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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270475
details
property
value
status
complete
benchmark
h56.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n142.star.cs.uiowa.edu
space
Hamana_Kikuchi_18
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
7.28873 seconds
cpu usage
7.26513
user time
7.15134
system time
0.113786
max virtual memory
186828.0
max residence set size
70232.0
stage attributes
key
value
starexec-result
YES
output
7.15/7.26 YES 7.15/7.27 We consider the system theBenchmark. 7.15/7.27 7.15/7.27 Alphabet: 7.15/7.27 7.15/7.27 cons : [a * b] --> b 7.15/7.27 foldr : [a -> b -> b * b * b] --> b 7.15/7.27 nil : [] --> b 7.15/7.27 xap : [a -> b -> b * a] --> b -> b 7.15/7.27 yap : [b -> b * b] --> b 7.15/7.27 7.15/7.27 Rules: 7.15/7.27 7.15/7.27 foldr(/\x./\y.yap(xap(f, x), y), z, nil) => z 7.15/7.27 foldr(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => yap(xap(f, u), foldr(/\w./\x'.yap(xap(f, w), x'), z, v)) 7.15/7.27 xap(f, x) => f x 7.15/7.27 yap(f, x) => f x 7.15/7.27 7.15/7.27 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 7.15/7.27 7.15/7.27 Symbol xap is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. This gives: 7.15/7.27 7.15/7.27 Alphabet: 7.15/7.27 7.15/7.27 cons : [a * b] --> b 7.15/7.27 foldr : [a -> b -> b * b * b] --> b 7.15/7.27 nil : [] --> b 7.15/7.27 yap : [b -> b * b] --> b 7.15/7.27 7.15/7.27 Rules: 7.15/7.27 7.15/7.27 foldr(/\x./\y.yap(F(x), y), X, nil) => X 7.15/7.27 foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 7.15/7.27 yap(F, X) => F X 7.15/7.27 7.15/7.27 We use rule removal, following [Kop12, Theorem 2.23]. 7.15/7.27 7.15/7.27 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 7.15/7.27 7.15/7.27 foldr(/\x./\y.yap(F(x), y), X, nil) >? X 7.15/7.27 foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 7.15/7.27 yap(F, X) >? F X 7.15/7.27 7.15/7.27 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 7.15/7.27 7.15/7.27 We choose Lex = {} and Mul = {@_{o -> o}, cons, foldr, nil, yap}, and the following precedence: foldr > nil > yap > @_{o -> o} > cons 7.15/7.27 7.15/7.27 With these choices, we have: 7.15/7.27 7.15/7.27 1] foldr(/\x./\y.yap(F(x), y), X, nil) > X because [2], by definition 7.15/7.27 2] foldr*(/\x./\y.yap(F(x), y), X, nil) >= X because [3], by (Select) 7.15/7.27 3] X >= X by (Meta) 7.15/7.27 7.15/7.27 4] foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) because [5], by (Star) 7.15/7.27 5] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) because foldr > yap, [6] and [13], by (Copy) 7.15/7.27 6] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y) because [7], by (Select) 7.15/7.27 7] /\x.yap(F(foldr*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y) because [8], by (Eta)[Kop13:2] 7.15/7.27 8] F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y) because [9], by (Meta) 7.15/7.27 9] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y because [10], by (Select) 7.15/7.27 10] cons(Y, Z) >= Y because [11], by (Star) 7.15/7.27 11] cons*(Y, Z) >= Y because [12], by (Select) 7.15/7.27 12] Y >= Y by (Meta) 7.15/7.27 13] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z) because foldr in Mul, [14], [20] and [21], by (Stat) 7.15/7.27 14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [15], by (Abs) 7.15/7.27 15] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [16], by (Abs) 7.15/7.27 16] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [17] and [19], by (Fun) 7.15/7.27 17] F(y) >= F(y) because [18], by (Meta) 7.15/7.27 18] y >= y by (Var) 7.15/7.27 19] x >= x by (Var) 7.15/7.27 20] X >= X by (Meta) 7.15/7.27 21] cons(Y, Z) > Z because [22], by definition 7.15/7.27 22] cons*(Y, Z) >= Z because [23], by (Select) 7.15/7.27 23] Z >= Z by (Meta) 7.15/7.27 7.15/7.27 24] yap(F, X) > @_{o -> o}(F, X) because [25], by definition 7.15/7.27 25] yap*(F, X) >= @_{o -> o}(F, X) because yap > @_{o -> o}, [26] and [28], by (Copy) 7.15/7.27 26] yap*(F, X) >= F because [27], by (Select) 7.15/7.27 27] F >= F by (Meta) 7.15/7.27 28] yap*(F, X) >= X because [29], by (Select) 7.15/7.27 29] X >= X by (Meta) 7.15/7.27 7.15/7.27 We can thus remove the following rules: 7.15/7.27 7.15/7.27 foldr(/\x./\y.yap(F(x), y), X, nil) => X 7.15/7.27 yap(F, X) => F X 7.15/7.27 7.15/7.27 We use rule removal, following [Kop12, Theorem 2.23]. 7.15/7.27 7.15/7.27 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 7.15/7.27 7.15/7.27 foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 7.15/7.27 7.15/7.27 We orient these requirements with a polynomial interpretation in the natural numbers. 7.15/7.27 7.15/7.27 The following interpretation satisfies the requirements: 7.15/7.27 7.15/7.27 cons = \y0y1.3 + y0 + 3y1 7.15/7.27 foldr = \G0y1y2.y1 + 3y2 + G0(y2,y1) + 3y2y2G0(y2,y2) + y1y2G0(y1,y2) 7.15/7.27 yap = \G0y1.2y1 + G0(0) 7.15/7.27
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