TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60043 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (51ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor BackwardInstantiation (2ms).
| | – Problem 5 remains open; application of the following processors failed [ForwardInstantiation (2ms), Propagation (2ms), ForwardNarrowing (0ms), BackwardInstantiation (2ms), ForwardInstantiation (1ms), Propagation (1ms)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| div#(x, y, z) | → | if#(ge(y, s(0)), ge(x, y), x, y, z) | | if#(true, true, x, y, z) | → | div#(minus(x, y), y, id_inc(z)) |
Rewrite Rules
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(x, 0) | → | x |
| minus(0, y) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| id_inc(x) | → | x | | id_inc(x) | → | s(x) |
| quot(x, y) | → | div(x, y, 0) | | div(x, y, z) | → | if(ge(y, s(0)), ge(x, y), x, y, z) |
| if(false, b, x, y, z) | → | div_by_zero | | if(true, false, x, y, z) | → | z |
| if(true, true, x, y, z) | → | div(minus(x, y), y, id_inc(z)) |
Original Signature
Termination of terms over the following signature is verified: div_by_zero, id_inc, minus, 0, s, if, div, false, true, ge, quot
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| div#(x, y, z) | → | ge#(x, y) | | div#(x, y, z) | → | if#(ge(y, s(0)), ge(x, y), x, y, z) |
| div#(x, y, z) | → | ge#(y, s(0)) | | ge#(s(x), s(y)) | → | ge#(x, y) |
| minus#(s(x), s(y)) | → | minus#(x, y) | | if#(true, true, x, y, z) | → | minus#(x, y) |
| if#(true, true, x, y, z) | → | div#(minus(x, y), y, id_inc(z)) | | quot#(x, y) | → | div#(x, y, 0) |
| if#(true, true, x, y, z) | → | id_inc#(z) |
Rewrite Rules
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(x, 0) | → | x |
| minus(0, y) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| id_inc(x) | → | x | | id_inc(x) | → | s(x) |
| quot(x, y) | → | div(x, y, 0) | | div(x, y, z) | → | if(ge(y, s(0)), ge(x, y), x, y, z) |
| if(false, b, x, y, z) | → | div_by_zero | | if(true, false, x, y, z) | → | z |
| if(true, true, x, y, z) | → | div(minus(x, y), y, id_inc(z)) |
Original Signature
Termination of terms over the following signature is verified: div_by_zero, 0, minus, id_inc, s, if, div, true, false, ge, quot
Strategy
The following SCCs where found
| minus#(s(x), s(y)) → minus#(x, y) |
| ge#(s(x), s(y)) → ge#(x, y) |
| div#(x, y, z) → if#(ge(y, s(0)), ge(x, y), x, y, z) | if#(true, true, x, y, z) → div#(minus(x, y), y, id_inc(z)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(s(x), s(y)) | → | minus#(x, y) |
Rewrite Rules
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(x, 0) | → | x |
| minus(0, y) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| id_inc(x) | → | x | | id_inc(x) | → | s(x) |
| quot(x, y) | → | div(x, y, 0) | | div(x, y, z) | → | if(ge(y, s(0)), ge(x, y), x, y, z) |
| if(false, b, x, y, z) | → | div_by_zero | | if(true, false, x, y, z) | → | z |
| if(true, true, x, y, z) | → | div(minus(x, y), y, id_inc(z)) |
Original Signature
Termination of terms over the following signature is verified: div_by_zero, 0, minus, id_inc, s, if, div, true, false, ge, quot
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(s(x), s(y)) | → | minus#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| ge#(s(x), s(y)) | → | ge#(x, y) |
Rewrite Rules
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(x, 0) | → | x |
| minus(0, y) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| id_inc(x) | → | x | | id_inc(x) | → | s(x) |
| quot(x, y) | → | div(x, y, 0) | | div(x, y, z) | → | if(ge(y, s(0)), ge(x, y), x, y, z) |
| if(false, b, x, y, z) | → | div_by_zero | | if(true, false, x, y, z) | → | z |
| if(true, true, x, y, z) | → | div(minus(x, y), y, id_inc(z)) |
Original Signature
Termination of terms over the following signature is verified: div_by_zero, 0, minus, id_inc, s, if, div, true, false, ge, quot
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| ge#(s(x), s(y)) | → | ge#(x, y) |
Problem 4: BackwardInstantiation
Dependency Pair Problem
Dependency Pairs
| div#(x, y, z) | → | if#(ge(y, s(0)), ge(x, y), x, y, z) | | if#(true, true, x, y, z) | → | div#(minus(x, y), y, id_inc(z)) |
Rewrite Rules
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(x, 0) | → | x |
| minus(0, y) | → | 0 | | minus(s(x), s(y)) | → | minus(x, y) |
| id_inc(x) | → | x | | id_inc(x) | → | s(x) |
| quot(x, y) | → | div(x, y, 0) | | div(x, y, z) | → | if(ge(y, s(0)), ge(x, y), x, y, z) |
| if(false, b, x, y, z) | → | div_by_zero | | if(true, false, x, y, z) | → | z |
| if(true, true, x, y, z) | → | div(minus(x, y), y, id_inc(z)) |
Original Signature
Termination of terms over the following signature is verified: div_by_zero, 0, minus, id_inc, s, if, div, true, false, ge, quot
Strategy
Instantiation
For all potential predecessors l → r of the rule div
#(
x,
y,
z) → if
#(ge(
y, s(0)), ge(
x,
y),
x,
y,
z) on dependency pair chains it holds that:
- div#(x, y, z) matches r,
- all variables of div#(x, y, z) are embedded in constructor contexts, i.e., each subterm of div#(x, y, z), containing a variable is rooted by a constructor symbol.
Thus, div
#(
x,
y,
z) → if
#(ge(
y, s(0)), ge(
x,
y),
x,
y,
z) is replaced by instances determined through the above matching. These instances are:
| div#(minus(_x, _y), _y, id_inc(_z)) → if#(ge(_y, s(0)), ge(minus(_x, _y), _y), minus(_x, _y), _y, id_inc(_z)) |