Problem AProVE 07 thiemann29

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann29

stdout:

MAYBE

Problem:
 le(s(x),0()) -> false()
 le(0(),y) -> true()
 le(s(x),s(y)) -> le(x,y)
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 times(0(),y) -> 0()
 times(s(x),y) -> plus(y,times(x,y))
 log(x,0()) -> baseError()
 log(x,s(0())) -> baseError()
 log(0(),s(s(b))) -> logZeroError()
 log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
 loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
 if(true(),x,b,y,z) -> z
 if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann29

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann29

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  le(s(x), 0()) -> false()
     , le(0(), y) -> true()
     , le(s(x), s(y)) -> le(x, y)
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , times(0(), y) -> 0()
     , times(s(x), y) -> plus(y, times(x, y))
     , log(x, 0()) -> baseError()
     , log(x, s(0())) -> baseError()
     , log(0(), s(s(b))) -> logZeroError()
     , log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0()), 0())
     , loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
     , if(true(), x, b, y, z) -> z
     , if(false(), x, b, y, z) -> loop(x, b, times(b, y), s(z))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: le^#(s(x), 0()) -> c_0()
              , 2: le^#(0(), y) -> c_1()
              , 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
              , 4: plus^#(0(), y) -> c_3()
              , 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
              , 6: times^#(0(), y) -> c_5()
              , 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
              , 8: log^#(x, 0()) -> c_7()
              , 9: log^#(x, s(0())) -> c_8()
              , 10: log^#(0(), s(s(b))) -> c_9()
              , 11: log^#(s(x), s(s(b))) ->
                    c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
              , 12: loop^#(x, s(s(b)), s(y), z) ->
                    c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
              , 13: if^#(true(), x, b, y, z) -> c_12()
              , 14: if^#(false(), x, b, y, z) ->
                    c_13(loop^#(x, b, times(b, y), s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                `->{12,14}                                               [     inherited      ]
                    |
                    `->{13}                                              [         NA         ]
             
             ->{10}                                                      [    YES(?,O(1))     ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [     inherited      ]
                    |
                    `->{4}                                               [         NA         ]
             
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^3))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [3 3 3]      [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                le^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                               [2 2 0]      [0 2 0]      [0]
                               [4 0 0]      [0 2 0]      [0]
                c_2(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {3}->{1}: YES(?,O(n^3))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_0()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
                               [0 0 0]      [0 0 2]      [0]
                               [0 1 0]      [0 2 4]      [0]
                c_0() = [1]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [6]
           
           * Path {3}->{2}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(0(), y) -> c_1()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 1]      [0]
                0() = [2]
                      [2]
                      [2]
                le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
                               [2 2 2]      [0 0 4]      [0]
                               [2 2 2]      [0 0 0]      [0]
                c_1() = [1]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                times^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                  [2 2 0]      [0 0 0]      [3]
                                  [2 2 2]      [0 0 0]      [3]
                c_5() = [0]
                        [1]
                        [1]
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{4}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
                  , plus^#(0(), y) -> c_3()
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}->{5}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{5}->{4}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(x, 0()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                log^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
                                [0 0 0]      [2 2 0]      [3]
                                [0 0 0]      [2 2 2]      [3]
                c_7() = [0]
                        [1]
                        [1]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(x, s(0())) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 3 2] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                log^#(x1, x2) = [0 0 0] x1 + [2 0 0] x2 + [3]
                                [0 0 0]      [0 0 0]      [7]
                                [0 0 0]      [0 0 0]      [7]
                c_8() = [0]
                        [1]
                        [1]
           
           * Path {10}: YES(?,O(1))
             ----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 2 2] x1 + [2]
                        [0 0 2]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                log^#(x1, x2) = [0 2 2] x1 + [2 2 0] x2 + [3]
                                [0 0 0]      [2 0 0]      [7]
                                [2 2 2]      [0 0 0]      [3]
                c_9() = [0]
                        [1]
                        [1]
           
           * Path {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}->{13}: NA
             ----------------------------
             
             The usable rules for this path are:
             
               {  le(s(x), 0()) -> false()
                , le(0(), y) -> true()
                , le(s(x), s(y)) -> le(x, y)
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: le^#(s(x), 0()) -> c_0()
              , 2: le^#(0(), y) -> c_1()
              , 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
              , 4: plus^#(0(), y) -> c_3()
              , 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
              , 6: times^#(0(), y) -> c_5()
              , 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
              , 8: log^#(x, 0()) -> c_7()
              , 9: log^#(x, s(0())) -> c_8()
              , 10: log^#(0(), s(s(b))) -> c_9()
              , 11: log^#(s(x), s(s(b))) ->
                    c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
              , 12: loop^#(x, s(s(b)), s(y), z) ->
                    c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
              , 13: if^#(true(), x, b, y, z) -> c_12()
              , 14: if^#(false(), x, b, y, z) ->
                    c_13(loop^#(x, b, times(b, y), s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                `->{12,14}                                               [     inherited      ]
                    |
                    `->{13}                                              [         NA         ]
             
