The rewrite relation of the following TRS is considered.
There are 111 ruless (increase limit for explicit display).
The evaluation strategy is innermost.prec(U11) | = | 1 | stat(U11) | = | lex | |
prec(tt) | = | 2 | stat(tt) | = | mul | |
prec(U12) | = | 0 | stat(U12) | = | mul | |
prec(U21) | = | 4 | stat(U21) | = | lex | |
prec(U22) | = | 3 | stat(U22) | = | mul | |
prec(U31) | = | 7 | stat(U31) | = | lex | |
prec(U32) | = | 6 | stat(U32) | = | mul | |
prec(U33) | = | 5 | stat(U33) | = | mul | |
prec(U41) | = | 8 | stat(U41) | = | mul | |
prec(U51) | = | 11 | stat(U51) | = | lex | |
prec(s) | = | 9 | stat(s) | = | mul | |
prec(plus) | = | 11 | stat(plus) | = | lex | |
prec(U61) | = | 12 | stat(U61) | = | mul | |
prec(0) | = | 12 | stat(0) | = | mul | |
prec(U71) | = | 13 | stat(U71) | = | lex | |
prec(x) | = | 13 | stat(x) | = | lex | |
prec(and) | = | 10 | stat(and) | = | mul |
π(active) | = | 1 |
π(U11) | = | [1,2,3] |
π(tt) | = | [] |
π(mark) | = | 1 |
π(U12) | = | [1,2] |
π(isNat) | = | 1 |
π(U13) | = | 1 |
π(U21) | = | [1,2] |
π(U22) | = | [1] |
π(U31) | = | [3,1,2] |
π(U32) | = | [1,2] |
π(U33) | = | [1] |
π(U41) | = | [1,2] |
π(U51) | = | [2,3,1] |
π(s) | = | [1] |
π(plus) | = | [2,1] |
π(U61) | = | [1] |
π(0) | = | [] |
π(U71) | = | [3,2,1] |
π(x) | = | [1,2] |
π(and) | = | [1,2] |
π(isNatKind) | = | 1 |
active(U11(tt,V1,V2)) | → | mark(U12(isNat(V1),V2)) | (1) |
active(U12(tt,V2)) | → | mark(U13(isNat(V2))) | (2) |
active(U21(tt,V1)) | → | mark(U22(isNat(V1))) | (4) |
active(U22(tt)) | → | mark(tt) | (5) |
active(U31(tt,V1,V2)) | → | mark(U32(isNat(V1),V2)) | (6) |
active(U32(tt,V2)) | → | mark(U33(isNat(V2))) | (7) |
active(U33(tt)) | → | mark(tt) | (8) |
active(U41(tt,N)) | → | mark(N) | (9) |
active(U51(tt,M,N)) | → | mark(s(plus(N,M))) | (10) |
active(U61(tt)) | → | mark(0) | (11) |
active(U71(tt,M,N)) | → | mark(plus(x(N,M),N)) | (12) |
active(and(tt,X)) | → | mark(X) | (13) |
active(isNat(0)) | → | mark(tt) | (14) |
active(isNat(plus(V1,V2))) | → | mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) | (15) |
active(isNat(s(V1))) | → | mark(U21(isNatKind(V1),V1)) | (16) |
active(isNat(x(V1,V2))) | → | mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)) | (17) |
active(isNatKind(0)) | → | mark(tt) | (18) |
active(isNatKind(plus(V1,V2))) | → | mark(and(isNatKind(V1),isNatKind(V2))) | (19) |
active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | (20) |
active(isNatKind(x(V1,V2))) | → | mark(and(isNatKind(V1),isNatKind(V2))) | (21) |
active(plus(N,0)) | → | mark(U41(and(isNat(N),isNatKind(N)),N)) | (22) |
active(plus(N,s(M))) | → | mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) | (23) |
active(x(N,0)) | → | mark(U61(and(isNat(N),isNatKind(N)))) | (24) |
active(x(N,s(M))) | → | mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) | (25) |
[active(x1)] | = | 1 + 1 · x1 |
[U13(x1)] | = | 2 + 1 · x1 |
