A() = (3,4)
            add_A(x1,x2) = ((0,0),(1,1)) x1 + ((0,0),(0,1)) x2 + (5,5)
            rm#_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(0,1)) x2 + (6,6)
            eq_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (11,6)
            true_A() = (3,2)
            |0|_A() = (4,4)
            s_A(x1) = x1
            le_A(x1,x2) = (4,5)
            app_A(x1,x2) = x1 + x2 + (9,0)
            nil_A() = (1,0)
            min_A(x1) = ((1,1),(0,1)) x1 + (5,7)
            if_min_A(x1,x2) = x1 + ((1,1),(0,1)) x2
            rm_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,1),(0,1)) x2 + (2,0)
            if_rm_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((0,1),(0,1)) x3 + (1,0)
            minsort_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (6,6)
            if_minsort_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((0,0),(0,1)) x3 + (5,6)
            rand_A(x1) = x1 + (1,0)
    
      precedence:
      
        nil > if_rm > app > minsort > if_minsort > add = min = if_min > true > eq > false = s > rm > |0| > if_rm# = rm# > le > rand
      
      partial status:
    
        pi(if_rm#) = [2, 3]
        pi(false) = []
        pi(add) = []
        pi(rm#) = [1]
        pi(eq) = [2]
        pi(true) = []
        pi(|0|) = []
        pi(s) = [1]
        pi(le) = []
        pi(app) = [1, 2]
        pi(nil) = []
        pi(min) = [1]
        pi(if_min) = [1, 2]
        pi(rm) = []
        pi(if_rm) = []
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            if_rm#_A(x1,x2,x3) = ((0,0),(1,0)) x3 + (2,6)
            false_A() = (2,3)
            add_A(x1,x2) = (3,9)
            rm#_A(x1,x2) = ((0,0),(1,0)) x1 + (6,8)
            eq_A(x1,x2) = ((1,1),(1,1)) x2 + (4,2)
            true_A() = (5,10)
            |0|_A() = (6,1)
            s_A(x1) = ((0,0),(1,1)) x1 + (1,1)
            le_A(x1,x2) = (2,2)
            app_A(x1,x2) = x2 + (4,10)
            nil_A() = (2,0)
            min_A(x1) = x1
            if_min_A(x1,x2) = x2 + (1,0)
            rm_A(x1,x2) = (6,8)
            if_rm_A(x1,x2,x3) = (6,8)
            minsort_A(x1,x2) = (3,9)
            if_minsort_A(x1,x2,x3) = (3,9)
            rand_A(x1) = (2,1)
    
      precedence:
      
        le > if_rm# = false = eq = true = s = minsort = if_minsort = rand > app > add = nil = if_min = rm = if_rm > |0| > min > rm#
      
      partial status:
    
        pi(if_rm#) = []
        pi(false) = []
        pi(add) = []
        pi(rm#) = []
        pi(eq) = []
        pi(true) = []
        pi(|0|) = []
        pi(s) = []
        pi(le) = []
        pi(app) = [2]
        pi(nil) = []
        pi(min) = [1]
        pi(if_min) = [2]
        pi(rm) = []
        pi(if_rm) = []
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1, p2, p3

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

-- Reduction pair.

Consider the non-minimal 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          eq#_A(x1,x2) = x2 + 1
          s_A(x1) = x1
          eq_A(x1,x2) = max{x1 - 2, x2 + 7}
          |0|_A = 37
          true_A = 36
          false_A = 31
          le_A(x1,x2) = max{50, x1 + 3, x2 + 37}
          app_A(x1,x2) = max{x1, x2}
          nil_A = 0
          add_A(x1,x2) = max{13, x1 + 8, x2}
          min_A(x1) = max{9, x1 - 6}
          if_min_A(x1,x2) = max{9, x2 - 6}
          rm_A(x1,x2) = max{19, x1 + 8, x2}
          if_rm_A(x1,x2,x3) = max{x1 - 12, x2 + 8, x3}
          minsort_A(x1,x2) = max{x1 + 12, x2 + 12}
          if_minsort_A(x1,x2,x3) = max{x1 - 5, x2 + 12, x3 + 12}
          rand_A(x1) = x1 + 2
    
      precedence:
      
        |0| > eq# = true = le = rm = if_rm = minsort = if_minsort > eq = nil > min = if_min = rand > false = app = add > s
      
      partial status:
    
        pi(eq#) = [2]
        pi(s) = [1]
        pi(eq) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(le) = [2]
        pi(app) = [1, 2]
        pi(nil) = []
        pi(add) = []
        pi(min) = []
        pi(if_min) = []
        pi(rm) = [1]
        pi(if_rm) = [2]
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          eq#_A(x1,x2) = 0
          s_A(x1) = x1 + 5
          eq_A(x1,x2) = 78
          |0|_A = 0
          true_A = 11
          false_A = 4
          le_A(x1,x2) = max{102, x2 + 97}
          app_A(x1,x2) = max{x1 + 62, x2 + 32}
          nil_A = 7
          add_A(x1,x2) = 6
          min_A(x1) = 21
          if_min_A(x1,x2) = 21
          rm_A(x1,x2) = x1 + 8
          if_rm_A(x1,x2,x3) = x2 + 8
          minsort_A(x1,x2) = 13
          if_minsort_A(x1,x2,x3) = 13
          rand_A(x1) = 1
    
