/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES

We split firstr-order part and higher-order part, and do modular checking by a general modularity.

******** FO SN check ********
Check SN using NaTT (Nagoya Termination Tool)
Input TRS:
    1: eq(0(),0()) -> true()
    2: eq(0(),s(X)) -> false()
    3: eq(s(Y),0()) -> false()
    4: eq(s(V),s(U)) -> eq(V,U)
    5: le(0(),W) -> true()
    6: le(s(P),0()) -> false()
    7: le(s(Y1),s(X1)) -> le(Y1,X1)
    8: min(cons(0(),nil())) -> 0()
    9: min(cons(s(U1),nil())) -> s(U1)
    10: min(cons(W1,cons(V1,P1))) -> ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
    11: ifxb6220min(true(),cons(Y2,cons(X2,U2))) -> min(cons(Y2,U2))
    12: ifxb6220min(false(),cons(W2,cons(V2,P2))) -> min(cons(V2,P2))
    13: replace(Y3,X3,nil()) -> nil()
    14: replace(W3,V3,cons(U3,P3)) -> ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
    15: ifxb6220replace(true(),U4,Y4,cons(X4,V4)) -> cons(Y4,V4)
    16: ifxb6220replace(false(),X5,P4,cons(W4,Y5)) -> cons(W4,replace(X5,P4,Y5))
    17: sort(nil()) -> nil()
    18: sort(cons(U5,V5)) -> cons(min(cons(U5,V5)),sort(replace(min(cons(U5,V5)),U5,V5)))
    19: _(X1,X2) -> X1
    20: _(X1,X2) -> X2
Number of strict rules: 20
Direct POLO(bPol) ... failed.
Uncurrying min
1: eq(0(),0()) -> true()
2: eq(0(),s(X)) -> false()
3: eq(s(Y),0()) -> false()
4: eq(s(V),s(U)) -> eq(V,U)
5: le(0(),W) -> true()
6: le(s(P),0()) -> false()
7: le(s(Y1),s(X1)) -> le(Y1,X1)
8: min^1_cons(0(),nil()) -> 0()
9: min^1_cons(s(U1),nil()) -> s(U1)
10: min^1_cons(W1,cons(V1,P1)) -> ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
11: ifxb6220min(true(),cons(Y2,cons(X2,U2))) -> min^1_cons(Y2,U2)
12: ifxb6220min(false(),cons(W2,cons(V2,P2))) -> min^1_cons(V2,P2)
13: replace(Y3,X3,nil()) -> nil()
14: replace(W3,V3,cons(U3,P3)) -> ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
15: ifxb6220replace(true(),U4,Y4,cons(X4,V4)) -> cons(Y4,V4)
16: ifxb6220replace(false(),X5,P4,cons(W4,Y5)) -> cons(W4,replace(X5,P4,Y5))
17: sort(nil()) -> nil()
18: sort(cons(U5,V5)) -> cons(min^1_cons(U5,V5),sort(replace(min^1_cons(U5,V5),U5,V5)))
19: _(X1,X2) -> X1
20: _(X1,X2) -> X2
21: min(cons(_1,_2)) ->= min^1_cons(_1,_2)
Number of strict rules: 20
Direct POLO(bPol) ... failed.
Dependency Pairs:
   #1: #ifxb6220min(true(),cons(Y2,cons(X2,U2))) -> #min^1_cons(Y2,U2)
   #2: #ifxb6220min(false(),cons(W2,cons(V2,P2))) -> #min^1_cons(V2,P2)
   #3: #replace(W3,V3,cons(U3,P3)) -> #ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
   #4: #replace(W3,V3,cons(U3,P3)) -> #eq(W3,U3)
   #5: #le(s(Y1),s(X1)) -> #le(Y1,X1)
   #6: #min^1_cons(W1,cons(V1,P1)) -> #ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
   #7: #min^1_cons(W1,cons(V1,P1)) -> #le(W1,V1)
   #8: #min(cons(_1,_2)) ->? #min^1_cons(_1,_2)
   #9: #ifxb6220replace(false(),X5,P4,cons(W4,Y5)) -> #replace(X5,P4,Y5)
   #10: #eq(s(V),s(U)) -> #eq(V,U)
   #11: #sort(cons(U5,V5)) -> #min^1_cons(U5,V5)
   #12: #sort(cons(U5,V5)) -> #sort(replace(min^1_cons(U5,V5),U5,V5))
   #13: #sort(cons(U5,V5)) -> #replace(min^1_cons(U5,V5),U5,V5)
   #14: #sort(cons(U5,V5)) -> #min^1_cons(U5,V5)
Number of SCCs: 5, DPs: 8
  SCC { #10 }
POLO(Sum)... succeeded.
