The rewrite relation of the following TRS is considered.
a__U11(tt,V1,V2) | → | a__U12(a__isNat(V1),V2) | (1) |
a__U12(tt,V2) | → | a__U13(a__isNat(V2)) | (2) |
a__U13(tt) | → | tt | (3) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__U22(tt) | → | tt | (5) |
a__U31(tt,V1,V2) | → | a__U32(a__isNat(V1),V2) | (6) |
a__U32(tt,V2) | → | a__U33(a__isNat(V2)) | (7) |
a__U33(tt) | → | tt | (8) |
a__U41(tt,N) | → | mark(N) | (9) |
a__U51(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (10) |
a__U61(tt) | → | 0 | (11) |
a__U71(tt,M,N) | → | a__plus(a__x(mark(N),mark(M)),mark(N)) | (12) |
a__and(tt,X) | → | mark(X) | (13) |
a__isNat(0) | → | tt | (14) |
a__isNat(plus(V1,V2)) | → | a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (15) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (16) |
a__isNat(x(V1,V2)) | → | a__U31(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (17) |
a__isNatKind(0) | → | tt | (18) |
a__isNatKind(plus(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (19) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (20) |
a__isNatKind(x(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (21) |
a__plus(N,0) | → | a__U41(a__and(a__isNat(N),isNatKind(N)),N) | (22) |
a__plus(N,s(M)) | → | a__U51(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (23) |
a__x(N,0) | → | a__U61(a__and(a__isNat(N),isNatKind(N))) | (24) |
a__x(N,s(M)) | → | a__U71(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (25) |
mark(U11(X1,X2,X3)) | → | a__U11(mark(X1),X2,X3) | (26) |
mark(U12(X1,X2)) | → | a__U12(mark(X1),X2) | (27) |
mark(isNat(X)) | → | a__isNat(X) | (28) |
mark(U13(X)) | → | a__U13(mark(X)) | (29) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (30) |
mark(U22(X)) | → | a__U22(mark(X)) | (31) |
mark(U31(X1,X2,X3)) | → | a__U31(mark(X1),X2,X3) | (32) |
mark(U32(X1,X2)) | → | a__U32(mark(X1),X2) | (33) |
mark(U33(X)) | → | a__U33(mark(X)) | (34) |
mark(U41(X1,X2)) | → | a__U41(mark(X1),X2) | (35) |
mark(U51(X1,X2,X3)) | → | a__U51(mark(X1),X2,X3) | (36) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (37) |
mark(U61(X)) | → | a__U61(mark(X)) | (38) |
mark(U71(X1,X2,X3)) | → | a__U71(mark(X1),X2,X3) | (39) |
mark(x(X1,X2)) | → | a__x(mark(X1),mark(X2)) | (40) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (41) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (42) |
mark(tt) | → | tt | (43) |
mark(s(X)) | → | s(mark(X)) | (44) |
mark(0) | → | 0 | (45) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (46) |
a__U12(X1,X2) | → | U12(X1,X2) | (47) |
a__isNat(X) | → | isNat(X) | (48) |
a__U13(X) | → | U13(X) | (49) |
a__U21(X1,X2) | → | U21(X1,X2) | (50) |
a__U22(X) | → | U22(X) | (51) |
a__U31(X1,X2,X3) | → | U31(X1,X2,X3) | (52) |
a__U32(X1,X2) | → | U32(X1,X2) | (53) |
a__U33(X) | → | U33(X) | (54) |
a__U41(X1,X2) | → | U41(X1,X2) | (55) |
a__U51(X1,X2,X3) | → | U51(X1,X2,X3) | (56) |
a__plus(X1,X2) | → | plus(X1,X2) | (57) |
a__U61(X) | → | U61(X) | (58) |
a__U71(X1,X2,X3) | → | U71(X1,X2,X3) | (59) |
a__x(X1,X2) | → | x(X1,X2) | (60) |
a__and(X1,X2) | → | and(X1,X2) | (61) |
