The rewrite relation of the following TRS is considered.
le(0,y) | → | true | (1) |
le(s(x),0) | → | false | (2) |
le(s(x),s(y)) | → | le(x,y) | (3) |
minus(0,y) | → | 0 | (4) |
minus(s(x),y) | → | if_minus(le(s(x),y),s(x),y) | (5) |
if_minus(true,s(x),y) | → | 0 | (6) |
if_minus(false,s(x),y) | → | s(minus(x,y)) | (7) |
gcd(0,y) | → | y | (8) |
gcd(s(x),0) | → | s(x) | (9) |
gcd(s(x),s(y)) | → | if_gcd(le(y,x),s(x),s(y)) | (10) |
if_gcd(true,s(x),s(y)) | → | gcd(minus(x,y),s(y)) | (11) |
if_gcd(false,s(x),s(y)) | → | gcd(minus(y,x),s(x)) | (12) |
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le#(0,z0) |
le#(s(z0),0) |
le#(s(z0),s(z1)) |
minus#(0,z0) |
minus#(s(z0),z1) |
if_minus#(true,s(z0),z1) |
if_minus#(false,s(z0),z1) |
gcd#(0,z0) |
gcd#(s(z0),0) |
gcd#(s(z0),s(z1)) |
if_gcd#(true,s(z0),s(z1)) |
if_gcd#(false,s(z0),s(z1)) |
gcd(0,z0) | → | z0 | (27) |
gcd(s(z0),0) | → | s(z0) | (29) |
gcd(s(z0),s(z1)) | → | if_gcd(le(z1,z0),s(z0),s(z1)) | (31) |
if_gcd(true,s(z0),s(z1)) | → | gcd(minus(z0,z1),s(z1)) | (33) |
if_gcd(false,s(z0),s(z1)) | → | gcd(minus(z1,z0),s(z0)) | (35) |
gcd#(s(z0),0) | → | c8 | (30) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 1 · x2 + 0 |
[minus(x1, x2)] | = | 1 + 1 · x2 |
[if_minus(x1, x2, x3)] | = | 1 + 1 · x2 + 1 · x3 |
[le#(x1, x2)] | = | 0 |
[minus#(x1, x2)] | = | 0 |
[if_minus#(x1, x2, x3)] | = | 1 · x2 + 0 |
[gcd#(x1, x2)] | = | 1 · x2 + 0 |
[if_gcd#(x1, x2, x3)] | = | 1 · x2 + 0 + 1 · x3 |
[0] | = | 1 |
[s(x1)] | = | 0 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
gcd#(0,z0) | → | c7 | (28) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 1 · x2 + 0 |
[minus(x1, x2)] | = | 1 + 1 · x2 |
[if_minus(x1, x2, x3)] | = | 1 + 1 · x2 + 1 · x3 |
[le#(x1, x2)] | = | 0 |
[minus#(x1, x2)] | = | 0 |
[if_minus#(x1, x2, x3)] | = | 0 |
[gcd#(x1, x2)] | = | 1 + 1 · x2 |
[if_gcd#(x1, x2, x3)] | = | 1 + 1 · x2 |
[0] | = | 1 |
[s(x1)] | = | 1 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 1 · x2 + 0 |
[minus(x1, x2)] | = | 1 · x1 + 0 |
[if_minus(x1, x2, x3)] | = | 1 · x2 + 0 |
[le#(x1, x2)] | = | 0 |
[minus#(x1, x2)] | = | 0 |
[if_minus#(x1, x2, x3)] | = | 0 |
[gcd#(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[if_gcd#(x1, x2, x3)] | = | 1 · x2 + 0 + 1 · x3 |
[0] | = | 0 |
[s(x1)] | = | 1 + 1 · x1 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
minus(s(z0),z1) | → | if_minus(le(s(z0),z1),s(z0),z1) | (21) |
if_minus(false,s(z0),z1) | → | s(minus(z0,z1)) | (25) |
if_minus(true,s(z0),z1) | → | 0 | (23) |
minus(0,z0) | → | 0 | (19) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 1 · x2 + 0 |
[minus(x1, x2)] | = | 1 · x1 + 0 |
[if_minus(x1, x2, x3)] | = | 1 · x2 + 0 |
[le#(x1, x2)] | = | 0 |
[minus#(x1, x2)] | = | 0 |
[if_minus#(x1, x2, x3)] | = | 0 |
[gcd#(x1, x2)] | = | 1 + 1 · x1 + 1 · x2 |
[if_gcd#(x1, x2, x3)] | = | 1 · x2 + 0 + 1 · x3 |
[0] | = | 0 |
[s(x1)] | = | 1 + 1 · x1 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
minus(s(z0),z1) | → | if_minus(le(s(z0),z1),s(z0),z1) | (21) |
if_minus(false,s(z0),z1) | → | s(minus(z0,z1)) | (25) |
if_minus(true,s(z0),z1) | → | 0 | (23) |
minus(0,z0) | → | 0 | (19) |
minus#(0,z0) | → | c3 | (20) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 1 · x2 + 0 |
[minus(x1, x2)] | = | 1 · x1 + 0 |
[if_minus(x1, x2, x3)] | = | 1 · x2 + 0 |
[le#(x1, x2)] | = | 0 |
[minus#(x1, x2)] | = | 1 |
[if_minus#(x1, x2, x3)] | = | 1 |