      le	w: 0
      s 	w: x1 + 1
      #le	w: 0
      ifxb6220replace	w: 0
      #ifxb6220min	w: 0
      min^1_cons	w: 0
      eq	w: 0
      false	w: 0
      #ifxb6220replace	w: 0
      #min	w: 0
      _ 	w: 0
      true	w: 0
      #eq	w: x1 + x2
      #sort	w: 0
      ifxb6220min	w: 0
      0 	w: 0
      nil	w: 0
      sort	w: 0
      #replace	w: 0
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: 0
      replace	w: 0
    USABLE RULES: { }
    Removed DPs: #10
Number of SCCs: 4, DPs: 7
  SCC { #5 }
POLO(Sum)... succeeded.
      le	w: 0
      s 	w: x1 + 1
      #le	w: x2
      ifxb6220replace	w: 0
      #ifxb6220min	w: 0
      min^1_cons	w: 0
      eq	w: 0
      false	w: 0
      #ifxb6220replace	w: 0
      #min	w: 0
      _ 	w: 0
      true	w: 0
      #eq	w: 0
      #sort	w: 0
      ifxb6220min	w: 0
      0 	w: 0
      nil	w: 0
      sort	w: 0
      #replace	w: 0
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: 0
      replace	w: 0
    USABLE RULES: { }
    Removed DPs: #5
Number of SCCs: 3, DPs: 6
  SCC { #12 }
POLO(Sum)... succeeded.
      le	w: x2 + 1
      s 	w: 1
      #le	w: 0
      ifxb6220replace	w: x2 + x4 + 1797
      #ifxb6220min	w: 0
      min^1_cons	w: 1
      eq	w: x1 + 1
      false	w: 3
      #ifxb6220replace	w: 0
      #min	w: 0
      _ 	w: 0
      true	w: 3
      #eq	w: 0
      #sort	w: x1
      ifxb6220min	w: 1
      0 	w: 1
      nil	w: 8366
      sort	w: 0
      #replace	w: 0
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: x2 + 1799
      replace	w: x1 + x3 + 1797
    USABLE RULES: { 8..16 }
    Removed DPs: #12
Number of SCCs: 2, DPs: 5
  SCC { #3 #9 }
POLO(Sum)... succeeded.
      le	w: x2 + 1
      s 	w: 1
      #le	w: 0
      ifxb6220replace	w: x2 + x4 + 1797
      #ifxb6220min	w: 0
      min^1_cons	w: 1
      eq	w: 2
      false	w: 2
      #ifxb6220replace	w: x1 + x4
      #min	w: 0
      _ 	w: 0
      true	w: 2
      #eq	w: 0
      #sort	w: x1
      ifxb6220min	w: 1
      0 	w: 0
      nil	w: 8366
      sort	w: 0
      #replace	w: x3 + 3
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: x2 + 1799
      replace	w: x1 + x3 + 1797
    USABLE RULES: { 1..4 8..16 }
    Removed DPs: #3 #9
Number of SCCs: 1, DPs: 3
  SCC { #1 #2 #6 }
POLO(Sum)... succeeded.
      le	w: x1
      s 	w: x1 + 1
      #le	w: 0
      ifxb6220replace	w: x2 + x3 + 3596
      #ifxb6220min	w: x2
      min^1_cons	w: 1
      eq	w: x2 + 2
      false	w: 0
      #ifxb6220replace	w: x1
      #min	w: 0
      _ 	w: 0
      true	w: 1
      #eq	w: 0
      #sort	w: 0
      ifxb6220min	w: x1 + 1
      0 	w: 1
      nil	w: 8366
      sort	w: 0
      #replace	w: 3
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: x1 + x2 + 3597
      cons	w: x1 + x2 + 1799
      replace	w: x1 + x2 + x3 + 1797
    USABLE RULES: { 1..8 13 }
    Removed DPs: #1 #2 #6
Number of SCCs: 0, DPs: 0

...
Input TRS:
    1: eq(0(),0()) -> true()
    2: eq(0(),s(X)) -> false()
    3: eq(s(Y),0()) -> false()
    4: eq(s(V),s(U)) -> eq(V,U)
    5: le(0(),W) -> true()
    6: le(s(P),0()) -> false()
    7: le(s(Y1),s(X1)) -> le(Y1,X1)
    8: min(cons(0(),nil())) -> 0()
    9: min(cons(s(U1),nil())) -> s(U1)
    10: min(cons(W1,cons(V1,P1))) -> ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
    11: ifxb6220min(true(),cons(Y2,cons(X2,U2))) -> min(cons(Y2,U2))
    12: ifxb6220min(false(),cons(W2,cons(V2,P2))) -> min(cons(V2,P2))
    13: replace(Y3,X3,nil()) -> nil()
    14: replace(W3,V3,cons(U3,P3)) -> ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
    15: ifxb6220replace(true(),U4,Y4,cons(X4,V4)) -> cons(Y4,V4)
    16: ifxb6220replace(false(),X5,P4,cons(W4,Y5)) -> cons(W4,replace(X5,P4,Y5))
    17: sort(nil()) -> nil()
    18: sort(cons(U5,V5)) -> cons(min(cons(U5,V5)),sort(replace(min(cons(U5,V5)),U5,V5)))
    19: _(X1,X2) -> X1
    20: _(X1,X2) -> X2
Number of strict rules: 20
Direct POLO(bPol) ... failed.
