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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270317
details
property
value
status
complete
benchmark
process.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n083.star.cs.uiowa.edu
space
Mixed_HO_10
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
4.46522 seconds
cpu usage
4.46653
user time
4.14534
system time
0.321187
max virtual memory
242440.0
max residence set size
127164.0
stage attributes
key
value
starexec-result
YES
output
4.31/4.35 YES 4.37/4.45 We consider the system theBenchmark. 4.37/4.45 4.37/4.45 Alphabet: 4.37/4.45 4.37/4.45 !plus : [proc * proc] --> proc 4.37/4.45 !times : [proc * proc] --> proc 4.37/4.45 delta : [] --> proc 4.37/4.45 sigma : [data -> proc] --> proc 4.37/4.45 4.37/4.45 Rules: 4.37/4.45 4.37/4.45 !plus(x, x) => x 4.37/4.45 !times(!plus(x, y), z) => !plus(!times(x, z), !times(y, z)) 4.37/4.45 !times(!times(x, y), z) => !times(x, !times(y, z)) 4.37/4.45 !plus(x, delta) => x 4.37/4.45 !times(delta, x) => delta 4.37/4.45 sigma(/\x.y) => y 4.37/4.45 !plus(sigma(/\x.f x), f y) => sigma(/\z.f z) 4.37/4.45 sigma(/\x.!plus(f x, g x)) => !plus(sigma(/\y.f y), sigma(/\z.g z)) 4.37/4.45 !times(sigma(/\x.f x), y) => sigma(/\z.!times(f z, y)) 4.37/4.45 4.37/4.45 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. 4.37/4.45 4.37/4.45 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: 4.37/4.45 4.37/4.45 Alphabet: 4.37/4.45 4.37/4.45 !plus : [proc * proc] --> proc 4.37/4.45 !times : [proc * proc] --> proc 4.37/4.45 delta : [] --> proc 4.37/4.45 sigma : [data -> proc] --> proc 4.37/4.45 ~AP1 : [data -> proc * data] --> proc 4.37/4.45 4.37/4.45 Rules: 4.37/4.45 4.37/4.45 !plus(X, X) => X 4.37/4.45 !times(!plus(X, Y), Z) => !plus(!times(X, Z), !times(Y, Z)) 4.37/4.45 !times(!times(X, Y), Z) => !times(X, !times(Y, Z)) 4.37/4.45 !plus(X, delta) => X 4.37/4.45 !times(delta, X) => delta 4.37/4.45 sigma(/\x.X) => X 4.37/4.45 !plus(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) => sigma(/\y.~AP1(F, y)) 4.37/4.45 sigma(/\x.!plus(~AP1(F, x), ~AP1(G, x))) => !plus(sigma(/\y.~AP1(F, y)), sigma(/\z.~AP1(G, z))) 4.37/4.45 !times(sigma(/\x.~AP1(F, x)), X) => sigma(/\y.!times(~AP1(F, y), X)) 4.37/4.45 ~AP1(F, X) => F X 4.37/4.45 4.37/4.45 We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs. 4.37/4.45 4.37/4.45 After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, all): 4.37/4.45 4.37/4.45 Dependency Pairs P_0: 4.37/4.45 4.37/4.45 0] !times#(!plus(X, Y), Z) =#> !plus#(!times(X, Z), !times(Y, Z)) 4.37/4.45 1] !times#(!plus(X, Y), Z) =#> !times#(X, Z) 4.37/4.45 2] !times#(!plus(X, Y), Z) =#> !times#(Y, Z) 4.37/4.45 3] !times#(!times(X, Y), Z) =#> !times#(X, !times(Y, Z)) 4.37/4.45 4] !times#(!times(X, Y), Z) =#> !times#(Y, Z) 4.37/4.45 5] !plus#(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) =#> sigma#(/\y.~AP1(F, y)) 4.37/4.45 6] !plus#(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) =#> ~AP1#(F, y) 4.37/4.45 7] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> !plus#(sigma(/\y.~AP1(F, y)), sigma(/\z.~AP1(G, z))) 4.37/4.45 8] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> sigma#(/\y.~AP1(F, y)) 4.37/4.45 9] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> ~AP1#(F, y) 4.37/4.45 10] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> sigma#(/\y.~AP1(G, y)) 4.37/4.45 11] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> ~AP1#(G, y) 4.37/4.45 12] !times#(sigma(/\x.~AP1(F, x)), X) =#> sigma#(/\y.!times(~AP1(F, y), X)) 4.37/4.45 13] !times#(sigma(/\x.~AP1(F, x)), X) =#> !times#(~AP1(F, y), X) 4.37/4.45 14] !times#(sigma(/\x.~AP1(F, x)), X) =#> ~AP1#(F, y) 4.37/4.45 15] ~AP1#(F, X) =#> F(X) 4.37/4.45 4.37/4.45 Rules R_0: 4.37/4.45 4.37/4.45 !plus(X, X) => X 4.37/4.45 !times(!plus(X, Y), Z) => !plus(!times(X, Z), !times(Y, Z)) 4.37/4.45 !times(!times(X, Y), Z) => !times(X, !times(Y, Z)) 4.37/4.45 !plus(X, delta) => X 4.37/4.45 !times(delta, X) => delta 4.37/4.45 sigma(/\x.X) => X 4.37/4.45 !plus(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) => sigma(/\y.~AP1(F, y)) 4.37/4.45 sigma(/\x.!plus(~AP1(F, x), ~AP1(G, x))) => !plus(sigma(/\y.~AP1(F, y)), sigma(/\z.~AP1(G, z))) 4.37/4.45 !times(sigma(/\x.~AP1(F, x)), X) => sigma(/\y.!times(~AP1(F, y), X)) 4.37/4.45 ~AP1(F, X) => F X 4.37/4.45 4.37/4.45 Thus, the original system is terminating if (P_0, R_0, minimal, all) is finite. 4.37/4.45 4.37/4.45 We consider the dependency pair problem (P_0, R_0, minimal, all). 4.37/4.45 4.37/4.45 We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 4.37/4.45 4.37/4.45 * 0 : 5, 6 4.37/4.45 * 1 : 0, 1, 2, 3, 4, 12, 13, 14 4.37/4.45 * 2 : 0, 1, 2, 3, 4, 12, 13, 14 4.37/4.45 * 3 : 0, 1, 2, 3, 4, 12, 13, 14 4.37/4.45 * 4 : 0, 1, 2, 3, 4, 12, 13, 14 4.37/4.45 * 5 : 7, 8, 9, 10, 11 4.37/4.45 * 6 : 4.37/4.45 * 7 : 5, 6 4.37/4.45 * 8 : 7, 8, 9, 10, 11 4.37/4.45 * 9 : 4.37/4.45 * 10 : 7, 8, 9, 10, 11
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