R
↳Dependency Pair Analysis
+'(O(x), O(y)) -> O'(+(x, y))
+'(O(x), O(y)) -> +'(x, y)
+'(O(x), I(y)) -> +'(x, y)
+'(I(x), O(y)) -> +'(x, y)
+'(I(x), I(y)) -> O'(+(+(x, y), I(0)))
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), I(y)) -> +'(x, y)
*'(O(x), y) -> O'(*(x, y))
*'(O(x), y) -> *'(x, y)
*'(I(x), y) -> +'(O(*(x, y)), y)
*'(I(x), y) -> O'(*(x, y))
*'(I(x), y) -> *'(x, y)
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
→DP Problem 2
↳UsableRules
+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), O(y)) -> +'(x, y)
+'(O(x), I(y)) -> +'(x, y)
+'(O(x), O(y)) -> +'(x, y)
O(0) -> 0
+(0, x) -> x
+(x, 0) -> x
+(O(x), O(y)) -> O(+(x, y))
+(O(x), I(y)) -> I(+(x, y))
+(I(x), O(y)) -> I(+(x, y))
+(I(x), I(y)) -> O(+(+(x, y), I(0)))
*(0, x) -> 0
*(x, 0) -> 0
*(O(x), y) -> O(*(x, y))
*(I(x), y) -> +(O(*(x, y)), y)
innermost
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 3
↳Modular Removal of Rules
→DP Problem 2
↳UsableRules
+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), O(y)) -> +'(x, y)
+'(O(x), I(y)) -> +'(x, y)
+'(O(x), O(y)) -> +'(x, y)
+(I(x), O(y)) -> I(+(x, y))
+(0, x) -> x
+(x, 0) -> x
+(I(x), I(y)) -> O(+(+(x, y), I(0)))
+(O(x), O(y)) -> O(+(x, y))
+(O(x), I(y)) -> I(+(x, y))
O(0) -> 0
innermost
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
+(I(x), O(y)) -> I(+(x, y))
+(0, x) -> x
+(x, 0) -> x
+(I(x), I(y)) -> O(+(+(x, y), I(0)))
+(O(x), O(y)) -> O(+(x, y))
+(O(x), I(y)) -> I(+(x, y))
O(0) -> 0
POL(I(x1)) = 1 + x1 POL(0) = 0 POL(O(x1)) = x1 POL(+(x1, x2)) = x1 + x2 POL(+'(x1, x2)) = 1 + x1 + x2
+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), O(y)) -> +'(x, y)
+'(O(x), I(y)) -> +'(x, y)
+(I(x), I(y)) -> O(+(+(x, y), I(0)))
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 3
↳MRR
...
→DP Problem 4
↳Modular Removal of Rules
→DP Problem 2
↳UsableRules
+'(O(x), O(y)) -> +'(x, y)
+(I(x), O(y)) -> I(+(x, y))
+(0, x) -> x
+(x, 0) -> x
+(O(x), O(y)) -> O(+(x, y))
+(O(x), I(y)) -> I(+(x, y))
O(0) -> 0
innermost
POL(O(x1)) = x1 POL(+'(x1, x2)) = x1 + x2
+'(O(x), O(y)) -> +'(x, y)
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Usable Rules (Innermost)
*'(I(x), y) -> *'(x, y)
*'(O(x), y) -> *'(x, y)
O(0) -> 0
+(0, x) -> x
+(x, 0) -> x
+(O(x), O(y)) -> O(+(x, y))
+(O(x), I(y)) -> I(+(x, y))
+(I(x), O(y)) -> I(+(x, y))
+(I(x), I(y)) -> O(+(+(x, y), I(0)))
*(0, x) -> 0
*(x, 0) -> 0
*(O(x), y) -> O(*(x, y))
*(I(x), y) -> +(O(*(x, y)), y)
innermost
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
→DP Problem 5
↳Size-Change Principle
*'(I(x), y) -> *'(x, y)
*'(O(x), y) -> *'(x, y)
none
innermost
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trivial
I(x1) -> I(x1)
O(x1) -> O(x1)