Uncurrying min
1: eq(0(),0()) -> true()
2: eq(0(),s(X)) -> false()
3: eq(s(Y),0()) -> false()
4: eq(s(V),s(U)) -> eq(V,U)
5: le(0(),W) -> true()
6: le(s(P),0()) -> false()
7: le(s(Y1),s(X1)) -> le(Y1,X1)
8: min^1_cons(0(),nil()) -> 0()
9: min^1_cons(s(U1),nil()) -> s(U1)
10: min^1_cons(W1,cons(V1,P1)) -> ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
11: ifxb6220min(true(),cons(Y2,cons(X2,U2))) -> min^1_cons(Y2,U2)
12: ifxb6220min(false(),cons(W2,cons(V2,P2))) -> min^1_cons(V2,P2)
13: replace(Y3,X3,nil()) -> nil()
14: replace(W3,V3,cons(U3,P3)) -> ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
15: ifxb6220replace(true(),U4,Y4,cons(X4,V4)) -> cons(Y4,V4)
16: ifxb6220replace(false(),X5,P4,cons(W4,Y5)) -> cons(W4,replace(X5,P4,Y5))
17: sort(nil()) -> nil()
18: sort(cons(U5,V5)) -> cons(min^1_cons(U5,V5),sort(replace(min^1_cons(U5,V5),U5,V5)))
19: _(X1,X2) -> X1
20: _(X1,X2) -> X2
21: min(cons(_1,_2)) ->= min^1_cons(_1,_2)
Number of strict rules: 20
Direct POLO(bPol) ... failed.
Dependency Pairs:
   #1: #ifxb6220min(true(),cons(Y2,cons(X2,U2))) -> #min^1_cons(Y2,U2)
   #2: #ifxb6220min(false(),cons(W2,cons(V2,P2))) -> #min^1_cons(V2,P2)
   #3: #replace(W3,V3,cons(U3,P3)) -> #ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
   #4: #replace(W3,V3,cons(U3,P3)) -> #eq(W3,U3)
   #5: #le(s(Y1),s(X1)) -> #le(Y1,X1)
   #6: #min^1_cons(W1,cons(V1,P1)) -> #ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
   #7: #min^1_cons(W1,cons(V1,P1)) -> #le(W1,V1)
   #8: #min(cons(_1,_2)) ->? #min^1_cons(_1,_2)
   #9: #ifxb6220replace(false(),X5,P4,cons(W4,Y5)) -> #replace(X5,P4,Y5)
   #10: #eq(s(V),s(U)) -> #eq(V,U)
   #11: #sort(cons(U5,V5)) -> #min^1_cons(U5,V5)
   #12: #sort(cons(U5,V5)) -> #sort(replace(min^1_cons(U5,V5),U5,V5))
   #13: #sort(cons(U5,V5)) -> #replace(min^1_cons(U5,V5),U5,V5)
   #14: #sort(cons(U5,V5)) -> #min^1_cons(U5,V5)
Number of SCCs: 5, DPs: 8
  SCC { #10 }
POLO(Sum)... succeeded.
      le	w: 0
      s 	w: x1 + 1
      #le	w: 0
      ifxb6220replace	w: 0
      #ifxb6220min	w: 0
      min^1_cons	w: 0
      eq	w: 0
      false	w: 0
      #ifxb6220replace	w: 0
      #min	w: 0
      _ 	w: 0
      true	w: 0
      #eq	w: x1 + x2
      #sort	w: 0
      ifxb6220min	w: 0
      0 	w: 0
      nil	w: 0
      sort	w: 0
      #replace	w: 0
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: 0
      replace	w: 0
    USABLE RULES: { }
    Removed DPs: #10
Number of SCCs: 4, DPs: 7
  SCC { #5 }
POLO(Sum)... succeeded.
      le	w: 0
      s 	w: x1 + 1
      #le	w: x2
      ifxb6220replace	w: 0
      #ifxb6220min	w: 0
      min^1_cons	w: 0
      eq	w: 0
      false	w: 0
      #ifxb6220replace	w: 0
      #min	w: 0
      _ 	w: 0
      true	w: 0
      #eq	w: 0
      #sort	w: 0
      ifxb6220min	w: 0
      0 	w: 0
      nil	w: 0
      sort	w: 0
      #replace	w: 0
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: 0
      replace	w: 0
    USABLE RULES: { }
    Removed DPs: #5
Number of SCCs: 3, DPs: 6
  SCC { #12 }
POLO(Sum)... succeeded.
