The rewrite relation of the following TRS is considered.
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,V2) | → | a__U32(a__isNat(V2)) | (4) |
a__U32(tt) | → | tt | (5) |
a__U41(tt,N) | → | mark(N) | (6) |
a__U51(tt,M,N) | → | a__U52(a__isNat(N),M,N) | (7) |
a__U52(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (8) |
a__U61(tt) | → | 0 | (9) |
a__U71(tt,M,N) | → | a__U72(a__isNat(N),M,N) | (10) |
a__U72(tt,M,N) | → | a__plus(a__x(mark(N),mark(M)),mark(N)) | (11) |
a__isNat(0) | → | tt | (12) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (13) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (14) |
a__isNat(x(V1,V2)) | → | a__U31(a__isNat(V1),V2) | (15) |
a__plus(N,0) | → | a__U41(a__isNat(N),N) | (16) |
a__plus(N,s(M)) | → | a__U51(a__isNat(M),M,N) | (17) |
a__x(N,0) | → | a__U61(a__isNat(N)) | (18) |
a__x(N,s(M)) | → | a__U71(a__isNat(M),M,N) | (19) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (20) |
mark(U12(X)) | → | a__U12(mark(X)) | (21) |
mark(isNat(X)) | → | a__isNat(X) | (22) |
mark(U21(X)) | → | a__U21(mark(X)) | (23) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (24) |
mark(U32(X)) | → | a__U32(mark(X)) | (25) |
mark(U41(X1,X2)) | → | a__U41(mark(X1),X2) | (26) |
mark(U51(X1,X2,X3)) | → | a__U51(mark(X1),X2,X3) | (27) |
mark(U52(X1,X2,X3)) | → | a__U52(mark(X1),X2,X3) | (28) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (29) |
mark(U61(X)) | → | a__U61(mark(X)) | (30) |
mark(U71(X1,X2,X3)) | → | a__U71(mark(X1),X2,X3) | (31) |
mark(U72(X1,X2,X3)) | → | a__U72(mark(X1),X2,X3) | (32) |
mark(x(X1,X2)) | → | a__x(mark(X1),mark(X2)) | (33) |
mark(tt) | → | tt | (34) |
mark(s(X)) | → | s(mark(X)) | (35) |
mark(0) | → | 0 | (36) |
a__U11(X1,X2) | → | U11(X1,X2) | (37) |
a__U12(X) | → | U12(X) | (38) |
a__isNat(X) | → | isNat(X) | (39) |
a__U21(X) | → | U21(X) | (40) |
a__U31(X1,X2) | → | U31(X1,X2) | (41) |
a__U32(X) | → | U32(X) | (42) |
a__U41(X1,X2) | → | U41(X1,X2) | (43) |
a__U51(X1,X2,X3) | → | U51(X1,X2,X3) | (44) |
a__U52(X1,X2,X3) | → | U52(X1,X2,X3) | (45) |
a__plus(X1,X2) | → | plus(X1,X2) | (46) |
a__U61(X) | → | U61(X) | (47) |
a__U71(X1,X2,X3) | → | U71(X1,X2,X3) | (48) |
a__U72(X1,X2,X3) | → | U72(X1,X2,X3) | (49) |
a__x(X1,X2) | → | x(X1,X2) | (50) |
a__U11#(tt,V2) | → | a__U12#(a__isNat(V2)) | (51) |
a__U11#(tt,V2) | → | a__isNat#(V2) | (52) |
a__U31#(tt,V2) | → | a__U32#(a__isNat(V2)) | (53) |
a__U31#(tt,V2) | → | a__isNat#(V2) | (54) |
a__U41#(tt,N) | → | mark#(N) | (55) |
a__U51#(tt,M,N) | → | a__U52#(a__isNat(N),M,N) | (56) |
a__U51#(tt,M,N) | → | a__isNat#(N) | (57) |
a__U52#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (58) |
