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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270457
details
property
value
status
complete
benchmark
h47.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n121.star.cs.uiowa.edu
space
Hamana_Kikuchi_18
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
7.7837 seconds
cpu usage
7.7832
user time
7.67768
system time
0.105519
max virtual memory
437888.0
max residence set size
95364.0
stage attributes
key
value
starexec-result
YES
output
7.69/7.75 YES 7.69/7.77 We consider the system theBenchmark. 7.69/7.77 7.69/7.77 Alphabet: 7.69/7.77 7.69/7.77 0 : [] --> nat 7.69/7.77 rec : [nat * nat * nat -> nat -> nat] --> nat 7.69/7.77 s : [nat] --> nat 7.69/7.77 xap : [nat -> nat -> nat * nat] --> nat -> nat 7.69/7.77 xplus : [nat * nat] --> nat 7.69/7.77 xtimes : [nat * nat] --> nat 7.69/7.77 yap : [nat -> nat * nat] --> nat 7.69/7.77 7.69/7.77 Rules: 7.69/7.77 7.69/7.77 xplus(x, 0) => x 7.69/7.77 xplus(x, s(y)) => s(xplus(x, y)) 7.69/7.77 rec(0, x, /\y./\z.yap(xap(f, y), z)) => x 7.69/7.77 rec(s(x), y, /\z./\u.yap(xap(f, z), u)) => yap(xap(f, x), rec(x, y, /\v./\w.yap(xap(f, v), w))) 7.69/7.77 xtimes(x, y) => rec(y, 0, /\z./\u.xplus(x, u)) 7.69/7.77 xap(f, x) => f x 7.69/7.77 yap(f, x) => f x 7.69/7.77 7.69/7.77 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 7.69/7.77 7.69/7.77 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: 7.69/7.77 7.69/7.77 Alphabet: 7.69/7.77 7.69/7.77 0 : [] --> nat 7.69/7.77 rec : [nat * nat * nat -> nat -> nat] --> nat 7.69/7.77 s : [nat] --> nat 7.69/7.77 xplus : [nat * nat] --> nat 7.69/7.77 xtimes : [nat * nat] --> nat 7.69/7.77 yap : [nat -> nat * nat] --> nat 7.69/7.77 7.69/7.77 Rules: 7.69/7.77 7.69/7.77 xplus(X, 0) => X 7.69/7.77 xplus(X, s(Y)) => s(xplus(X, Y)) 7.69/7.77 rec(0, X, /\x./\y.yap(F(x), y)) => X 7.69/7.77 rec(s(X), Y, /\x./\y.yap(F(x), y)) => yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 7.69/7.77 xtimes(X, Y) => rec(Y, 0, /\x./\y.xplus(X, y)) 7.69/7.77 yap(F, X) => F X 7.69/7.77 7.69/7.77 We use rule removal, following [Kop12, Theorem 2.23]. 7.69/7.77 7.69/7.77 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 7.69/7.77 7.69/7.77 xplus(X, 0) >? X 7.69/7.77 xplus(X, s(Y)) >? s(xplus(X, Y)) 7.69/7.77 rec(0, X, /\x./\y.yap(F(x), y)) >? X 7.69/7.77 rec(s(X), Y, /\x./\y.yap(F(x), y)) >? yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 7.69/7.77 xtimes(X, Y) >? rec(Y, 0, /\x./\y.xplus(X, y)) 7.69/7.77 yap(F, X) >? F X 7.69/7.77 7.69/7.77 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 7.69/7.77 7.69/7.77 Argument functions: 7.69/7.77 7.69/7.77 [[0]] = _|_ 7.69/7.77 7.69/7.77 We choose Lex = {} and Mul = {@_{o -> o}, rec, s, xplus, xtimes, yap}, and the following precedence: xtimes > xplus > s > rec > yap > @_{o -> o} 7.69/7.77 7.69/7.77 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 7.69/7.77 7.69/7.77 xplus(X, _|_) >= X 7.69/7.77 xplus(X, s(Y)) >= s(xplus(X, Y)) 7.69/7.77 rec(_|_, X, /\x./\y.yap(F(x), y)) >= X 7.69/7.77 rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) 7.69/7.77 xtimes(X, Y) > rec(Y, _|_, /\x./\y.xplus(X, y)) 7.69/7.77 yap(F, X) > @_{o -> o}(F, X) 7.69/7.77 7.69/7.77 With these choices, we have: 7.69/7.77 7.69/7.77 1] xplus(X, _|_) >= X because [2], by (Star) 7.69/7.77 2] xplus*(X, _|_) >= X because [3], by (Select) 7.69/7.77 3] X >= X by (Meta) 7.69/7.77 7.69/7.77 4] xplus(X, s(Y)) >= s(xplus(X, Y)) because [5], by (Star) 7.69/7.77 5] xplus*(X, s(Y)) >= s(xplus(X, Y)) because xplus > s and [6], by (Copy) 7.69/7.77 6] xplus*(X, s(Y)) >= xplus(X, Y) because xplus in Mul, [7] and [8], by (Stat) 7.69/7.77 7] X >= X by (Meta) 7.69/7.77 8] s(Y) > Y because [9], by definition 7.69/7.77 9] s*(Y) >= Y because [10], by (Select) 7.69/7.77 10] Y >= Y by (Meta) 7.69/7.77 7.69/7.77 11] rec(_|_, X, /\x./\y.yap(F(x), y)) >= X because [12], by (Star) 7.69/7.77 12] rec*(_|_, X, /\x./\y.yap(F(x), y)) >= X because [13], by (Select) 7.69/7.77 13] X >= X by (Meta) 7.69/7.77 7.69/7.77 14] rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) because [15], by (Star) 7.69/7.77 15] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) because [16], by (Select) 7.69/7.77 16] yap(F(rec*(s(X), Y, /\x./\y.yap(F(x), y))), rec*(s(X), Y, /\z./\u.yap(F(z), u))) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) because yap in Mul, [17] and [22], by (Fun) 7.69/7.77 17] F(rec*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X) because [18], by (Meta) 7.69/7.77 18] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= X because [19], by (Select) 7.69/7.77 19] s(X) >= X because [20], by (Star) 7.69/7.77 20] s*(X) >= X because [21], by (Select) 7.69/7.77 21] X >= X by (Meta) 7.69/7.77 22] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y)) because rec in Mul, [23], [25] and [26], by (Stat)
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