      le	w: x2 + 1
      s 	w: 1
      #le	w: 0
      ifxb6220replace	w: x2 + x4 + 1797
      #ifxb6220min	w: 0
      min^1_cons	w: 1
      eq	w: x1 + 1
      false	w: 3
      #ifxb6220replace	w: 0
      #min	w: 0
      _ 	w: 0
      true	w: 3
      #eq	w: 0
      #sort	w: x1
      ifxb6220min	w: 1
      0 	w: 1
      nil	w: 8366
      sort	w: 0
      #replace	w: 0
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: x2 + 1799
      replace	w: x1 + x3 + 1797
    USABLE RULES: { 8..16 }
    Removed DPs: #12
Number of SCCs: 2, DPs: 5
  SCC { #3 #9 }
POLO(Sum)... succeeded.
      le	w: x2 + 1
      s 	w: 1
      #le	w: 0
      ifxb6220replace	w: x2 + x4 + 1797
      #ifxb6220min	w: 0
      min^1_cons	w: 1
      eq	w: 2
      false	w: 2
      #ifxb6220replace	w: x1 + x4
      #min	w: 0
      _ 	w: 0
      true	w: 2
      #eq	w: 0
      #sort	w: x1
      ifxb6220min	w: 1
      0 	w: 0
      nil	w: 8366
      sort	w: 0
      #replace	w: x3 + 3
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: 0
      cons	w: x2 + 1799
      replace	w: x1 + x3 + 1797
    USABLE RULES: { 1..4 8..16 }
    Removed DPs: #3 #9
Number of SCCs: 1, DPs: 3
  SCC { #1 #2 #6 }
POLO(Sum)... succeeded.
      le	w: x1
      s 	w: x1 + 1
      #le	w: 0
      ifxb6220replace	w: x2 + x3 + 3596
      #ifxb6220min	w: x2
      min^1_cons	w: 1
      eq	w: x2 + 2
      false	w: 0
      #ifxb6220replace	w: x1
      #min	w: 0
      _ 	w: 0
      true	w: 1
      #eq	w: 0
      #sort	w: 0
      ifxb6220min	w: x1 + 1
      0 	w: 1
      nil	w: 8366
      sort	w: 0
      #replace	w: 3
      #_ 	w: 0
      min	w: 0
      #min^1_cons	w: x1 + x2 + 3597
      cons	w: x1 + x2 + 1799
      replace	w: x1 + x2 + x3 + 1797
    USABLE RULES: { 1..8 13 }
    Removed DPs: #1 #2 #6
Number of SCCs: 0, DPs: 0
>>YES

******** Signature ********
 map : ((b -> b),c) -> c
 nil : c
 cons : (b,c) -> c
 filter : ((b -> a),c) -> c
 filter2 : (a,(b -> a),b,c) -> c
 true : a
 false : a
******** Computation rules ********
 (19)  map(I5,nil) => nil
 (20)  map(J5,cons(X6,Y6)) => cons(J5[X6],map(J5,Y6))
 (21)  filter(G6,nil) => nil
 (22)  filter(H6,cons(W6,P6)) => filter2(H6[W6],H6,W6,P6)
 (23)  filter2(true,F7,Y7,U7) => cons(Y7,filter(F7,U7))
 (24)  filter2(false,H7,W7,P7) => filter(H7,P7)


******** General Schema criterion ********
Found constructors: 0, cons, false, nil, s, true
Checking type order >>OK
Checking positivity of constructors >>OK
Checking function dependency >>Regared as equal: filter2, filter
Checking  (1) eq(0,0) => true
 (fun eq>true) >>True
Checking  (2) eq(0,s(X)) => false
 (fun eq>false) >>True
Checking  (3) eq(s(Y),0) => false
 (fun eq>false) >>True
Checking  (4) eq(s(V),s(U)) => eq(V,U)
 (fun eq=eq) subterm comparison of args w. LR LR
 (meta V)[is acc in s(V),s(U)] [is positive in s(V)] [is acc in V] 
 (meta U)[is acc in s(V),s(U)] [is positive in s(V)] [is positive in s(U)] [is acc in U]  >>True
Checking  (5) le(0,W) => true
 (fun le>true) >>True
Checking  (6) le(s(P),0) => false
 (fun le>false) >>True
Checking  (7) le(s(Y1),s(X1)) => le(Y1,X1)
 (fun le=le) subterm comparison of args w. LR LR
 (meta Y1)[is acc in s(Y1),s(X1)] [is positive in s(Y1)] [is acc in Y1] 
 (meta X1)[is acc in s(Y1),s(X1)] [is positive in s(Y1)] [is positive in s(X1)] [is acc in X1]  >>True
Checking  (8) min(cons(0,nil)) => 0
 (fun min>0) >>True
Checking  (9) min(cons(s(U1),nil)) => s(U1)
 (fun min>s)
 (meta U1)[is acc in cons(s(U1),nil)] [is positive in cons(s(U1),nil)] [is positive in s(U1)] [is acc in U1]  >>True
Checking  (10) min(cons(W1,cons(V1,P1))) => ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
 (fun min>ifxb6220min)
 (fun min>le)
 (meta W1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is acc in W1] 
 (meta V1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in V1] 