a__U52#(tt,M,N) | → | mark#(N) | (59) |
a__U52#(tt,M,N) | → | mark#(M) | (60) |
a__U71#(tt,M,N) | → | a__U72#(a__isNat(N),M,N) | (61) |
a__U71#(tt,M,N) | → | a__isNat#(N) | (62) |
a__U72#(tt,M,N) | → | a__plus#(a__x(mark(N),mark(M)),mark(N)) | (63) |
a__U72#(tt,M,N) | → | a__x#(mark(N),mark(M)) | (64) |
a__U72#(tt,M,N) | → | mark#(N) | (65) |
a__U72#(tt,M,N) | → | mark#(M) | (66) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNat(V1),V2) | (67) |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (68) |
a__isNat#(s(V1)) | → | a__U21#(a__isNat(V1)) | (69) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (70) |
a__isNat#(x(V1,V2)) | → | a__U31#(a__isNat(V1),V2) | (71) |
a__isNat#(x(V1,V2)) | → | a__isNat#(V1) | (72) |
a__plus#(N,0) | → | a__U41#(a__isNat(N),N) | (73) |
a__plus#(N,0) | → | a__isNat#(N) | (74) |
a__plus#(N,s(M)) | → | a__U51#(a__isNat(M),M,N) | (75) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (76) |
a__x#(N,0) | → | a__U61#(a__isNat(N)) | (77) |
a__x#(N,0) | → | a__isNat#(N) | (78) |
a__x#(N,s(M)) | → | a__U71#(a__isNat(M),M,N) | (79) |
a__x#(N,s(M)) | → | a__isNat#(M) | (80) |
mark#(U11(X1,X2)) | → | a__U11#(mark(X1),X2) | (81) |
mark#(U11(X1,X2)) | → | mark#(X1) | (82) |
mark#(U12(X)) | → | a__U12#(mark(X)) | (83) |
mark#(U12(X)) | → | mark#(X) | (84) |
mark#(isNat(X)) | → | a__isNat#(X) | (85) |
mark#(U21(X)) | → | a__U21#(mark(X)) | (86) |
mark#(U21(X)) | → | mark#(X) | (87) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (88) |
mark#(U31(X1,X2)) | → | mark#(X1) | (89) |
mark#(U32(X)) | → | a__U32#(mark(X)) | (90) |
mark#(U32(X)) | → | mark#(X) | (91) |
mark#(U41(X1,X2)) | → | a__U41#(mark(X1),X2) | (92) |
mark#(U41(X1,X2)) | → | mark#(X1) | (93) |
mark#(U51(X1,X2,X3)) | → | a__U51#(mark(X1),X2,X3) | (94) |
mark#(U51(X1,X2,X3)) | → | mark#(X1) | (95) |
mark#(U52(X1,X2,X3)) | → | a__U52#(mark(X1),X2,X3) | (96) |
mark#(U52(X1,X2,X3)) | → | mark#(X1) | (97) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (98) |
mark#(plus(X1,X2)) | → | mark#(X1) | (99) |
mark#(plus(X1,X2)) | → | mark#(X2) | (100) |
mark#(U61(X)) | → | a__U61#(mark(X)) | (101) |
mark#(U61(X)) | → | mark#(X) | (102) |
mark#(U71(X1,X2,X3)) | → | a__U71#(mark(X1),X2,X3) | (103) |
mark#(U71(X1,X2,X3)) | → | mark#(X1) | (104) |
mark#(U72(X1,X2,X3)) | → | a__U72#(mark(X1),X2,X3) | (105) |
mark#(U72(X1,X2,X3)) | → | mark#(X1) | (106) |
mark#(x(X1,X2)) | → | a__x#(mark(X1),mark(X2)) | (107) |
mark#(x(X1,X2)) | → | mark#(X1) | (108) |
mark#(x(X1,X2)) | → | mark#(X2) | (109) |
mark#(s(X)) | → | mark#(X) | (110) |
The dependency pairs are split into 2 components.