      precedence:
      
        eq# = s = eq = |0| = true = le = app = rand > false = nil = add = rm = if_rm > min = if_min > minsort = if_minsort
      
      partial status:
    
        pi(eq#) = []
        pi(s) = [1]
        pi(eq) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(le) = []
        pi(app) = []
        pi(nil) = []
        pi(add) = []
        pi(min) = []
        pi(if_min) = []
        pi(rm) = [1]
        pi(if_rm) = [2]
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

Consider the non-minimal 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            if_min#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,0)) x2 + (0,7)
            false_A() = (11,9)
            add_A(x1,x2) = ((0,1),(1,0)) x1 + x2 + (12,8)
            min#_A(x1) = ((0,1),(0,0)) x1 + (10,16)
            le_A(x1,x2) = ((1,0),(0,0)) x2 + (47,9)
            true_A() = (11,9)
            eq_A(x1,x2) = x1 + ((1,1),(1,0)) x2 + (1,11)
            |0|_A() = (12,9)
            s_A(x1) = x1
            app_A(x1,x2) = x1 + x2 + (0,9)
            nil_A() = (0,1)
            min_A(x1) = ((1,1),(1,1)) x1 + (6,18)
            if_min_A(x1,x2) = ((1,1),(1,1)) x2
            rm_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (0,2)
            if_rm_A(x1,x2,x3) = ((0,0),(0,1)) x2 + x3 + (0,2)
            minsort_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (13,17)
            if_minsort_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (13,17)
            rand_A(x1) = x1 + (1,1)
    
      precedence:
      
        rm = if_rm > if_min# = min# = |0| = app = min = if_min > s > le > false = eq > true > add = nil = minsort = if_minsort = rand
      
      partial status:
    
        pi(if_min#) = []
        pi(false) = []
        pi(add) = []
        pi(min#) = []
        pi(le) = []
        pi(true) = []
        pi(eq) = [1]
        pi(|0|) = []
        pi(s) = [1]
        pi(app) = [2]
        pi(nil) = []
        pi(min) = []
        pi(if_min) = []
        pi(rm) = [2]
        pi(if_rm) = [3]
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            if_min#_A(x1,x2) = (4,4)
            false_A() = (5,3)
            add_A(x1,x2) = (1,0)
            min#_A(x1) = (4,4)
            le_A(x1,x2) = (7,5)
            true_A() = (3,2)
            eq_A(x1,x2) = ((0,1),(1,0)) x1 + (4,4)
            |0|_A() = (3,2)
            s_A(x1) = x1 + (6,4)
            app_A(x1,x2) = (2,0)
            nil_A() = (1,1)
            min_A(x1) = (6,4)
            if_min_A(x1,x2) = (6,4)
            rm_A(x1,x2) = (3,2)
            if_rm_A(x1,x2,x3) = ((0,1),(0,0)) x3 + (3,1)
            minsort_A(x1,x2) = (1,3)
            if_minsort_A(x1,x2,x3) = (1,3)
            rand_A(x1) = (0,0)
    
      precedence:
      
        |0| > true = min = if_min > app > minsort = if_minsort > nil > if_min# = add > min# > false = le = s = rm > eq > if_rm = rand
      
      partial status:
    
        pi(if_min#) = []
        pi(false) = []
        pi(add) = []
        pi(min#) = []
        pi(le) = []
        pi(true) = []
        pi(eq) = []
        pi(|0|) = []
        pi(s) = [1]
        pi(app) = []
        pi(nil) = []
        pi(min) = []
        pi(if_min) = []
        pi(rm) = []
        pi(if_rm) = []
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p2, p3

We remove them from the problem.

-- SCC decomposition.

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

p1: if_min#(false(),add(n,add(m,x))) -> min#(add(m,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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

Consider the non-minimal 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          le#_A(x1,x2) = x2 + 1
          s_A(x1) = x1
          eq_A(x1,x2) = 32
          |0|_A = 32
          true_A = 20
          false_A = 31
          le_A(x1,x2) = max{x1 + 17, x2 + 21}
          app_A(x1,x2) = max{x1, x2}
          nil_A = 0
          add_A(x1,x2) = max{x1 + 12, x2}
          min_A(x1) = max{8, x1 + 4}
          if_min_A(x1,x2) = max{x1 - 6, x2 + 4}
          rm_A(x1,x2) = max{x1 + 12, x2}
          if_rm_A(x1,x2,x3) = max{x1 - 21, x2 + 12, x3}
          minsort_A(x1,x2) = max{7, x1, x2}
          if_minsort_A(x1,x2,x3) = max{12, x2, x3}
          rand_A(x1) = x1 + 2
    