 (fun min>cons)
 (meta W1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is acc in W1] 
 (fun min>cons)
 (meta V1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in V1] 
 (meta P1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in P1]  >>True
Checking  (11) ifxb6220min(true,cons(Y2,cons(X2,U2))) => min(cons(Y2,U2))
 (fun ifxb6220min>min)
 (fun ifxb6220min>cons)
 (meta Y2)[is acc in true,cons(Y2,cons(X2,U2))] [is positive in true] [is positive in cons(Y2,cons(X2,U2))] [is acc in Y2] 
 (meta U2)[is acc in true,cons(Y2,cons(X2,U2))] [is positive in true] [is positive in cons(Y2,cons(X2,U2))] [is positive in cons(X2,U2)] [is acc in U2]  >>True
Checking  (12) ifxb6220min(false,cons(W2,cons(V2,P2))) => min(cons(V2,P2))
 (fun ifxb6220min>min)
 (fun ifxb6220min>cons)
 (meta V2)[is acc in false,cons(W2,cons(V2,P2))] [is positive in false] [is positive in cons(W2,cons(V2,P2))] [is positive in cons(V2,P2)] [is acc in V2] 
 (meta P2)[is acc in false,cons(W2,cons(V2,P2))] [is positive in false] [is positive in cons(W2,cons(V2,P2))] [is positive in cons(V2,P2)] [is acc in P2]  >>True
Checking  (13) replace(Y3,X3,nil) => nil
 (fun replace>nil) >>True
Checking  (14) replace(W3,V3,cons(U3,P3)) => ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
 (fun replace>ifxb6220replace)
 (fun replace>eq)
 (meta W3)[is acc in W3,V3,cons(U3,P3)] [is acc in W3] 
 (meta U3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in U3] 
 (meta W3)[is acc in W3,V3,cons(U3,P3)] [is acc in W3] 
 (meta V3)[is acc in W3,V3,cons(U3,P3)] [is acc in V3] 
 (fun replace>cons)
 (meta U3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in U3] 
 (meta P3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in P3]  >>True
Checking  (15) ifxb6220replace(true,U4,Y4,cons(X4,V4)) => cons(Y4,V4)
 (fun ifxb6220replace>cons)
 (meta Y4)[is acc in true,U4,Y4,cons(X4,V4)] [is positive in true] [is acc in Y4] 
 (meta V4)[is acc in true,U4,Y4,cons(X4,V4)] [is positive in true] [is positive in cons(X4,V4)] [is acc in V4]  >>True
Checking  (16) ifxb6220replace(false,X5,P4,cons(W4,Y5)) => cons(W4,replace(X5,P4,Y5))
 (fun ifxb6220replace>cons)
 (meta W4)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is positive in cons(W4,Y5)] [is acc in W4] 
 (fun ifxb6220replace>replace)
 (meta X5)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is acc in X5] 
 (meta P4)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is acc in P4] 
 (meta Y5)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is positive in cons(W4,Y5)] [is acc in Y5]  >>True
Checking  (17) sort(nil) => nil
 (fun sort>nil) >>True
Checking  (18) sort(cons(U5,V5)) => cons(min(cons(U5,V5)),sort(replace(min(cons(U5,V5)),U5,V5)))
 (fun sort>cons)
 (fun sort>min)
 (fun sort>cons)
 (meta U5)[is acc in cons(U5,V5)] [is positive in cons(U5,V5)] [is acc in U5] 
 (meta V5)[is acc in cons(U5,V5)] [is positive in cons(U5,V5)] [is acc in V5] 
 (fun sort=sort) subterm comparison of args w. LR LR >>False

Try again using status RL
Checking  (1) eq(0,0) => true
 (fun eq>true) >>True
Checking  (2) eq(0,s(X)) => false
 (fun eq>false) >>True
Checking  (3) eq(s(Y),0) => false
 (fun eq>false) >>True
Checking  (4) eq(s(V),s(U)) => eq(V,U)
 (fun eq=eq) subterm comparison of args w. RL RL
 (meta V)[is acc in s(V),s(U)] [is positive in s(V)] [is acc in V] 
 (meta U)[is acc in s(V),s(U)] [is positive in s(V)] [is positive in s(U)] [is acc in U]  >>True
Checking  (5) le(0,W) => true
 (fun le>true) >>True
Checking  (6) le(s(P),0) => false
 (fun le>false) >>True
Checking  (7) le(s(Y1),s(X1)) => le(Y1,X1)
 (fun le=le) subterm comparison of args w. RL RL
 (meta Y1)[is acc in s(Y1),s(X1)] [is positive in s(Y1)] [is acc in Y1] 
