YES
The TRS could be proven terminating. The proof took 44997 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (433ms).
| – Problem 2 was processed with processor SubtermCriterion (6ms).
| – Problem 3 was processed with processor ReductionPairSAT (8543ms).
| | – Problem 10 was processed with processor ReductionPairSAT (3337ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| – Problem 9 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| minus#(ok(X1), ok(X2)) | → | minus#(X1, X2) | | top#(ok(X)) | → | top#(active(X)) |
| proper#(geq(X1, X2)) | → | geq#(proper(X1), proper(X2)) | | proper#(minus(X1, X2)) | → | proper#(X1) |
| geq#(ok(X1), ok(X2)) | → | geq#(X1, X2) | | proper#(div(X1, X2)) | → | div#(proper(X1), proper(X2)) |
| top#(ok(X)) | → | active#(X) | | active#(div(s(X), s(Y))) | → | minus#(X, Y) |
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(div(s(X), s(Y))) | → | geq#(X, Y) |
| active#(div(X1, X2)) | → | div#(active(X1), X2) | | active#(div(s(X), s(Y))) | → | div#(minus(X, Y), s(Y)) |
| proper#(geq(X1, X2)) | → | proper#(X1) | | proper#(minus(X1, X2)) | → | proper#(X2) |
| if#(ok(X1), ok(X2), ok(X3)) | → | if#(X1, X2, X3) | | top#(mark(X)) | → | proper#(X) |
| active#(div(s(X), s(Y))) | → | s#(div(minus(X, Y), s(Y))) | | top#(mark(X)) | → | top#(proper(X)) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X2) |
| active#(div(s(X), s(Y))) | → | s#(Y) | | active#(s(X)) | → | s#(active(X)) |
| s#(ok(X)) | → | s#(X) | | active#(geq(s(X), s(Y))) | → | geq#(X, Y) |
| s#(mark(X)) | → | s#(X) | | proper#(s(X)) | → | proper#(X) |
| proper#(geq(X1, X2)) | → | proper#(X2) | | active#(minus(s(X), s(Y))) | → | minus#(X, Y) |
| proper#(minus(X1, X2)) | → | minus#(proper(X1), proper(X2)) | | active#(div(s(X), s(Y))) | → | if#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) |
| active#(s(X)) | → | active#(X) | | proper#(s(X)) | → | s#(proper(X)) |
| proper#(if(X1, X2, X3)) | → | proper#(X3) | | div#(ok(X1), ok(X2)) | → | div#(X1, X2) |
| active#(div(X1, X2)) | → | active#(X1) | | if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) |
| div#(mark(X1), X2) | → | div#(X1, X2) | | proper#(div(X1, X2)) | → | proper#(X1) |
| active#(if(X1, X2, X3)) | → | if#(active(X1), X2, X3) | | proper#(div(X1, X2)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | if#(proper(X1), proper(X2), proper(X3)) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
The following SCCs where found
| active#(if(X1, X2, X3)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(div(X1, X2)) → active#(X1) |
| minus#(ok(X1), ok(X2)) → minus#(X1, X2) |
| div#(ok(X1), ok(X2)) → div#(X1, X2) | div#(mark(X1), X2) → div#(X1, X2) |
| if#(mark(X1), X2, X3) → if#(X1, X2, X3) | if#(ok(X1), ok(X2), ok(X3)) → if#(X1, X2, X3) |
| geq#(ok(X1), ok(X2)) → geq#(X1, X2) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| proper#(s(X)) → proper#(X) | proper#(geq(X1, X2)) → proper#(X2) |
| proper#(if(X1, X2, X3)) → proper#(X1) | proper#(if(X1, X2, X3)) → proper#(X2) |
| proper#(if(X1, X2, X3)) → proper#(X3) | proper#(div(X1, X2)) → proper#(X1) |
| proper#(minus(X1, X2)) → proper#(X1) | proper#(geq(X1, X2)) → proper#(X1) |
| proper#(minus(X1, X2)) → proper#(X2) | proper#(div(X1, X2)) → proper#(X2) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| div#(ok(X1), ok(X2)) | → | div#(X1, X2) | | div#(mark(X1), X2) | → | div#(X1, X2) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| div#(ok(X1), ok(X2)) | → | div#(X1, X2) | | div#(mark(X1), X2) | → | div#(X1, X2) |
Problem 3: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Function Precedence
active < div < 0 < geq < s = if = false < minus = true = mark = ok = proper = top = top#
Argument Filtering
geq: 1 2
minus: 1
div: 1 2
true: all arguments are removed from true
mark: 1
0: all arguments are removed from 0
s: 1
if: 1 2 3
false: all arguments are removed from false
active: collapses to 1
ok: collapses to 1
proper: collapses to 1
top: collapses to 1
top#: 1
Status
geq: lexicographic with permutation 1 → 1 2 → 2
minus: multiset
div: lexicographic with permutation 1 → 2 2 → 1
true: multiset
mark: multiset
0: multiset
s: lexicographic with permutation 1 → 1
if: lexicographic with permutation 1 → 1 2 → 3 3 → 2
false: multiset
top#: multiset
Usable Rules
| proper(false) → ok(false) | proper(div(X1, X2)) → div(proper(X1), proper(X2)) |
| active(s(X)) → s(active(X)) | active(div(0, s(Y))) → mark(0) |
| active(if(X1, X2, X3)) → if(active(X1), X2, X3) | active(minus(0, Y)) → mark(0) |
| s(mark(X)) → mark(s(X)) | proper(s(X)) → s(proper(X)) |
| proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3)) | geq(ok(X1), ok(X2)) → ok(geq(X1, X2)) |
| proper(geq(X1, X2)) → geq(proper(X1), proper(X2)) | proper(minus(X1, X2)) → minus(proper(X1), proper(X2)) |
| active(minus(s(X), s(Y))) → mark(minus(X, Y)) | if(mark(X1), X2, X3) → mark(if(X1, X2, X3)) |
