We consider the TRS containing the following rules:
| c | → | f(a,h(b)) | (1) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| c | → | c | (3) |
| g(g(a)) | → | f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))) | (4) |
| f(x,y) | → | f(y,x) | (5) |
The underlying signature is as follows:
{c/0, f/2, a/0, h/1, b/0, g/1}To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| f(x,y) | → | f(y,x) | (5) |
| g(g(a)) | → | f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))) | (4) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| c | → | f(a,h(b)) | (1) |
All redundant rules that were added or removed can be simulated in 1 steps .
To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| f(x,y) | → | f(y,x) | (5) |
| g(g(a)) | → | f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))) | (4) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| c | → | f(a,h(b)) | (1) |
| f(h(f(f(a,a),h(a))),g(f(g(b),x))) | → | c | (6) |
| f(h(f(h(a),f(a,a))),g(f(x,g(b)))) | → | c | (7) |
| f(g(f(x,g(b))),h(f(f(a,a),h(a)))) | → | c | (8) |
| f(g(f(x61,g(b))),h(f(f(a,a),h(a)))) | → | c | (9) |
All redundant rules that were added or removed can be simulated in 2 steps .
To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| f(g(f(x61,g(b))),h(f(f(a,a),h(a)))) | → | c | (9) |
| f(h(f(h(a),f(a,a))),g(f(x,g(b)))) | → | c | (7) |
| f(h(f(f(a,a),h(a))),g(f(g(b),x))) | → | c | (6) |
| c | → | f(a,h(b)) | (1) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| g(g(a)) | → | f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))) | (4) |
| f(x,y) | → | f(y,x) | (5) |
All redundant rules that were added or removed can be simulated in 1 steps .
To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| f(g(f(x61,g(b))),h(f(f(a,a),h(a)))) | → | c | (9) |
| f(h(f(h(a),f(a,a))),g(f(x,g(b)))) | → | c | (7) |
| f(h(f(f(a,a),h(a))),g(f(g(b),x))) | → | c | (6) |
| c | → | f(a,h(b)) | (1) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| g(g(a)) | → | f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))) | (4) |
| f(x,y) | → | f(y,x) | (5) |
| f(h(f(h(a),f(a,a))),g(f(g(b),x))) | → | c | (10) |
| f(g(f(g(b),x)),h(f(f(a,a),h(a)))) | → | c | (11) |
| f(g(f(x,g(b))),h(f(h(a),f(a,a)))) | → | c | (12) |
| f(g(f(x61,g(b))),h(f(h(a),f(a,a)))) | → | c | (13) |
| f(g(f(g(b),x61)),h(f(f(a,a),h(a)))) | → | c | (14) |
| f(g(f(g(b),x383)),h(f(f(a,a),h(a)))) | → | c | (15) |
| f(g(f(x381,g(b))),h(f(h(a),f(a,a)))) | → | c | (16) |
All redundant rules that were added or removed can be simulated in 2 steps .
To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| f(g(f(x381,g(b))),h(f(h(a),f(a,a)))) | → | c | (16) |
| f(g(f(g(b),x383)),h(f(f(a,a),h(a)))) | → | c | (15) |
| f(h(f(h(a),f(a,a))),g(f(g(b),x))) | → | c | (10) |
| f(x,y) | → | f(y,x) | (5) |
| g(g(a)) | → | f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))) | (4) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| c | → | f(a,h(b)) | (1) |
| f(h(f(f(a,a),h(a))),g(f(g(b),x))) | → | c | (6) |
| f(h(f(h(a),f(a,a))),g(f(x,g(b)))) | → | c | (7) |
| f(g(f(x61,g(b))),h(f(f(a,a),h(a)))) | → | c | (9) |
All redundant rules that were added or removed can be simulated in 1 steps .
To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| f(g(f(x381,g(b))),h(f(h(a),f(a,a)))) | → | c | (16) |
| f(g(f(g(b),x383)),h(f(f(a,a),h(a)))) | → | c | (15) |
| f(h(f(h(a),f(a,a))),g(f(g(b),x))) | → | c | (10) |
| f(x,y) | → | f(y,x) | (5) |
| g(g(a)) | → | f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))) | (4) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| c | → | f(a,h(b)) | (1) |
| f(h(f(f(a,a),h(a))),g(f(g(b),x))) | → | c | (6) |
| f(h(f(h(a),f(a,a))),g(f(x,g(b)))) | → | c | (7) |
| f(g(f(x61,g(b))),h(f(f(a,a),h(a)))) | → | c | (9) |
| f(g(f(g(b),x)),h(f(h(a),f(a,a)))) | → | c | (17) |
| f(g(f(g(b),x383)),h(f(h(a),f(a,a)))) | → | c | (18) |
| f(g(f(g(b),x381)),h(f(h(a),f(a,a)))) | → | c | (19) |
| f(g(f(g(b),x1222)),h(f(h(a),f(a,a)))) | → | c | (20) |
All redundant rules that were added or removed can be simulated in 2 steps .
Confluence is proven using the following terminating critical-pair-closing-system R:
| f(h(f(h(a),f(a,a))),g(f(x,g(b)))) | → | c | (7) |
| f(h(f(f(a,a),h(a))),g(f(g(b),x))) | → | c | (6) |
| f(g(f(g(b),x)),h(f(h(a),f(a,a)))) | → | c | (17) |
| f(g(f(x61,g(b))),h(f(f(a,a),h(a)))) | → | c | (9) |
| f(g(f(x381,g(b))),h(f(h(a),f(a,a)))) | → | c | (16) |
| f(g(f(g(b),x383)),h(f(f(a,a),h(a)))) | → | c | (15) |
| f(h(f(h(a),f(a,a))),g(f(g(b),x))) | → | c | (10) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
| [f(x1, x2)] | = | 1 · x1 + 1 · x2 + 0 |
| [a] | = | 0 |
| [h(x1)] | = | 4 · x1 + 0 |
| [g(x1)] | = | 2 · x1 + 2 |
| [c] | = | 0 |
| [b] | = | 2 |
| f(h(f(h(a),f(a,a))),g(f(x,g(b)))) | → | c | (7) |
| f(h(f(f(a,a),h(a))),g(f(g(b),x))) | → | c | (6) |
| f(g(f(g(b),x)),h(f(h(a),f(a,a)))) | → | c | (17) |
| f(g(f(x61,g(b))),h(f(f(a,a),h(a)))) | → | c | (9) |
| f(g(f(x381,g(b))),h(f(h(a),f(a,a)))) | → | c | (16) |
| f(g(f(g(b),x383)),h(f(f(a,a),h(a)))) | → | c | (15) |
| f(h(f(h(a),f(a,a))),g(f(g(b),x))) | → | c | (10) |
| f(h(f(f(a,a),h(a))),g(f(x,g(b)))) | → | c | (2) |
There are no rules in the TRS. Hence, it is terminating.