 (meta X1)[is acc in s(Y1),s(X1)] [is positive in s(Y1)] [is positive in s(X1)] [is acc in X1]  >>True
Checking  (8) min(cons(0,nil)) => 0
 (fun min>0) >>True
Checking  (9) min(cons(s(U1),nil)) => s(U1)
 (fun min>s)
 (meta U1)[is acc in cons(s(U1),nil)] [is positive in cons(s(U1),nil)] [is positive in s(U1)] [is acc in U1]  >>True
Checking  (10) min(cons(W1,cons(V1,P1))) => ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
 (fun min>ifxb6220min)
 (fun min>le)
 (meta W1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is acc in W1] 
 (meta V1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in V1] 
 (fun min>cons)
 (meta W1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is acc in W1] 
 (fun min>cons)
 (meta V1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in V1] 
 (meta P1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in P1]  >>True
Checking  (11) ifxb6220min(true,cons(Y2,cons(X2,U2))) => min(cons(Y2,U2))
 (fun ifxb6220min>min)
 (fun ifxb6220min>cons)
 (meta Y2)[is acc in true,cons(Y2,cons(X2,U2))] [is positive in true] [is positive in cons(Y2,cons(X2,U2))] [is acc in Y2] 
 (meta U2)[is acc in true,cons(Y2,cons(X2,U2))] [is positive in true] [is positive in cons(Y2,cons(X2,U2))] [is positive in cons(X2,U2)] [is acc in U2]  >>True
Checking  (12) ifxb6220min(false,cons(W2,cons(V2,P2))) => min(cons(V2,P2))
 (fun ifxb6220min>min)
 (fun ifxb6220min>cons)
 (meta V2)[is acc in false,cons(W2,cons(V2,P2))] [is positive in false] [is positive in cons(W2,cons(V2,P2))] [is positive in cons(V2,P2)] [is acc in V2] 
 (meta P2)[is acc in false,cons(W2,cons(V2,P2))] [is positive in false] [is positive in cons(W2,cons(V2,P2))] [is positive in cons(V2,P2)] [is acc in P2]  >>True
Checking  (13) replace(Y3,X3,nil) => nil
 (fun replace>nil) >>True
Checking  (14) replace(W3,V3,cons(U3,P3)) => ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
 (fun replace>ifxb6220replace)
 (fun replace>eq)
 (meta W3)[is acc in W3,V3,cons(U3,P3)] [is acc in W3] 
 (meta U3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in U3] 
 (meta W3)[is acc in W3,V3,cons(U3,P3)] [is acc in W3] 
 (meta V3)[is acc in W3,V3,cons(U3,P3)] [is acc in V3] 
 (fun replace>cons)
 (meta U3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in U3] 
 (meta P3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in P3]  >>True
Checking  (15) ifxb6220replace(true,U4,Y4,cons(X4,V4)) => cons(Y4,V4)
 (fun ifxb6220replace>cons)
 (meta Y4)[is acc in true,U4,Y4,cons(X4,V4)] [is positive in true] [is acc in Y4] 
 (meta V4)[is acc in true,U4,Y4,cons(X4,V4)] [is positive in true] [is positive in cons(X4,V4)] [is acc in V4]  >>True
Checking  (16) ifxb6220replace(false,X5,P4,cons(W4,Y5)) => cons(W4,replace(X5,P4,Y5))
 (fun ifxb6220replace>cons)
 (meta W4)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is positive in cons(W4,Y5)] [is acc in W4] 
 (fun ifxb6220replace>replace)
 (meta X5)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is acc in X5] 
 (meta P4)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is acc in P4] 
 (meta Y5)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is positive in cons(W4,Y5)] [is acc in Y5]  >>True
Checking  (17) sort(nil) => nil
 (fun sort>nil) >>True
Checking  (18) sort(cons(U5,V5)) => cons(min(cons(U5,V5)),sort(replace(min(cons(U5,V5)),U5,V5)))
 (fun sort>cons)
 (fun sort>min)
 (fun sort>cons)
 (meta U5)[is acc in cons(U5,V5)] [is positive in cons(U5,V5)] [is acc in U5] 
 (meta V5)[is acc in cons(U5,V5)] [is positive in cons(U5,V5)] [is acc in V5] 
 (fun sort=sort) subterm comparison of args w. RL RL >>False

Try again using status Mul
Checking  (1) eq(0,0) => true
 (fun eq>true) >>True
Checking  (2) eq(0,s(X)) => false
 (fun eq>false) >>True
Checking  (3) eq(s(Y),0) => false
 (fun eq>false) >>True
Checking  (4) eq(s(V),s(U)) => eq(V,U)