| active(geq(X, 0)) → mark(true) | if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3)) |
| active(geq(0, s(Y))) → mark(false) | s(ok(X)) → ok(s(X)) |
| active(div(X1, X2)) → div(active(X1), X2) | active(if(false, X, Y)) → mark(Y) |
| proper(true) → ok(true) | div(mark(X1), X2) → mark(div(X1, X2)) |
| active(geq(s(X), s(Y))) → mark(geq(X, Y)) | active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) |
| div(ok(X1), ok(X2)) → ok(div(X1, X2)) | proper(0) → ok(0) |
| active(if(true, X, Y)) → mark(X) | minus(ok(X1), ok(X2)) → ok(minus(X1, X2)) |
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| top#(mark(X)) → top#(proper(X)) |
Problem 10: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Function Precedence
false < geq = if < true = s < 0 < active = top# < minus = div = mark = ok = proper = top
Argument Filtering
geq: 1 2
minus: 2
div: 2
true: all arguments are removed from true
mark: all arguments are removed from mark
0: all arguments are removed from 0
s: collapses to 1
if: collapses to 1
false: all arguments are removed from false
active: collapses to 1
ok: 1
proper: all arguments are removed from proper
top: 1
top#: 1
Status
geq: lexicographic with permutation 1 → 2 2 → 1
minus: lexicographic with permutation 2 → 1
div: lexicographic with permutation 2 → 1
true: multiset
mark: multiset
0: multiset
false: multiset
ok: lexicographic with permutation 1 → 1
proper: multiset
top: lexicographic with permutation 1 → 1
top#: lexicographic with permutation 1 → 1
Usable Rules
| if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3)) | active(s(X)) → s(active(X)) |
| active(geq(0, s(Y))) → mark(false) | s(ok(X)) → ok(s(X)) |
| active(div(0, s(Y))) → mark(0) | active(if(X1, X2, X3)) → if(active(X1), X2, X3) |
| active(minus(0, Y)) → mark(0) | active(if(false, X, Y)) → mark(Y) |
| active(div(X1, X2)) → div(active(X1), X2) | div(mark(X1), X2) → mark(div(X1, X2)) |
| active(geq(s(X), s(Y))) → mark(geq(X, Y)) | active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) |
| s(mark(X)) → mark(s(X)) | geq(ok(X1), ok(X2)) → ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) → ok(div(X1, X2)) | active(minus(s(X), s(Y))) → mark(minus(X, Y)) |
| active(if(true, X, Y)) → mark(X) | if(mark(X1), X2, X3) → mark(if(X1, X2, X3)) |
| minus(ok(X1), ok(X2)) → ok(minus(X1, X2)) | active(geq(X, 0)) → mark(true) |
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| top#(ok(X)) → top#(active(X)) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(div(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(div(X1, X2)) | → | active#(X1) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(s(X)) | → | proper#(X) | | proper#(geq(X1, X2)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X3) | | proper#(div(X1, X2)) | → | proper#(X1) |
| proper#(minus(X1, X2)) | → | proper#(X1) | | proper#(geq(X1, X2)) | → | proper#(X1) |
| proper#(minus(X1, X2)) | → | proper#(X2) | | proper#(div(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(s(X)) | → | proper#(X) | | proper#(geq(X1, X2)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X3) | | proper#(div(X1, X2)) | → | proper#(X1) |
| proper#(geq(X1, X2)) | → | proper#(X1) | | proper#(minus(X1, X2)) | → | proper#(X1) |
| proper#(minus(X1, X2)) | → | proper#(X2) | | proper#(div(X1, X2)) | → | proper#(X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| geq#(ok(X1), ok(X2)) | → | geq#(X1, X2) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| geq#(ok(X1), ok(X2)) | → | geq#(X1, X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | if#(ok(X1), ok(X2), ok(X3)) | → | if#(X1, X2, X3) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | if#(ok(X1), ok(X2), ok(X3)) | → | if#(X1, X2, X3) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(ok(X1), ok(X2)) | → | minus#(X1, X2) |
Rewrite Rules
| active(minus(0, Y)) | → | mark(0) | | active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) |
| active(geq(X, 0)) | → | mark(true) | | active(geq(0, s(Y))) | → | mark(false) |
| active(geq(s(X), s(Y))) | → | mark(geq(X, Y)) | | active(div(0, s(Y))) | → | mark(0) |
| active(div(s(X), s(Y))) | → | mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(s(X)) | → | s(active(X)) |
| active(div(X1, X2)) | → | div(active(X1), X2) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| s(mark(X)) | → | mark(s(X)) | | div(mark(X1), X2) | → | mark(div(X1, X2)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(geq(X1, X2)) | → | geq(proper(X1), proper(X2)) | | proper(true) | → | ok(true) |
| proper(false) | → | ok(false) | | proper(div(X1, X2)) | → | div(proper(X1), proper(X2)) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | geq(ok(X1), ok(X2)) | → | ok(geq(X1, X2)) |
| div(ok(X1), ok(X2)) | → | ok(div(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: geq, minus, div, true, mark, 0, s, if, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(ok(X1), ok(X2)) | → | minus#(X1, X2) |