mark#(U11(X1,X2)) | → | mark#(X1) | (82) |
mark#(U12(X)) | → | mark#(X) | (84) |
mark#(U21(X)) | → | mark#(X) | (87) |
mark#(U31(X1,X2)) | → | mark#(X1) | (89) |
mark#(U32(X)) | → | mark#(X) | (91) |
mark#(U41(X1,X2)) | → | a__U41#(mark(X1),X2) | (92) |
a__U41#(tt,N) | → | mark#(N) | (55) |
mark#(U41(X1,X2)) | → | mark#(X1) | (93) |
mark#(U51(X1,X2,X3)) | → | a__U51#(mark(X1),X2,X3) | (94) |
a__U51#(tt,M,N) | → | a__U52#(a__isNat(N),M,N) | (56) |
a__U52#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (58) |
a__plus#(N,0) | → | a__U41#(a__isNat(N),N) | (73) |
a__plus#(N,s(M)) | → | a__U51#(a__isNat(M),M,N) | (75) |
a__U52#(tt,M,N) | → | mark#(N) | (59) |
mark#(U51(X1,X2,X3)) | → | mark#(X1) | (95) |
mark#(U52(X1,X2,X3)) | → | a__U52#(mark(X1),X2,X3) | (96) |
a__U52#(tt,M,N) | → | mark#(M) | (60) |
mark#(U52(X1,X2,X3)) | → | mark#(X1) | (97) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (98) |
mark#(plus(X1,X2)) | → | mark#(X1) | (99) |
mark#(plus(X1,X2)) | → | mark#(X2) | (100) |
mark#(U61(X)) | → | mark#(X) | (102) |
mark#(U71(X1,X2,X3)) | → | a__U71#(mark(X1),X2,X3) | (103) |
a__U71#(tt,M,N) | → | a__U72#(a__isNat(N),M,N) | (61) |
a__U72#(tt,M,N) | → | a__plus#(a__x(mark(N),mark(M)),mark(N)) | (63) |
a__U72#(tt,M,N) | → | a__x#(mark(N),mark(M)) | (64) |
a__x#(N,s(M)) | → | a__U71#(a__isNat(M),M,N) | (79) |
a__U72#(tt,M,N) | → | mark#(N) | (65) |
mark#(U71(X1,X2,X3)) | → | mark#(X1) | (104) |
mark#(U72(X1,X2,X3)) | → | a__U72#(mark(X1),X2,X3) | (105) |
a__U72#(tt,M,N) | → | mark#(M) | (66) |
mark#(U72(X1,X2,X3)) | → | mark#(X1) | (106) |
mark#(x(X1,X2)) | → | a__x#(mark(X1),mark(X2)) | (107) |
mark#(x(X1,X2)) | → | mark#(X1) | (108) |
mark#(x(X1,X2)) | → | mark#(X2) | (109) |
mark#(s(X)) | → | mark#(X) | (110) |
prec(mark#) | = | 1 | stat(mark#) | = | mul | |
prec(U41) | = | 0 | stat(U41) | = | mul | |
prec(a__U41#) | = | 1 | stat(a__U41#) | = | mul | |
prec(tt) | = | 2 | stat(tt) | = | mul | |
prec(U51) | = | 3 | stat(U51) | = | mul | |
prec(a__U51#) | = | 1 | stat(a__U51#) | = | mul | |
prec(a__U52#) | = | 1 | stat(a__U52#) | = | mul | |
prec(a__isNat) | = | 2 | stat(a__isNat) | = | mul | |
prec(a__plus#) | = | 1 | stat(a__plus#) | = | mul | |
prec(0) | = | 1 | stat(0) | = | mul | |
prec(s) | = | 2 | stat(s) | = | mul | |
prec(U52) | = | 3 | stat(U52) | = | mul | |
prec(plus) | = | 3 | stat(plus) | = | mul | |
prec(U71) | = | 4 | stat(U71) | = | lex | |
prec(a__U71#) | = | 4 | stat(a__U71#) | = | lex | |
prec(a__U72#) | = | 4 | stat(a__U72#) | = | lex | |
prec(a__x) | = | 4 | stat(a__x) | = | lex | |
prec(a__x#) | = | 4 | stat(a__x#) | = | lex | |
prec(U72) | = | 4 | stat(U72) | = | lex | |