 (fun eq=eq) subterm comparison of args w. Mul Mul
 (meta V)[is acc in s(V),s(U)] [is positive in s(V)] [is acc in V] 
 (meta U)[is acc in s(V),s(U)] [is positive in s(V)] [is positive in s(U)] [is acc in U]  >>True
Checking  (5) le(0,W) => true
 (fun le>true) >>True
Checking  (6) le(s(P),0) => false
 (fun le>false) >>True
Checking  (7) le(s(Y1),s(X1)) => le(Y1,X1)
 (fun le=le) subterm comparison of args w. Mul Mul
 (meta Y1)[is acc in s(Y1),s(X1)] [is positive in s(Y1)] [is acc in Y1] 
 (meta X1)[is acc in s(Y1),s(X1)] [is positive in s(Y1)] [is positive in s(X1)] [is acc in X1]  >>True
Checking  (8) min(cons(0,nil)) => 0
 (fun min>0) >>True
Checking  (9) min(cons(s(U1),nil)) => s(U1)
 (fun min>s)
 (meta U1)[is acc in cons(s(U1),nil)] [is positive in cons(s(U1),nil)] [is positive in s(U1)] [is acc in U1]  >>True
Checking  (10) min(cons(W1,cons(V1,P1))) => ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
 (fun min>ifxb6220min)
 (fun min>le)
 (meta W1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is acc in W1] 
 (meta V1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in V1] 
 (fun min>cons)
 (meta W1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is acc in W1] 
 (fun min>cons)
 (meta V1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in V1] 
 (meta P1)[is acc in cons(W1,cons(V1,P1))] [is positive in cons(W1,cons(V1,P1))] [is positive in cons(V1,P1)] [is acc in P1]  >>True
Checking  (11) ifxb6220min(true,cons(Y2,cons(X2,U2))) => min(cons(Y2,U2))
 (fun ifxb6220min>min)
 (fun ifxb6220min>cons)
 (meta Y2)[is acc in true,cons(Y2,cons(X2,U2))] [is positive in true] [is positive in cons(Y2,cons(X2,U2))] [is acc in Y2] 
 (meta U2)[is acc in true,cons(Y2,cons(X2,U2))] [is positive in true] [is positive in cons(Y2,cons(X2,U2))] [is positive in cons(X2,U2)] [is acc in U2]  >>True
Checking  (12) ifxb6220min(false,cons(W2,cons(V2,P2))) => min(cons(V2,P2))
 (fun ifxb6220min>min)
 (fun ifxb6220min>cons)
 (meta V2)[is acc in false,cons(W2,cons(V2,P2))] [is positive in false] [is positive in cons(W2,cons(V2,P2))] [is positive in cons(V2,P2)] [is acc in V2] 
 (meta P2)[is acc in false,cons(W2,cons(V2,P2))] [is positive in false] [is positive in cons(W2,cons(V2,P2))] [is positive in cons(V2,P2)] [is acc in P2]  >>True
Checking  (13) replace(Y3,X3,nil) => nil
 (fun replace>nil) >>True
Checking  (14) replace(W3,V3,cons(U3,P3)) => ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
 (fun replace>ifxb6220replace)
 (fun replace>eq)
 (meta W3)[is acc in W3,V3,cons(U3,P3)] [is acc in W3] 
 (meta U3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in U3] 
 (meta W3)[is acc in W3,V3,cons(U3,P3)] [is acc in W3] 
 (meta V3)[is acc in W3,V3,cons(U3,P3)] [is acc in V3] 
 (fun replace>cons)
 (meta U3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in U3] 
 (meta P3)[is acc in W3,V3,cons(U3,P3)] [is positive in cons(U3,P3)] [is acc in P3]  >>True
Checking  (15) ifxb6220replace(true,U4,Y4,cons(X4,V4)) => cons(Y4,V4)
 (fun ifxb6220replace>cons)
 (meta Y4)[is acc in true,U4,Y4,cons(X4,V4)] [is positive in true] [is acc in Y4] 
 (meta V4)[is acc in true,U4,Y4,cons(X4,V4)] [is positive in true] [is positive in cons(X4,V4)] [is acc in V4]  >>True
Checking  (16) ifxb6220replace(false,X5,P4,cons(W4,Y5)) => cons(W4,replace(X5,P4,Y5))
 (fun ifxb6220replace>cons)
 (meta W4)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is positive in cons(W4,Y5)] [is acc in W4] 
 (fun ifxb6220replace>replace)
 (meta X5)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is acc in X5] 
 (meta P4)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is acc in P4] 
 (meta Y5)[is acc in false,X5,P4,cons(W4,Y5)] [is positive in false] [is positive in cons(W4,Y5)] [is acc in Y5]  >>True
Checking  (17) sort(nil) => nil
