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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270325
details
property
value
status
complete
benchmark
sdu.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n165.star.cs.uiowa.edu
space
Mixed_HO_10
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
0.421608 seconds
cpu usage
0.421723
user time
0.395387
system time
0.026336
max virtual memory
113176.0
max residence set size
9704.0
stage attributes
key
value
starexec-result
YES
output
0.00/0.40 YES 0.00/0.42 We consider the system theBenchmark. 0.00/0.42 0.00/0.42 Alphabet: 0.00/0.42 0.00/0.42 casea : [u * a -> a * b -> a] --> a 0.00/0.42 caseb : [u * a -> b * b -> b] --> b 0.00/0.42 caseu : [u * a -> u * b -> u] --> u 0.00/0.42 inl : [a] --> u 0.00/0.42 inr : [b] --> u 0.00/0.42 0.00/0.42 Rules: 0.00/0.42 0.00/0.42 casea(inl(x), f, g) => f x 0.00/0.42 casea(inr(x), f, g) => g x 0.00/0.42 casea(x, /\y.f inl(y), /\z.f inr(z)) => f x 0.00/0.42 caseb(inl(x), f, g) => f x 0.00/0.42 caseb(inr(x), f, g) => g x 0.00/0.42 caseb(x, /\y.f inl(y), /\z.f inr(z)) => f x 0.00/0.42 caseu(inl(x), f, g) => f x 0.00/0.42 caseu(inr(x), f, g) => g x 0.00/0.42 caseu(x, /\y.f inl(y), /\z.f inr(z)) => f x 0.00/0.42 0.00/0.42 Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. 0.00/0.42 0.00/0.42 We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: 0.00/0.42 0.00/0.42 Alphabet: 0.00/0.42 0.00/0.42 casea : [u * a -> a * b -> a] --> a 0.00/0.42 caseb : [u * a -> b * b -> b] --> b 0.00/0.42 caseu : [u * a -> u * b -> u] --> u 0.00/0.42 inl : [a] --> u 0.00/0.42 inr : [b] --> u 0.00/0.42 ~AP1 : [u -> a * u] --> a 0.00/0.42 ~AP2 : [u -> b * u] --> b 0.00/0.42 ~AP3 : [u -> u * u] --> u 0.00/0.42 0.00/0.42 Rules: 0.00/0.42 0.00/0.42 casea(inl(X), F, G) => F X 0.00/0.42 casea(inr(X), F, G) => G X 0.00/0.42 casea(X, /\x.~AP1(F, inl(x)), /\y.~AP1(F, inr(y))) => ~AP1(F, X) 0.00/0.42 caseb(inl(X), F, G) => F X 0.00/0.42 caseb(inr(X), F, G) => G X 0.00/0.42 caseb(X, /\x.~AP2(F, inl(x)), /\y.~AP2(F, inr(y))) => ~AP2(F, X) 0.00/0.42 caseu(inl(X), F, G) => F X 0.00/0.42 caseu(inr(X), F, G) => G X 0.00/0.42 caseu(X, /\x.~AP3(F, inl(x)), /\y.~AP3(F, inr(y))) => ~AP3(F, X) 0.00/0.42 ~AP1(F, X) => F X 0.00/0.42 ~AP2(F, X) => F X 0.00/0.42 ~AP3(F, X) => F X 0.00/0.42 0.00/0.42 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.42 0.00/0.42 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.42 0.00/0.42 casea(inl(X), F, G) >? F X 0.00/0.42 casea(inr(X), F, G) >? G X 0.00/0.42 casea(X, /\x.~AP1(F, inl(x)), /\y.~AP1(F, inr(y))) >? ~AP1(F, X) 0.00/0.42 caseb(inl(X), F, G) >? F X 0.00/0.42 caseb(inr(X), F, G) >? G X 0.00/0.42 caseb(X, /\x.~AP2(F, inl(x)), /\y.