[gcd#(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[if_gcd#(x1, x2, x3)] | = | 1 · x2 + 0 + 1 · x3 |
[0] | = | 0 |
[s(x1)] | = | 1 + 1 · x1 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
minus(s(z0),z1) | → | if_minus(le(s(z0),z1),s(z0),z1) | (21) |
if_minus(false,s(z0),z1) | → | s(minus(z0,z1)) | (25) |
if_minus(true,s(z0),z1) | → | 0 | (23) |
minus(0,z0) | → | 0 | (19) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 0 |
[minus(x1, x2)] | = | 1 · x1 + 0 |
[if_minus(x1, x2, x3)] | = | 1 · x2 + 0 |
[le#(x1, x2)] | = | 0 |
[minus#(x1, x2)] | = | 1 + 1 · x1 |
[if_minus#(x1, x2, x3)] | = | 1 · x2 + 0 |
[gcd#(x1, x2)] | = | 1 + 1 · x2 · x2 + 1 · x1 · x1 |
[if_gcd#(x1, x2, x3)] | = | 1 · x3 · x3 + 0 + 1 · x2 · x2 |
[0] | = | 0 |
[s(x1)] | = | 2 + 1 · x1 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
minus(s(z0),z1) | → | if_minus(le(s(z0),z1),s(z0),z1) | (21) |
if_minus(false,s(z0),z1) | → | s(minus(z0,z1)) | (25) |
if_minus(true,s(z0),z1) | → | 0 | (23) |
minus(0,z0) | → | 0 | (19) |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 0 |
[minus(x1, x2)] | = | 1 · x1 + 0 |
[if_minus(x1, x2, x3)] | = | 1 · x2 + 0 |
[le#(x1, x2)] | = | 1 |
[minus#(x1, x2)] | = | 2 + 2 · x1 |
[if_minus#(x1, x2, x3)] | = | 1 + 2 · x2 |
[gcd#(x1, x2)] | = | 2 + 1 · x2 · x2 + 1 · x1 · x1 |
[if_gcd#(x1, x2, x3)] | = | 1 + 1 · x3 · x3 + 1 · x2 · x2 |
[0] | = | 0 |
[s(x1)] | = | 2 + 1 · x1 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
minus(s(z0),z1) | → | if_minus(le(s(z0),z1),s(z0),z1) | (21) |
if_minus(false,s(z0),z1) | → | s(minus(z0,z1)) | (25) |
if_minus(true,s(z0),z1) | → | 0 | (23) |
minus(0,z0) | → | 0 | (19) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
[c] | = | 0 |
[c1] | = | 0 |
[c2(x1)] | = | 1 · x1 + 0 |
[c3] | = | 0 |
[c4(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c5] | = | 0 |
[c6(x1)] | = | 1 · x1 + 0 |
[c7] | = | 0 |
[c8] | = | 0 |
[c9(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c10(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[c11(x1, x2)] | = | 1 · x1 + 0 + 1 · x2 |
[le(x1, x2)] | = | 0 |
[minus(x1, x2)] | = | 1 · x1 + 0 |
[if_minus(x1, x2, x3)] | = | 1 · x2 + 0 |
[le#(x1, x2)] | = | 1 · x1 + 0 |
[minus#(x1, x2)] | = | 1 · x1 + 0 + 1 · x1 · x1 |
[if_minus#(x1, x2, x3)] | = | 1 · x2 · x2 + 0 |
[gcd#(x1, x2)] | = | 1 + 1 · x1 + 1 · x2 + 1 · x1 · x2 + 1 · x1 · x1 · x1 + 1 · x2 · x2 · x2 |
[if_gcd#(x1, x2, x3)] | = | 1 · x2 · x3 + 0 + 1 · x2 · x2 · x2 + 1 · x3 · x3 · x3 |
[0] | = | 0 |
[s(x1)] | = | 1 + 1 · x1 |
[false] | = | 0 |
[true] | = | 0 |
le#(0,z0) | → | c | (14) |
le#(s(z0),0) | → | c1 | (16) |
le#(s(z0),s(z1)) | → | c2(le#(z0,z1)) | (18) |
minus#(0,z0) | → | c3 | (20) |
minus#(s(z0),z1) | → | c4(if_minus#(le(s(z0),z1),s(z0),z1),le#(s(z0),z1)) | (22) |
if_minus#(true,s(z0),z1) | → | c5 | (24) |
if_minus#(false,s(z0),z1) | → | c6(minus#(z0,z1)) | (26) |
gcd#(0,z0) | → | c7 | (28) |
gcd#(s(z0),0) | → | c8 | (30) |
gcd#(s(z0),s(z1)) | → | c9(if_gcd#(le(z1,z0),s(z0),s(z1)),le#(z1,z0)) | (32) |
if_gcd#(true,s(z0),s(z1)) | → | c10(gcd#(minus(z0,z1),s(z1)),minus#(z0,z1)) | (34) |
if_gcd#(false,s(z0),s(z1)) | → | c11(gcd#(minus(z1,z0),s(z0)),minus#(z1,z0)) | (36) |
minus(s(z0),z1) | → | if_minus(le(s(z0),z1),s(z0),z1) | (21) |
if_minus(false,s(z0),z1) | → | s(minus(z0,z1)) | (25) |
if_minus(true,s(z0),z1) | → | 0 | (23) |
minus(0,z0) | → | 0 | (19) |
There are no rules in the TRS R. Hence, R/S has complexity O(1).