(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
rec(rec(x)) → rec(sent(x))
sent(rec(x)) → rec(sent(x))
no(rec(x)) → rec(sent(x))
bot'(rec(x)) → bot'(sent(up(x)))
up(rec(x)) → rec(up(x))
up(sent(x)) → sent(up(x))
up(no(x)) → no(up(x))
up(rec(top(x))) → rec(check(top(x)))
up(sent(top(x))) → rec(check(top(x)))
up(no(top(x))) → rec(check(top(x)))
up(check(x)) → check(up(x))
sent(check(x)) → check(sent(x))
rec(check(x)) → check(rec(x))
no(check(x)) → check(no(x))
no(check(x)) → no(x)
Q is empty.
(3) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
rec(rec(x)) → rec(sent(x))
sent(rec(x)) → rec(sent(x))
no(rec(x)) → rec(sent(x))
bot'(rec(x)) → bot'(sent(up(x)))
up(rec(x)) → rec(up(x))
up(sent(x)) → sent(up(x))
up(no(x)) → no(up(x))
up(rec(top(x))) → rec(check(top(x)))
up(sent(top(x))) → rec(check(top(x)))
up(no(top(x))) → rec(check(top(x)))
up(check(x)) → check(up(x))
sent(check(x)) → check(sent(x))
rec(check(x)) → check(rec(x))
no(check(x)) → check(no(x))
no(check(x)) → no(x)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(bot) = 0
POL(check(x1)) = x1
POL(no(x1)) = 1 + x1
POL(rec(x1)) = 1 + x1
POL(sent(x1)) = x1
POL(top(x1)) = x1
POL(up(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
rec(rec(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
top(rec(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(sent(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
(7) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
REC(sent(x)) → SENT(rec(x))
REC(sent(x)) → REC(x)
REC(bot) → SENT(bot)
REC(up(x)) → REC(x)
SENT(up(x)) → SENT(x)
NO(up(x)) → NO(x)
TOP(sent(up(x))) → TOP(check(rec(x)))
TOP(sent(up(x))) → CHECK(rec(x))
TOP(sent(up(x))) → REC(x)
CHECK(up(x)) → CHECK(x)
CHECK(sent(x)) → SENT(check(x))
CHECK(sent(x)) → CHECK(x)
CHECK(rec(x)) → REC(check(x))
CHECK(rec(x)) → CHECK(x)
CHECK(no(x)) → NO(check(x))
CHECK(no(x)) → CHECK(x)
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(sent(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 7 less nodes.
(10) Complex Obligation (AND)
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
NO(up(x)) → NO(x)
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(sent(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(13) Obligation:
Q DP problem:
The TRS P consists of the following rules:
NO(up(x)) → NO(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(14) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
NO(up(x)) → NO(x)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(NO(x1)) = 2·x1
POL(up(x1)) = 2·x1
(15) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(16) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(17) TRUE
(18) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(19) Obligation:
Q DP problem:
The TRS P consists of the following rules:
NO(up(x)) → NO(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(20) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SENT(up(x)) → SENT(x)
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(sent(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(21) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SENT(up(x)) → SENT(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(23) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
SENT(up(x)) → SENT(x)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(SENT(x1)) = 2·x1
POL(up(x1)) = 2·x1
(24) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(25) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(26) TRUE
(27) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(28) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SENT(up(x)) → SENT(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(29) Obligation:
Q DP problem:
The TRS P consists of the following rules:
REC(up(x)) → REC(x)
REC(sent(x)) → REC(x)
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(sent(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(30) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(31) Obligation:
Q DP problem:
The TRS P consists of the following rules:
REC(up(x)) → REC(x)
REC(sent(x)) → REC(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(32) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
REC(up(x)) → REC(x)
REC(sent(x)) → REC(x)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(REC(x1)) = 2·x1
POL(sent(x1)) = 2·x1
POL(up(x1)) = 2·x1
(33) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(34) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(35) TRUE
(36) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(37) Obligation:
Q DP problem:
The TRS P consists of the following rules:
REC(up(x)) → REC(x)
REC(sent(x)) → REC(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(38) Obligation:
Q DP problem:
The TRS P consists of the following rules:
CHECK(sent(x)) → CHECK(x)
CHECK(up(x)) → CHECK(x)
CHECK(rec(x)) → CHECK(x)
CHECK(no(x)) → CHECK(x)
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(sent(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(39) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(40) Obligation:
Q DP problem:
The TRS P consists of the following rules:
CHECK(sent(x)) → CHECK(x)
CHECK(up(x)) → CHECK(x)
CHECK(rec(x)) → CHECK(x)
CHECK(no(x)) → CHECK(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(41) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(42) Obligation:
Q DP problem:
The TRS P consists of the following rules:
CHECK(sent(x)) → CHECK(x)
CHECK(up(x)) → CHECK(x)
CHECK(rec(x)) → CHECK(x)
CHECK(no(x)) → CHECK(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(43) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
CHECK(sent(x)) → CHECK(x)
CHECK(up(x)) → CHECK(x)
CHECK(rec(x)) → CHECK(x)
CHECK(no(x)) → CHECK(x)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(CHECK(x1)) = 2·x1
POL(no(x1)) = 2·x1
POL(rec(x1)) = 2·x1
POL(sent(x1)) = 2·x1
POL(up(x1)) = 2·x1
(44) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(45) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(46) TRUE
(47) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(sent(up(x))) → TOP(check(rec(x)))
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(sent(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(48) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(49) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(sent(up(x))) → TOP(check(rec(x)))
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
no(up(x)) → up(no(x))
sent(up(x)) → up(sent(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(50) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
no(up(x)) → up(no(x))
Used ordering: Polynomial interpretation [POLO]:
POL(TOP(x1)) = 2·x1
POL(bot) = 0
POL(check(x1)) = x1
POL(no(x1)) = 2·x1
POL(rec(x1)) = 1 + 2·x1
POL(sent(x1)) = x1
POL(up(x1)) = 1 + 2·x1
(51) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(sent(up(x))) → TOP(check(rec(x)))
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
sent(up(x)) → up(sent(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(52) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
check(no(x)) → no(check(x))
check(no(x)) → no(x)
Used ordering: Polynomial interpretation [POLO]:
POL(TOP(x1)) = 2·x1
POL(bot) = 0
POL(check(x1)) = 2·x1
POL(no(x1)) = 1 + 2·x1
POL(rec(x1)) = x1
POL(sent(x1)) = x1
POL(up(x1)) = 2·x1
(53) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(sent(up(x))) → TOP(check(rec(x)))
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
sent(up(x)) → up(sent(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(54) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
TOP(sent(up(x))) → TOP(check(rec(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
The following usable rules [FROCOS05] were oriented:
check(up(x)) → up(check(x))
rec(up(x)) → up(rec(x))
rec(bot) → up(sent(bot))
rec(sent(x)) → sent(rec(x))
sent(up(x)) → up(sent(x))
check(rec(x)) → rec(check(x))
check(sent(x)) → sent(check(x))
(55) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
sent(up(x)) → up(sent(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(56) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(57) TRUE
(58) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(59) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(sent(up(x))) → TOP(check(rec(x)))
The TRS R consists of the following rules:
rec(sent(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)
no(up(x)) → up(no(x))
sent(up(x)) → up(sent(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.