~AP2(F, inr(y))) >? ~AP2(F, X) 0.00/0.42 caseu(inl(X), F, G) >? F X 0.00/0.42 caseu(inr(X), F, G) >? G X 0.00/0.42 caseu(X, /\x.~AP3(F, inl(x)), /\y.~AP3(F, inr(y))) >? ~AP3(F, X) 0.00/0.42 ~AP1(F, X) >? F X 0.00/0.42 ~AP2(F, X) >? F X 0.00/0.42 ~AP3(F, X) >? F X 0.00/0.42 0.00/0.42 We orient these requirements with a polynomial interpretation in the natural numbers. 0.00/0.42 0.00/0.42 The following interpretation satisfies the requirements: 0.00/0.42 0.00/0.42 casea = \y0G1G2.3 + 3y0 + G1(y0) + G2(y0) + 2y0G2(y0) 0.00/0.42 caseb = \y0G1G2.3 + 3y0 + 3G1(y0) + 3G2(0) + 3G2(y0) + y0G2(y0) 0.00/0.42 caseu = \y0G1G2.3 + 3y0 + 3y0G1(y0) + 3G1(0) + 3G1(y0) + 3G2(0) + 3G2(y0) 0.00/0.42 inl = \y0.3 + 3y0 0.00/0.42 inr = \y0.3 + 3y0 0.00/0.42 ~AP1 = \G0y1.2 + y1 + 2G0(y1) 0.00/0.42 ~AP2 = \G0y1.2y1 + 3G0(0) + 3G0(y1) + y1G0(y1) 0.00/0.42 ~AP3 = \G0y1.2y1 + 2y1G0(y1) + 3G0(0) + 3G0(y1) 0.00/0.42 0.00/0.42 Using this interpretation, the requirements translate to: 0.00/0.42 0.00/0.42 [[casea(inl(_x0), _F1, _F2)]] = 12 + 9x0 + F1(3 + 3x0) + 6x0F2(3 + 3x0) + 7F2(3 + 3x0) > x0 + F1(x0) = [[_F1 _x0]] 0.00/0.42 [[casea(inr(_x0), _F1, _F2)]] = 12 + 9x0 + F1(3 + 3x0) + 6x0F2(3 + 3x0) + 7F2(3 + 3x0) > x0 + F2(x0) = [[_F2 _x0]] 0.00/0.42 [[casea(_x0, /\x.~AP1(_F1, inl(x)), /\y.~AP1(_F1, inr(y)))]] = 13 + 6x0x0 + 19x0 + 4x0F1(3 + 3x0) + 4F1(3 + 3x0) > 2 + x0 + 2F1(x0) = [[~AP1(_F1, _x0)]] 0.00/0.42 [[caseb(inl(_x0), _F1, _F2)]] = 12 + 9x0 + 3x0F2(3 + 3x0) + 3F1(3 + 3x0) + 3F2(0) + 6F2(3 + 3x0) > x0 + F1(x0) = [[_F1 _x0]] 0.00/0.42 [[caseb(inr(_x0), _F1, _F2)]] = 12 + 9x0 + 3x0F2(3 + 3x0) + 3F1(3 + 3x0) + 3F2(0) + 6F2(3 + 3x0) > x0 + F2(x0) = [[_F2 _x0]] 0.00/0.42 [[caseb(_x0, /\x.~AP2(_F1, inl(x)), /\y.~AP2(_F1, inr(y)))]] = 57 + 6x0x0 + 45x0 + 3x0x0F1(3 + 3x0) + 3x0F1(0) + 18F1(3) + 24x0F1(3 + 3x0) + 27F1(0) + 36F1(3 + 3x0) > 2x0 + 3F1(0) + 3F1(x0) + x0F1(x0) = [[~AP2(_F1, _x0)]] 0.00/0.42 [[caseu(inl(_x0), _F1, _F2)]] = 12 + 9x0 + 3F1(0) + 3F2(0) + 3F2(3 + 3x0) + 9x0F1(3 + 3x0) + 12F1(3 + 3x0) > x0 + F1(x0) = [[_F1 _x0]] 0.00/0.42 [[caseu(inr(_x0), _F1, _F2)]] = 12 + 9x0 + 3F1(0) + 3F2(0) + 3F2(3 + 3x0) + 9x0F1(3 + 3x0) + 12F1(3 + 3x0) > x0 + F2(x0) = [[_F2 _x0]] 0.00/0.42 [[caseu(_x0, /\x.~AP3(_F1, inl(x)), /\y.~AP3(_F1, inr(y)))]] = 75 + 18x0x0 + 57x0 + 9x0F1(0) + 18x0x0F1(3 + 3x0) + 36F1(0) + 54F1(3) + 54F1(3 + 3x0) + 63x0F1(3 + 3x0) > 2x0 + 2x0F1(x0) + 3F1(0) + 3F1(x0) = [[~AP3(_F1, _x0)]] 0.00/0.42 [[~AP1(_F0, _x1)]] = 2 + x1 + 2F0(x1) > x1 + F0(x1) = [[_F0 _x1]] 0.00/0.42 [[~AP2(_F0, _x1)]] = 2x1 + 3F0(0) + 3F0(x1) + x1F0(x1) >= x1 + F0(x1) = [[_F0 _x1]] 0.00/0.42 [[~AP3(_F0, _x1)]] = 2x1 + 2x1F0(x1) + 3F0(0) + 3F0(x1) >= x1 + F0(x1) = [[_F0 _x1]] 0.00/0.42 0.00/0.42 We can thus remove the following rules: 0.00/0.42
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