prec(x) | = | 4 | stat(x) | = | lex | |
prec(isNat) | = | 2 | stat(isNat) | = | mul | |
prec(a__U41) | = | 0 | stat(a__U41) | = | mul | |
prec(a__plus) | = | 3 | stat(a__plus) | = | mul | |
prec(a__U71) | = | 4 | stat(a__U71) | = | lex | |
prec(a__U72) | = | 4 | stat(a__U72) | = | lex | |
prec(a__U51) | = | 3 | stat(a__U51) | = | mul | |
prec(a__U52) | = | 3 | stat(a__U52) | = | mul |
π(mark#) | = | [1] |
π(U11) | = | 1 |
π(U12) | = | 1 |
π(U21) | = | 1 |
π(U31) | = | 1 |
π(U32) | = | 1 |
π(U41) | = | [1,2] |
π(a__U41#) | = | [2] |
π(mark) | = | 1 |
π(tt) | = | [] |
π(U51) | = | [1,2,3] |
π(a__U51#) | = | [1,2,3] |
π(a__U52#) | = | [1,2,3] |
π(a__isNat) | = | [] |
π(a__plus#) | = | [1,2] |
π(0) | = | [] |
π(s) | = | [1] |
π(U52) | = | [1,2,3] |
π(plus) | = | [1,2] |
π(U61) | = | 1 |
π(U71) | = | [2,3,1] |
π(a__U71#) | = | [2,3,1] |
π(a__U72#) | = | [2,3,1] |
π(a__x) | = | [2,1] |
π(a__x#) | = | [2,1] |
π(U72) | = | [2,3,1] |
π(x) | = | [2,1] |
π(a__U11) | = | 1 |
π(a__U12) | = | 1 |
π(isNat) | = | [] |
π(a__U21) | = | 1 |
π(a__U31) | = | 1 |
π(a__U32) | = | 1 |
π(a__U41) | = | [1,2] |
π(a__plus) | = | [1,2] |
π(a__U71) | = | [2,3,1] |
π(a__U72) | = | [2,3,1] |
π(a__U51) | = | [1,2,3] |
π(a__U52) | = | [1,2,3] |
π(a__U61) | = | 1 |
mark#(U41(X1,X2)) | → | a__U41#(mark(X1),X2) | (92) |
mark#(U41(X1,X2)) | → | mark#(X1) | (93) |
mark#(U51(X1,X2,X3)) | → | a__U51#(mark(X1),X2,X3) | (94) |
a__U52#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (58) |
a__plus#(N,0) | → | a__U41#(a__isNat(N),N) | (73) |
a__plus#(N,s(M)) | → | a__U51#(a__isNat(M),M,N) | (75) |
a__U52#(tt,M,N) | → | mark#(N) | (59) |
mark#(U51(X1,X2,X3)) | → | mark#(X1) | (95) |
mark#(U52(X1,X2,X3)) | → | a__U52#(mark(X1),X2,X3) | (96) |
a__U52#(tt,M,N) | → | mark#(M) | (60) |
mark#(U52(X1,X2,X3)) | → | mark#(X1) | (97) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (98) |
mark#(plus(X1,X2)) | → | mark#(X1) | (99) |
mark#(plus(X1,X2)) | → | mark#(X2) | (100) |
mark#(U71(X1,X2,X3)) | → | a__U71#(mark(X1),X2,X3) | (103) |
a__U72#(tt,M,N) | → | a__plus#(a__x(mark(N),mark(M)),mark(N)) | (63) |
a__U72#(tt,M,N) | → | a__x#(mark(N),mark(M)) | (64) |
a__x#(N,s(M)) | → | a__U71#(a__isNat(M),M,N) | (79) |
a__U72#(tt,M,N) | → | mark#(N) | (65) |
mark#(U71(X1,X2,X3)) | → | mark#(X1) | (104) |
mark#(U72(X1,X2,X3)) | → | a__U72#(mark(X1),X2,X3) | (105) |
a__U72#(tt,M,N) | → | mark#(M) | (66) |
mark#(U72(X1,X2,X3)) | → | mark#(X1) | (106) |
mark#(x(X1,X2)) | → | a__x#(mark(X1),mark(X2)) | (107) |
mark#(x(X1,X2)) | → | mark#(X1) | (108) |
mark#(x(X1,X2)) | → | mark#(X2) | (109) |
mark#(s(X)) | → | mark#(X) | (110) |
The dependency pairs are split into 1 component.
