(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
+(x, +(y, z)) → +(+(x, y), z)
+(*(x, y), +(x, z)) → *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(*(z0, z1), +(z0, z2)) → *(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), z3)) → +(*(z0, +(z1, z2)), z3)
Tuples:
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
S tuples:
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
K tuples:none
Defined Rule Symbols:
+
Defined Pair Symbols:
+'
Compound Symbols:
c, c1, c2
(3) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
We considered the (Usable) Rules:
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(*(z0, z1), +(z0, z2)) → *(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), z3)) → +(*(z0, +(z1, z2)), z3)
And the Tuples:
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(*(x1, x2)) = [1] + x2
POL(+(x1, x2)) = [2]x1 + [4]x2
POL(+'(x1, x2)) = [4]x2
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(*(z0, z1), +(z0, z2)) → *(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), z3)) → +(*(z0, +(z1, z2)), z3)
Tuples:
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
S tuples:
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
K tuples:
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
Defined Rule Symbols:
+
Defined Pair Symbols:
+'
Compound Symbols:
c, c1, c2
(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
We considered the (Usable) Rules:
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(*(z0, z1), +(z0, z2)) → *(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), z3)) → +(*(z0, +(z1, z2)), z3)
And the Tuples:
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(*(x1, x2)) = x2
POL(+(x1, x2)) = [2] + [2]x1 + x2
POL(+'(x1, x2)) = [2]x2
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(*(z0, z1), +(z0, z2)) → *(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), z3)) → +(*(z0, +(z1, z2)), z3)
Tuples:
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
S tuples:none
K tuples:
+'(*(z0, z1), +(*(z0, z2), z3)) → c2(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
+'(z0, +(z1, z2)) → c(+'(+(z0, z1), z2), +'(z0, z1))
+'(*(z0, z1), +(z0, z2)) → c1(+'(z1, z2))
Defined Rule Symbols:
+
Defined Pair Symbols:
+'
Compound Symbols:
c, c1, c2
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))