[tt] | = | 2 |
[mark(x1)] | = | 2 · x1 |
[U11(x1, x2, x3)] | = | 2 + 1 · x1 + 1 · x2 + 1 · x3 |
[U12(x1, x2)] | = | 2 + 1 · x1 + 1 · x2 |
[isNat(x1)] | = | 2 + 1 · x1 |
[U21(x1, x2)] | = | 2 + 1 · x1 + 1 · x2 |
[U22(x1)] | = | 2 + 1 · x1 |
[U31(x1, x2, x3)] | = | 2 + 1 · x1 + 1 · x2 + 1 · x3 |
[U32(x1, x2)] | = | 2 + 1 · x1 + 1 · x2 |
[U33(x1)] | = | 2 + 1 · x1 |
[U41(x1, x2)] | = | 2 + 1 · x1 + 1 · x2 |
[U51(x1, x2, x3)] | = | 2 + 1 · x1 + 1 · x2 + 1 · x3 |
[s(x1)] | = | 2 + 1 · x1 |
[plus(x1, x2)] | = | 2 + 1 · x1 + 1 · x2 |
[U61(x1)] | = | 2 + 1 · x1 |
[0] | = | 2 |
[U71(x1, x2, x3)] | = | 2 + 1 · x1 + 1 · x2 + 2 · x3 |
[x(x1, x2)] | = | 2 + 1 · x1 + 1 · x2 |
[and(x1, x2)] | = | 2 + 2 · x1 + 1 · x2 |
[isNatKind(x1)] | = | 2 + 1 · x1 |
active(U13(tt)) | → | mark(tt) | (3) |
mark(U11(X1,X2,X3)) | → | active(U11(mark(X1),X2,X3)) | (26) |
mark(tt) | → | active(tt) | (27) |
mark(U12(X1,X2)) | → | active(U12(mark(X1),X2)) | (28) |
mark(isNat(X)) | → | active(isNat(X)) | (29) |
mark(U13(X)) | → | active(U13(mark(X))) | (30) |
mark(U21(X1,X2)) | → | active(U21(mark(X1),X2)) | (31) |
mark(U22(X)) | → | active(U22(mark(X))) | (32) |
mark(U31(X1,X2,X3)) | → | active(U31(mark(X1),X2,X3)) | (33) |
mark(U32(X1,X2)) | → | active(U32(mark(X1),X2)) | (34) |
mark(U33(X)) | → | active(U33(mark(X))) | (35) |
mark(U41(X1,X2)) | → | active(U41(mark(X1),X2)) | (36) |
mark(U51(X1,X2,X3)) | → | active(U51(mark(X1),X2,X3)) | (37) |
mark(s(X)) | → | active(s(mark(X))) | (38) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (39) |
mark(U61(X)) | → | active(U61(mark(X))) | (40) |
mark(0) | → | active(0) | (41) |
mark(U71(X1,X2,X3)) | → | active(U71(mark(X1),X2,X3)) | (42) |
mark(x(X1,X2)) | → | active(x(mark(X1),mark(X2))) | (43) |
mark(and(X1,X2)) | → | active(and(mark(X1),X2)) | (44) |
mark(isNatKind(X)) | → | active(isNatKind(X)) | (45) |
U11(active(X1),X2,X3) | → | U11(X1,X2,X3) | (49) |
U11(X1,active(X2),X3) | → | U11(X1,X2,X3) | (50) |
U11(X1,X2,active(X3)) | → | U11(X1,X2,X3) | (51) |
U12(active(X1),X2) | → | U12(X1,X2) | (54) |
U12(X1,active(X2)) | → | U12(X1,X2) | (55) |
isNat(active(X)) | → | isNat(X) | (57) |
U13(active(X)) | → | U13(X) | (59) |
U21(active(X1),X2) | → | U21(X1,X2) | (62) |
U21(X1,active(X2)) | → | U21(X1,X2) | (63) |
U22(active(X)) | → | U22(X) | (65) |
U31(active(X1),X2,X3) | → | U31(X1,X2,X3) | (69) |
U31(X1,active(X2),X3) | → | U31(X1,X2,X3) | (70) |
U31(X1,X2,active(X3)) | → | U31(X1,X2,X3) | (71) |
U32(active(X1),X2) | → | U32(X1,X2) | (74) |
U32(X1,active(X2)) | → | U32(X1,X2) | (75) |
U33(active(X)) | → | U33(X) | (77) |
U41(active(X1),X2) | → | U41(X1,X2) | (80) |
U41(X1,active(X2)) | → | U41(X1,X2) | (81) |
U51(active(X1),X2,X3) | → | U51(X1,X2,X3) | (85) |
U51(X1,active(X2),X3) | → | U51(X1,X2,X3) | (86) |
U51(X1,X2,active(X3)) | → | U51(X1,X2,X3) | (87) |
s(active(X)) | → | s(X) | (89) |
plus(active(X1),X2) | → | plus(X1,X2) | (92) |