             ->{10}                                                      [    YES(?,O(1))     ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [     inherited      ]
                    |
                    `->{4}                                               [         NA         ]
             
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12() = [0]
                         [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_2(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {3}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12() = [0]
                         [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_0()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4] x1 + [2]
                        [0 0]      [0]
                0() = [0]
                      [0]
                le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
                               [0 0]      [0 2]      [0]
                c_0() = [1]
                        [0]
                c_2(x1) = [1 2] x1 + [3]
                          [0 0]      [0]
           
           * Path {3}->{2}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12() = [0]
                         [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(0(), y) -> c_1()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                               [4 1]      [2 0]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12() = [0]
                         [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                  [2 2]      [0 0]      [7]
                c_5() = [0]
                        [1]
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{4}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
                  , plus^#(0(), y) -> c_3()
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}->{5}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{5}->{4}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12() = [0]
                         [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(x, 0()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                log^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
                                [0 0]      [2 2]      [7]
                c_7() = [0]
                        [1]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12() = [0]
                         [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(x, s(0())) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 1] x1 + [2]
                        [0 0]      [2]
                0() = [0]
                      [2]
                log^#(x1, x2) = [0 0] x1 + [2 2] x2 + [3]
                                [0 0]      [2 2]      [3]
                c_8() = [0]
                        [1]
           
           * Path {10}: YES(?,O(1))
             ----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12() = [0]
                         [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 2] x1 + [2]
                        [0 0]      [0]
                0() = [0]
                      [2]
                log^#(x1, x2) = [0 2] x1 + [2 0] x2 + [7]
                                [0 0]      [0 0]      [7]
                c_9() = [0]
                        [1]
           
           * Path {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}->{13}: NA
             ----------------------------
             
             The usable rules for this path are:
             
               {  le(s(x), 0()) -> false()
                , le(0(), y) -> true()
                , le(s(x), s(y)) -> le(x, y)
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: le^#(s(x), 0()) -> c_0()
              , 2: le^#(0(), y) -> c_1()
              , 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
              , 4: plus^#(0(), y) -> c_3()
              , 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
              , 6: times^#(0(), y) -> c_5()
              , 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
              , 8: log^#(x, 0()) -> c_7()
              , 9: log^#(x, s(0())) -> c_8()
              , 10: log^#(0(), s(s(b))) -> c_9()
              , 11: log^#(s(x), s(s(b))) ->
                    c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
              , 12: loop^#(x, s(s(b)), s(y), z) ->
                    c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
              , 13: if^#(true(), x, b, y, z) -> c_12()
              , 14: if^#(false(), x, b, y, z) ->
                    c_13(loop^#(x, b, times(b, y), s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                `->{12,14}                                               [     inherited      ]
                    |
                    `->{13}                                              [         NA         ]
             
             ->{10}                                                      [    YES(?,O(1))     ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [     inherited      ]
                    |
                    `->{4}                                               [         NA         ]
             
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12() = [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12() = [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_0()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_0() = [1]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12() = [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(0(), y) -> c_1()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_1() = [1]
                c_2(x1) = [1] x1 + [2]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12() = [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                times^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_5() = [1]
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{4}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
                  , plus^#(0(), y) -> c_3()
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}->{5}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{5}->{4}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12() = [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(x, 0()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                log^#(x1, x2) = [0] x1 + [1] x2 + [7]
                c_7() = [1]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12() = [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(x, s(0())) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [2]
                0() = [0]
                log^#(x1, x2) = [0] x1 + [2] x2 + [7]
                c_8() = [0]
           
           * Path {10}: YES(?,O(1))
             ----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_4) = {}, Uargs(times^#) = {}, Uargs(c_6) = {},
                 Uargs(log^#) = {}, Uargs(c_10) = {}, Uargs(loop^#) = {},
                 Uargs(c_11) = {}, Uargs(if^#) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12() = [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [2]
                0() = [2]
                log^#(x1, x2) = [2] x1 + [2] x2 + [7]
                c_9() = [0]
           
           * Path {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}->{13}: NA
             ----------------------------
             
             The usable rules for this path are:
             
               {  le(s(x), 0()) -> false()
                , le(0(), y) -> true()
                , le(s(x), s(y)) -> le(x, y)
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann29