      precedence:
      
        le# = |0| = le = minsort = if_minsort > rm = if_rm > nil > eq = false = app = add = min = if_min > true > rand > s
      
      partial status:
    
        pi(le#) = [2]
        pi(s) = [1]
        pi(eq) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(le) = [1, 2]
        pi(app) = [1, 2]
        pi(nil) = []
        pi(add) = []
        pi(min) = []
        pi(if_min) = []
        pi(rm) = [1]
        pi(if_rm) = [2]
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          le#_A(x1,x2) = 0
          s_A(x1) = x1 + 5
          eq_A(x1,x2) = 103
          |0|_A = 0
          true_A = 36
          false_A = 4
          le_A(x1,x2) = max{x1 + 123, x2 + 141}
          app_A(x1,x2) = max{x1 + 55, x2 + 35}
          nil_A = 85
          add_A(x1,x2) = 34
          min_A(x1) = 64
          if_min_A(x1,x2) = 64
          rm_A(x1,x2) = x1 + 33
          if_rm_A(x1,x2,x3) = x2 + 33
          minsort_A(x1,x2) = 34
          if_minsort_A(x1,x2,x3) = 34
          rand_A(x1) = 1
    
      precedence:
      
        app > le# = s = eq = |0| = true = false = le = nil = rand > min = if_min > rm = if_rm > minsort = if_minsort > add
      
      partial status:
    
        pi(le#) = []
        pi(s) = [1]
        pi(eq) = []
        pi(|0|) = []
        pi(true) = []
        pi(false) = []
        pi(le) = [1, 2]
        pi(app) = []
        pi(nil) = []
        pi(add) = []
        pi(min) = []
        pi(if_min) = []
        pi(rm) = [1]
        pi(if_rm) = [2]
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

Consider the non-minimal 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))
r22: rand(x) -> x
r23: rand(x) -> rand(s(x))

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            app#_A(x1,x2) = x1 + x2 + (4,4)
            add_A(x1,x2) = ((1,1),(1,0)) x1 + x2 + (3,3)
            eq_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (0,4)
            |0|_A() = (0,1)
            true_A() = (0,0)
            s_A(x1) = x1
            false_A() = (1,1)
            le_A(x1,x2) = (2,3)
            app_A(x1,x2) = x1 + x2
            nil_A() = (0,0)
            min_A(x1) = ((1,1),(1,0)) x1 + (5,4)
            if_min_A(x1,x2) = x1 + ((1,1),(1,0)) x2 + (1,1)
            rm_A(x1,x2) = ((1,0),(0,0)) x1 + x2
            if_rm_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3
            minsort_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (4,1)
            if_minsort_A(x1,x2,x3) = ((1,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (4,1)
            rand_A(x1) = x1 + (1,0)
    
      precedence:
      
        app > rm = if_rm > minsort = if_minsort > add > min = if_min > nil > eq = s > le > rand > |0| > false > true > app#
      
      partial status:
    
        pi(app#) = [1, 2]
        pi(add) = []
        pi(eq) = []
        pi(|0|) = []
        pi(true) = []
        pi(s) = [1]
        pi(false) = []
        pi(le) = []
        pi(app) = [1, 2]
        pi(nil) = []
        pi(min) = []
        pi(if_min) = [1]
        pi(rm) = []
        pi(if_rm) = []
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            app#_A(x1,x2) = (2,1)
            add_A(x1,x2) = (2,2)
            eq_A(x1,x2) = (2,2)
            |0|_A() = (0,0)
            true_A() = (2,1)
            s_A(x1) = (0,1)
            false_A() = (1,2)
            le_A(x1,x2) = (3,2)
            app_A(x1,x2) = ((1,0),(1,1)) x1 + (3,1)
            nil_A() = (0,0)
            min_A(x1) = (4,2)
            if_min_A(x1,x2) = (5,2)
            rm_A(x1,x2) = (3,2)
            if_rm_A(x1,x2,x3) = (3,2)
            minsort_A(x1,x2) = (0,0)
            if_minsort_A(x1,x2,x3) = (0,0)
            rand_A(x1) = (1,1)
    
      precedence:
      
        rand > app > if_min > eq = true > false = le > add = min = rm = if_rm > |0| > app# = nil > s = minsort = if_minsort
      
      partial status:
    
        pi(app#) = []
        pi(add) = []
        pi(eq) = []
        pi(|0|) = []
        pi(true) = []
        pi(s) = []
        pi(false) = []
        pi(le) = []
        pi(app) = [1]
        pi(nil) = []
        pi(min) = []
        pi(if_min) = []
        pi(rm) = []
        pi(if_rm) = []
        pi(minsort) = []
        pi(if_minsort) = []
        pi(rand) = []
    

The next rules are strictly ordered:

  p1

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