plus(X1,active(X2)) | → | plus(X1,X2) | (93) |
U61(active(X)) | → | U61(X) | (95) |
U71(active(X1),X2,X3) | → | U71(X1,X2,X3) | (99) |
U71(X1,active(X2),X3) | → | U71(X1,X2,X3) | (100) |
U71(X1,X2,active(X3)) | → | U71(X1,X2,X3) | (101) |
x(active(X1),X2) | → | x(X1,X2) | (104) |
x(X1,active(X2)) | → | x(X1,X2) | (105) |
and(active(X1),X2) | → | and(X1,X2) | (108) |
and(X1,active(X2)) | → | and(X1,X2) | (109) |
isNatKind(active(X)) | → | isNatKind(X) | (111) |
prec(mark) | = | 18 | weight(mark) | = | 0 | ||||
prec(isNat) | = | 1 | weight(isNat) | = | 1 | ||||
prec(U13) | = | 2 | weight(U13) | = | 1 | ||||
prec(U22) | = | 4 | weight(U22) | = | 1 | ||||
prec(U33) | = | 7 | weight(U33) | = | 1 | ||||
prec(s) | = | 10 | weight(s) | = | 1 | ||||
prec(U61) | = | 12 | weight(U61) | = | 1 | ||||
prec(isNatKind) | = | 17 | weight(isNatKind) | = | 1 | ||||
prec(U11) | = | 14 | weight(U11) | = | 0 | ||||
prec(U12) | = | 0 | weight(U12) | = | 0 | ||||
prec(U21) | = | 3 | weight(U21) | = | 0 | ||||
prec(U31) | = | 5 | weight(U31) | = | 0 | ||||
prec(U32) | = | 6 | weight(U32) | = | 0 | ||||
prec(U41) | = | 8 | weight(U41) | = | 0 | ||||
prec(U51) | = | 9 | weight(U51) | = | 0 | ||||
prec(plus) | = | 11 | weight(plus) | = | 0 | ||||
prec(U71) | = | 13 | weight(U71) | = | 0 | ||||
prec(x) | = | 15 | weight(x) | = | 0 | ||||
prec(and) | = | 16 | weight(and) | = | 0 |
U11(mark(X1),X2,X3) | → | U11(X1,X2,X3) | (46) |
U11(X1,mark(X2),X3) | → | U11(X1,X2,X3) | (47) |
U11(X1,X2,mark(X3)) | → | U11(X1,X2,X3) | (48) |
U12(mark(X1),X2) | → | U12(X1,X2) | (52) |
U12(X1,mark(X2)) | → | U12(X1,X2) | (53) |
isNat(mark(X)) | → | isNat(X) | (56) |
U13(mark(X)) | → | U13(X) | (58) |
U21(mark(X1),X2) | → | U21(X1,X2) | (60) |
U21(X1,mark(X2)) | → | U21(X1,X2) | (61) |
U22(mark(X)) | → | U22(X) | (64) |
U31(mark(X1),X2,X3) | → | U31(X1,X2,X3) | (66) |
U31(X1,mark(X2),X3) | → | U31(X1,X2,X3) | (67) |
U31(X1,X2,mark(X3)) | → | U31(X1,X2,X3) | (68) |
U32(mark(X1),X2) | → | U32(X1,X2) | (72) |
U32(X1,mark(X2)) | → | U32(X1,X2) | (73) |
U33(mark(X)) | → | U33(X) | (76) |
U41(mark(X1),X2) | → | U41(X1,X2) | (78) |
U41(X1,mark(X2)) | → | U41(X1,X2) | (79) |
U51(mark(X1),X2,X3) | → | U51(X1,X2,X3) | (82) |
U51(X1,mark(X2),X3) | → | U51(X1,X2,X3) | (83) |
U51(X1,X2,mark(X3)) | → | U51(X1,X2,X3) | (84) |
s(mark(X)) | → | s(X) | (88) |
plus(mark(X1),X2) | → | plus(X1,X2) | (90) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (91) |
U61(mark(X)) | → | U61(X) | (94) |
U71(mark(X1),X2,X3) | → | U71(X1,X2,X3) | (96) |
U71(X1,mark(X2),X3) | → | U71(X1,X2,X3) | (97) |
U71(X1,X2,mark(X3)) | → | U71(X1,X2,X3) | (98) |
x(mark(X1),X2) | → | x(X1,X2) | (102) |
x(X1,mark(X2)) | → | x(X1,X2) | (103) |
and(mark(X1),X2) | → | and(X1,X2) | (106) |
and(X1,mark(X2)) | → | and(X1,X2) | (107) |
isNatKind(mark(X)) | → | isNatKind(X) | (110) |
There are no rules in the TRS. Hence, it is terminating.