mark#(U12(X)) | → | mark#(X) | (84) |
mark#(U11(X1,X2)) | → | mark#(X1) | (82) |
mark#(U21(X)) | → | mark#(X) | (87) |
mark#(U31(X1,X2)) | → | mark#(X1) | (89) |
mark#(U32(X)) | → | mark#(X) | (91) |
mark#(U61(X)) | → | mark#(X) | (102) |
[U12(x1)] | = | 1 · x1 |
[U11(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U21(x1)] | = | 1 · x1 |
[U31(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U32(x1)] | = | 1 · x1 |
[U61(x1)] | = | 1 · x1 |
[mark#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(U12(X)) | → | mark#(X) | (84) |
1 | > | 1 | |
mark#(U11(X1,X2)) | → | mark#(X1) | (82) |
1 | > | 1 | |
mark#(U21(X)) | → | mark#(X) | (87) |
1 | > | 1 | |
mark#(U31(X1,X2)) | → | mark#(X1) | (89) |
1 | > | 1 | |
mark#(U32(X)) | → | mark#(X) | (91) |
1 | > | 1 | |
mark#(U61(X)) | → | mark#(X) | (102) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__U11#(tt,V2) | → | a__isNat#(V2) | (52) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNat(V1),V2) | (67) |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (68) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (70) |
a__isNat#(x(V1,V2)) | → | a__U31#(a__isNat(V1),V2) | (71) |
a__U31#(tt,V2) | → | a__isNat#(V2) | (54) |
a__isNat#(x(V1,V2)) | → | a__isNat#(V1) | (72) |
[a__isNat(x1)] | = | 1 · x1 |
[0] | = | 0 |
[tt] | = | 0 |
[plus(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__U11(x1, x2)] | = | 1 · x1 + 1 · x2 |
[s(x1)] | = | 1 · x1 |
[a__U21(x1)] | = | 1 · x1 |
[x(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__U31(x1, x2)] | = | 1 · x1 + 1 · x2 |
[isNat(x1)] | = | 1 · x1 |
[a__U32(x1)] | = | 1 · x1 |
[U31(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U32(x1)] | = | 1 · x1 |
[U21(x1)] | = | 1 · x1 |
[a__U12(x1)] | = | 1 · x1 |
[U11(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U12(x1)] | = | 1 · x1 |
[a__isNat#(x1)] | = | 1 · x1 |
[a__U11#(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__U31#(x1, x2)] | = | 1 · x1 + 1 · x2 |
a__isNat(0) | → | tt | (12) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (13) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (14) |
a__isNat(x(V1,V2)) | → | a__U31(a__isNat(V1),V2) | (15) |
a__isNat(X) | → | isNat(X) | (39) |
a__U31(tt,V2) | → | a__U32(a__isNat(V2)) | (4) |
a__U31(X1,X2) | → | U31(X1,X2) | (41) |
a__U32(tt) | → | tt | (5) |
a__U32(X) | → | U32(X) | (42) |
a__U21(tt) | → | tt | (3) |
a__U21(X) | → | U21(X) | (40) |
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U11(X1,X2) | → | U11(X1,X2) | (37) |
a__U12(tt) | → | tt | (2) |
a__U12(X) | → | U12(X) | (38) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNat(V1),V2) | (67) |
1 | > | 2 | |
a__isNat#(x(V1,V2)) | → | a__U31#(a__isNat(V1),V2) | (71) |
1 | > | 2 | |
a__U11#(tt,V2) | → | a__isNat#(V2) | (52) |
2 | ≥ | 1 | |
a__U31#(tt,V2) | → | a__isNat#(V2) | (54) |
2 | ≥ | 1 | |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (68) |
1 | > | 1 | |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (70) |
1 | > | 1 | |
a__isNat#(x(V1,V2)) | → | a__isNat#(V1) | (72) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.