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann29

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  le(s(x), 0()) -> false()
     , le(0(), y) -> true()
     , le(s(x), s(y)) -> le(x, y)
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , times(0(), y) -> 0()
     , times(s(x), y) -> plus(y, times(x, y))
     , log(x, 0()) -> baseError()
     , log(x, s(0())) -> baseError()
     , log(0(), s(s(b))) -> logZeroError()
     , log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0()), 0())
     , loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
     , if(true(), x, b, y, z) -> z
     , if(false(), x, b, y, z) -> loop(x, b, times(b, y), s(z))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: le^#(s(x), 0()) -> c_0()
              , 2: le^#(0(), y) -> c_1()
              , 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
              , 4: plus^#(0(), y) -> c_3(y)
              , 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
              , 6: times^#(0(), y) -> c_5()
              , 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
              , 8: log^#(x, 0()) -> c_7()
              , 9: log^#(x, s(0())) -> c_8()
              , 10: log^#(0(), s(s(b))) -> c_9()
              , 11: log^#(s(x), s(s(b))) ->
                    c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
              , 12: loop^#(x, s(s(b)), s(y), z) ->
                    c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
              , 13: if^#(true(), x, b, y, z) -> c_12(z)
              , 14: if^#(false(), x, b, y, z) ->
                    c_13(loop^#(x, b, times(b, y), s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                `->{12,14}                                               [     inherited      ]
                    |
                    `->{13}                                              [         NA         ]
             
             ->{10}                                                      [    YES(?,O(1))     ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [     inherited      ]
                    |
                    `->{4}                                               [         NA         ]
             
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^3))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [3 3 3]      [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                le^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                               [2 2 0]      [0 2 0]      [0]
                               [4 0 0]      [0 2 0]      [0]
                c_2(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {3}->{1}: YES(?,O(n^3))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_0()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
                               [0 0 0]      [0 0 2]      [0]
                               [0 1 0]      [0 2 4]      [0]
                c_0() = [1]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [6]
           
           * Path {3}->{2}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(0(), y) -> c_1()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 1]      [0]
                0() = [2]
                      [2]
                      [2]
                le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
                               [2 2 2]      [0 0 4]      [0]
                               [2 2 2]      [0 0 0]      [0]
                c_1() = [1]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                times^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                  [2 2 0]      [0 0 0]      [3]
                                  [2 2 2]      [0 0 0]      [3]
                c_5() = [0]
                        [1]
                        [1]
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{4}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
                  , plus^#(0(), y) -> c_3(y)
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}->{5}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{5}->{4}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(x, 0()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                log^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
                                [0 0 0]      [2 2 0]      [3]
                                [0 0 0]      [2 2 2]      [3]
                c_7() = [0]
                        [1]
                        [1]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(x, s(0())) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 3 2] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                log^#(x1, x2) = [0 0 0] x1 + [2 0 0] x2 + [3]
                                [0 0 0]      [0 0 0]      [7]
                                [0 0 0]      [0 0 0]      [7]
                c_8() = [0]
                        [1]
                        [1]
           
           * Path {10}: YES(?,O(1))
             ----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                true() = [0]
                         [0]
                         [0]
                plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                log(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                baseError() = [0]
                              [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                log^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                         [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                           [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 2 2] x1 + [2]
                        [0 0 2]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                log^#(x1, x2) = [0 2 2] x1 + [2 2 0] x2 + [3]
                                [0 0 0]      [2 0 0]      [7]
                                [2 2 2]      [0 0 0]      [3]
                c_9() = [0]
                        [1]
                        [1]
           
           * Path {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}->{13}: NA
             ----------------------------
             
             The usable rules for this path are:
             
               {  le(s(x), 0()) -> false()
                , le(0(), y) -> true()
                , le(s(x), s(y)) -> le(x, y)
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: le^#(s(x), 0()) -> c_0()
              , 2: le^#(0(), y) -> c_1()
              , 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
              , 4: plus^#(0(), y) -> c_3(y)
              , 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
              , 6: times^#(0(), y) -> c_5()
              , 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
              , 8: log^#(x, 0()) -> c_7()
              , 9: log^#(x, s(0())) -> c_8()
              , 10: log^#(0(), s(s(b))) -> c_9()
              , 11: log^#(s(x), s(s(b))) ->
                    c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
              , 12: loop^#(x, s(s(b)), s(y), z) ->
                    c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
              , 13: if^#(true(), x, b, y, z) -> c_12(z)
              , 14: if^#(false(), x, b, y, z) ->
                    c_13(loop^#(x, b, times(b, y), s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                `->{12,14}                                               [     inherited      ]
                    |
                    `->{13}                                              [         NA         ]
             
             ->{10}                                                      [    YES(?,O(1))     ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [     inherited      ]
                    |
                    `->{4}                                               [         NA         ]
             
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_2(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {3}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_0()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4] x1 + [2]
                        [0 0]      [0]
                0() = [0]
                      [0]
                le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
                               [0 0]      [0 2]      [0]
                c_0() = [1]
                        [0]
                c_2(x1) = [1 2] x1 + [3]
                          [0 0]      [0]
           