 (fun sort>nil) >>True
Checking  (18) sort(cons(U5,V5)) => cons(min(cons(U5,V5)),sort(replace(min(cons(U5,V5)),U5,V5)))
 (fun sort>cons)
 (fun sort>min)
 (fun sort>cons)
 (meta U5)[is acc in cons(U5,V5)] [is positive in cons(U5,V5)] [is acc in U5] 
 (meta V5)[is acc in cons(U5,V5)] [is positive in cons(U5,V5)] [is acc in V5] 
 (fun sort=sort) subterm comparison of args w. Mul Mul >>False
Found constructors: nil, cons, true, false
Checking type order >>OK
Checking positivity of constructors >>OK
Checking function dependency >>Regared as equal: filter2, filter
Checking  (19) map(I5,nil) => nil
 (fun map>nil) >>True
Checking  (20) map(J5,cons(X6,Y6)) => cons(J5[X6],map(J5,Y6))
 (fun map>cons)
 (meta J5)[is acc in J5,cons(X6,Y6)] [is acc in J5] 
 (meta X6)[is acc in J5,cons(X6,Y6)] [is positive in cons(X6,Y6)] [is acc in X6] 
 (fun map=map) subterm comparison of args w. LR LR
 (meta J5)[is acc in J5,cons(X6,Y6)] [is acc in J5] 
 (meta Y6)[is acc in J5,cons(X6,Y6)] [is positive in cons(X6,Y6)] [is acc in Y6]  >>True
Checking  (21) filter(G6,nil) => nil
 (fun filter>nil) >>True
Checking  (22) filter(H6,cons(W6,P6)) => filter2(H6[W6],H6,W6,P6)
 (fun filter=filter2) subterm comparison of args w. Arg [1,2] Arg [2,4,3,1]
 (meta H6)[is acc in H6,cons(W6,P6)] [is acc in H6] 
 (meta W6)[is acc in H6,cons(W6,P6)] [is positive in cons(W6,P6)] [is acc in W6] 
 (meta H6)[is acc in H6,cons(W6,P6)] [is acc in H6] 
 (meta W6)[is acc in H6,cons(W6,P6)] [is positive in cons(W6,P6)] [is acc in W6] 
 (meta P6)[is acc in H6,cons(W6,P6)] [is positive in cons(W6,P6)] [is acc in P6]  >>True
Checking  (23) filter2(true,F7,Y7,U7) => cons(Y7,filter(F7,U7))
 (fun filter2>cons)
 (meta Y7)[is acc in true,F7,Y7,U7] [is positive in true] [is acc in Y7] 
 (fun filter2=filter) subterm comparison of args w. Arg [2,4,3,1] Arg [1,2]
 (meta F7)[is acc in true,F7,Y7,U7] [is positive in true] [is acc in F7] 
 (meta U7)[is acc in true,F7,Y7,U7] [is positive in true] [is acc in U7]  >>True
Checking  (24) filter2(false,H7,W7,P7) => filter(H7,P7)
 (fun filter2=filter) subterm comparison of args w. Arg [2,4,3,1] Arg [1,2]
 (meta H7)[is acc in false,H7,W7,P7] [is positive in false] [is acc in H7] 
 (meta P7)[is acc in false,H7,W7,P7] [is positive in false] [is acc in P7]  >>True
#SN!
******** Signature ********
 0 : b
 cons : (b,c) -> c
 eq : (b,b) -> a
 false : a
 filter : ((b -> a),c) -> c
 filter2 : (a,(b -> a),b,c) -> c
 ifxb6220min : (a,c) -> b
 ifxb6220replace : (a,b,b,c) -> c
 le : (b,b) -> a
 map : ((b -> b),c) -> c
 min : c -> b
 nil : c
 replace : (b,b,c) -> c
 s : b -> b
 sort : c -> c
 true : a
******** Computation Rules ********
 (1)  eq(0,0) => true
 (2)  eq(0,s(X)) => false
 (3)  eq(s(Y),0) => false
 (4)  eq(s(V),s(U)) => eq(V,U)
 (5)  le(0,W) => true
 (6)  le(s(P),0) => false
 (7)  le(s(Y1),s(X1)) => le(Y1,X1)
 (8)  min(cons(0,nil)) => 0
 (9)  min(cons(s(U1),nil)) => s(U1)
 (10)  min(cons(W1,cons(V1,P1))) => ifxb6220min(le(W1,V1),cons(W1,cons(V1,P1)))
 (11)  ifxb6220min(true,cons(Y2,cons(X2,U2))) => min(cons(Y2,U2))
 (12)  ifxb6220min(false,cons(W2,cons(V2,P2))) => min(cons(V2,P2))
 (13)  replace(Y3,X3,nil) => nil
 (14)  replace(W3,V3,cons(U3,P3)) => ifxb6220replace(eq(W3,U3),W3,V3,cons(U3,P3))
 (15)  ifxb6220replace(true,U4,Y4,cons(X4,V4)) => cons(Y4,V4)
 (16)  ifxb6220replace(false,X5,P4,cons(W4,Y5)) => cons(W4,replace(X5,P4,Y5))
 (17)  sort(nil) => nil
 (18)  sort(cons(U5,V5)) => cons(min(cons(U5,V5)),sort(replace(min(cons(U5,V5)),U5,V5)))
 (19)  map(I5,nil) => nil
 (20)  map(J5,cons(X6,Y6)) => cons(J5[X6],map(J5,Y6))
 (21)  filter(G6,nil) => nil
 (22)  filter(H6,cons(W6,P6)) => filter2(H6[W6],H6,W6,P6)
 (23)  filter2(true,F7,Y7,U7) => cons(Y7,filter(F7,U7))
 (24)  filter2(false,H7,W7,P7) => filter(H7,P7)

YES