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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270231
details
property
value
status
complete
benchmark
merge.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n123.star.cs.uiowa.edu
space
Kop_11
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
0.080757 seconds
cpu usage
0.080846
user time
0.067873
system time
0.012973
max virtual memory
113176.0
max residence set size
2688.0
stage attributes
key
value
starexec-result
YES
output
0.00/0.07 YES 0.00/0.08 We consider the system theBenchmark. 0.00/0.08 0.00/0.08 Alphabet: 0.00/0.08 0.00/0.08 cons : [nat * list] --> list 0.00/0.08 map : [nat -> nat * list] --> list 0.00/0.08 merge : [list * list * list] --> list 0.00/0.08 nil : [] --> list 0.00/0.08 0.00/0.08 Rules: 0.00/0.08 0.00/0.08 merge(nil, nil, x) => x 0.00/0.08 merge(nil, cons(x, y), z) => merge(y, nil, cons(x, z)) 0.00/0.08 merge(cons(x, y), z, u) => merge(z, y, cons(x, u)) 0.00/0.08 map(f, nil) => nil 0.00/0.08 map(f, cons(x, y)) => cons(f x, map(f, y)) 0.00/0.08 0.00/0.08 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 0.00/0.08 0.00/0.08 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.08 0.00/0.08 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.08 0.00/0.08 merge(nil, nil, X) >? X 0.00/0.08 merge(nil, cons(X, Y), Z) >? merge(Y, nil, cons(X, Z)) 0.00/0.08 merge(cons(X, Y), Z, U) >? merge(Z, Y, cons(X, U)) 0.00/0.08 map(F, nil) >? nil 0.00/0.08 map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 0.00/0.08 0.00/0.08 We orient these requirements with a polynomial interpretation in the natural numbers. 0.00/0.08 0.00/0.08 The following interpretation satisfies the requirements: 0.00/0.08 0.00/0.08 cons = \y0y1.1 + y0 + y1 0.00/0.08 map = \G0y1.3y1 + G0(0) + 2y1G0(y1) 0.00/0.08 merge = \y0y1y2.3 + y0 + y1 + y2 0.00/0.08 nil = 2 0.00/0.08 0.00/0.08 Using this interpretation, the requirements translate to: 0.00/0.08 0.00/0.08 [[merge(nil, nil, _x0)]] = 7 + x0 > x0 = [[_x0]] 0.00/0.08 [[merge(nil, cons(_x0, _x1), _x2)]] = 6 + x0 + x1 + x2 >= 6 + x0 + x1 + x2 = [[merge(_x1, nil, cons(_x0, _x2))]] 0.00/0.08 [[merge(cons(_x0, _x1), _x2, _x3)]] = 4 + x0 + x1 + x2 + x3 >= 4 + x0 + x1 + x2 + x3 = [[merge(_x2, _x1, cons(_x0, _x3))]] 0.00/0.08 [[map(_F0, nil)]] = 6 + F0(0) + 4F0(2) > 2 = [[nil]] 0.00/0.08 [[map(_F0, cons(_x1, _x2))]] = 3 + 3x1 + 3x2 + F0(0) + 2x1F0(1 + x1 + x2) + 2x2F0(1 + x1 + x2) + 2F0(1 + x1 + x2) > 1 + x1 + 3x2 + F0(0) + F0(x1) + 2x2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 0.00/0.08 0.00/0.08 We can thus remove the following rules: 0.00/0.08 0.00/0.08 merge(nil, nil, X) => X 0.00/0.08 map(F, nil) => nil 0.00/0.08 map(F, cons(X, Y)) => cons(F X, map(F, Y)) 0.00/0.08 0.00/0.08 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.08 0.00/0.08 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.08 0.00/0.08 merge(nil, cons(X, Y), Z) >? merge(Y, nil, cons(X, Z)) 0.00/0.08 merge(cons(X, Y), Z, U) >? merge(Z, Y, cons(X, U)) 0.00/0.08 0.00/0.08 We orient these requirements with a polynomial interpretation in the natural numbers. 0.00/0.08 0.00/0.08 The following interpretation satisfies the requirements: 0.00/0.08 0.00/0.08 cons = \y0y1.3 + y1 + 3y0 0.00/0.08 merge = \y0y1y2.2y2 + 3y0 + 3y1 0.00/0.08 nil = 0 0.00/0.08 0.00/0.08 Using this interpretation, the requirements translate to: 0.00/0.08 0.00/0.08 [[merge(nil, cons(_x0, _x1), _x2)]] = 9 + 2x2 + 3x1 + 9x0 > 6 + 2x2 + 3x1 + 6x0 = [[merge(_x1, nil, cons(_x0, _x2))]] 0.00/0.08 [[merge(cons(_x0, _x1), _x2, _x3)]] = 9 + 2x3 + 3x1 + 3x2 + 9x0 > 6 + 2x3 + 3x1 + 3x2 + 6x0 = [[merge(_x2, _x1, cons(_x0, _x3))]] 0.00/0.08 0.00/0.08 We can thus remove the following rules: 0.00/0.08 0.00/0.08 merge(nil, cons(X, Y), Z) => merge(Y, nil, cons(X, Z)) 0.00/0.08 merge(cons(X, Y), Z, U) => merge(Z, Y, cons(X, U)) 0.00/0.08 0.00/0.08 All rules were succesfully removed. Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold. 0.00/0.08 0.00/0.08 0.00/0.08 +++ Citations +++ 0.00/0.08 0.00/0.08 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 0.00/0.08 EOF
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