           * Path {3}->{2}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(0(), y) -> c_1()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                               [4 1]      [2 0]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                  [2 2]      [0 0]      [7]
                c_5() = [0]
                        [1]
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{4}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
                  , plus^#(0(), y) -> c_3(y)
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}->{5}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{5}->{4}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(x, 0()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                log^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
                                [0 0]      [2 2]      [7]
                c_7() = [0]
                        [1]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(x, s(0())) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 1] x1 + [2]
                        [0 0]      [2]
                0() = [0]
                      [2]
                log^#(x1, x2) = [0 0] x1 + [2 2] x2 + [3]
                                [0 0]      [2 2]      [3]
                c_8() = [0]
                        [1]
           
           * Path {10}: YES(?,O(1))
             ----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                true() = [0]
                         [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                baseError() = [0]
                              [0]
                logZeroError() = [0]
                                 [0]
                loop(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                       [0 0]      [0 0]      [0 0]      [0 0]      [0]
                if(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                loop^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                         [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
                                           [0 0]      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 2] x1 + [2]
                        [0 0]      [0]
                0() = [0]
                      [2]
                log^#(x1, x2) = [0 2] x1 + [2 0] x2 + [7]
                                [0 0]      [0 0]      [7]
                c_9() = [0]
                        [1]
           
           * Path {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}->{13}: NA
             ----------------------------
             
             The usable rules for this path are:
             
               {  le(s(x), 0()) -> false()
                , le(0(), y) -> true()
                , le(s(x), s(y)) -> le(x, y)
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: le^#(s(x), 0()) -> c_0()
              , 2: le^#(0(), y) -> c_1()
              , 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
              , 4: plus^#(0(), y) -> c_3(y)
              , 5: plus^#(s(x), y) -> c_4(plus^#(x, y))
              , 6: times^#(0(), y) -> c_5()
              , 7: times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
              , 8: log^#(x, 0()) -> c_7()
              , 9: log^#(x, s(0())) -> c_8()
              , 10: log^#(0(), s(s(b))) -> c_9()
              , 11: log^#(s(x), s(s(b))) ->
                    c_10(loop^#(s(x), s(s(b)), s(0()), 0()))
              , 12: loop^#(x, s(s(b)), s(y), z) ->
                    c_11(if^#(le(x, s(y)), x, s(s(b)), s(y), z))
              , 13: if^#(true(), x, b, y, z) -> c_12(z)
              , 14: if^#(false(), x, b, y, z) ->
                    c_13(loop^#(x, b, times(b, y), s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                `->{12,14}                                               [     inherited      ]
                    |
                    `->{13}                                              [         NA         ]
             
             ->{10}                                                      [    YES(?,O(1))     ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [     inherited      ]
                    |
                    `->{4}                                               [         NA         ]
             
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_0()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_0() = [1]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {1}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(0(), y) -> c_1()}
               Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_1() = [1]
                c_2(x1) = [1] x1 + [2]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                times^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_5() = [1]
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{4}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  times^#(s(x), y) -> c_6(plus^#(y, times(x, y)))
                  , plus^#(0(), y) -> c_3(y)
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}->{5}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {7}->{5}->{4}.
           
           * Path {7}->{5}->{4}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(x, 0()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                log^#(x1, x2) = [0] x1 + [1] x2 + [7]
                c_7() = [1]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(x, s(0())) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [2]
                0() = [0]
                log^#(x1, x2) = [0] x1 + [2] x2 + [7]
                c_8() = [0]
           
           * Path {10}: YES(?,O(1))
             ----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(le) = {}, Uargs(s) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(log) = {}, Uargs(loop) = {}, Uargs(if) = {},
                 Uargs(le^#) = {}, Uargs(c_2) = {}, Uargs(plus^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(times^#) = {},
                 Uargs(c_6) = {}, Uargs(log^#) = {}, Uargs(c_10) = {},
                 Uargs(loop^#) = {}, Uargs(c_11) = {}, Uargs(if^#) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                true() = [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                baseError() = [0]
                logZeroError() = [0]
                loop(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                loop^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_11(x1) = [0] x1 + [0]
                if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {log^#(0(), s(s(b))) -> c_9()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(log^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [2]
                0() = [2]
                log^#(x1, x2) = [2] x1 + [2] x2 + [7]
                c_9() = [0]
           
           * Path {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {11}->{12,14}->{13}.
           
           * Path {11}->{12,14}->{13}: NA
             ----------------------------
             
             The usable rules for this path are:
             
               {  le(s(x), 0()) -> false()
                , le(0(), y) -> true()
                , le(s(x), s(y)) -> le(x, y)
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.