a__isNatKind(X) | → | isNatKind(X) | (62) |
prec(a__U11) | = | 0 | stat(a__U11) | = | mul | |
prec(tt) | = | 4 | stat(tt) | = | mul | |
prec(a__U12) | = | 0 | stat(a__U12) | = | mul | |
prec(a__U21) | = | 5 | stat(a__U21) | = | mul | |
prec(a__U22) | = | 1 | stat(a__U22) | = | mul | |
prec(a__U31) | = | 6 | stat(a__U31) | = | mul | |
prec(a__U32) | = | 2 | stat(a__U32) | = | mul | |
prec(a__U41) | = | 3 | stat(a__U41) | = | mul | |
prec(a__U51) | = | 8 | stat(a__U51) | = | lex | |
prec(s) | = | 7 | stat(s) | = | mul | |
prec(a__plus) | = | 8 | stat(a__plus) | = | lex | |
prec(0) | = | 4 | stat(0) | = | mul | |
prec(a__U71) | = | 9 | stat(a__U71) | = | lex | |
prec(a__x) | = | 9 | stat(a__x) | = | lex | |
prec(a__and) | = | 7 | stat(a__and) | = | mul | |
prec(plus) | = | 8 | stat(plus) | = | lex | |
prec(x) | = | 9 | stat(x) | = | lex | |
prec(and) | = | 7 | stat(and) | = | mul | |
prec(U11) | = | 0 | stat(U11) | = | mul | |
prec(U12) | = | 0 | stat(U12) | = | mul | |
prec(U21) | = | 5 | stat(U21) | = | mul | |
prec(U22) | = | 1 | stat(U22) | = | mul | |
prec(U31) | = | 6 | stat(U31) | = | mul | |
prec(U32) | = | 2 | stat(U32) | = | mul | |
prec(U41) | = | 3 | stat(U41) | = | mul | |
prec(U51) | = | 8 | stat(U51) | = | lex | |
prec(U71) | = | 9 | stat(U71) | = | lex |
π(a__U11) | = | [1,2,3] |
π(tt) | = | [] |
π(a__U12) | = | [1,2] |
π(a__isNat) | = | 1 |
π(a__U13) | = | 1 |
π(a__U21) | = | [1,2] |
π(a__U22) | = | [1] |
π(a__U31) | = | [1,2,3] |
π(a__U32) | = | [1,2] |
π(a__U33) | = | 1 |
π(a__U41) | = | [1,2] |
π(mark) | = | 1 |
π(a__U51) | = | [3,2,1] |
π(s) | = | [1] |
π(a__plus) | = | [1,2] |
π(a__U61) | = | 1 |
π(0) | = | [] |
π(a__U71) | = | [2,3,1] |
π(a__x) | = | [2,1] |
π(a__and) | = | [1,2] |
π(plus) | = | [1,2] |
π(a__isNatKind) | = | 1 |
π(isNatKind) | = | 1 |
π(x) | = | [2,1] |
π(and) | = | [1,2] |
π(isNat) | = | 1 |
π(U11) | = | [1,2,3] |
π(U12) | = | [1,2] |
π(U13) | = | 1 |
π(U21) | = | [1,2] |
π(U22) | = | [1] |
π(U31) | = | [1,2,3] |
π(U32) | = | [1,2] |
π(U33) | = | 1 |
π(U41) | = | [1,2] |
π(U51) | = | [3,2,1] |
π(U61) | = | 1 |
π(U71) | = | [2,3,1] |
a__U11(tt,V1,V2) | → | a__U12(a__isNat(V1),V2) | (1) |
a__U12(tt,V2) | → | a__U13(a__isNat(V2)) | (2) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__U22(tt) | → | tt | (5) |
a__U31(tt,V1,V2) | → | a__U32(a__isNat(V1),V2) | (6) |
a__U32(tt,V2) | → | a__U33(a__isNat(V2)) | (7) |
a__U41(tt,N) | → | mark(N) | (9) |
a__U51(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (10) |
a__U71(tt,M,N) | → | a__plus(a__x(mark(N),mark(M)),mark(N)) | (12) |
a__and(tt,X) | → | mark(X) | (13) |
a__isNat(plus(V1,V2)) | → | a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (15) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (16) |
a__isNat(x(V1,V2)) | → | a__U31(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (17) |
a__isNatKind(plus(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (19) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (20) |
a__isNatKind(x(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (21) |
a__plus(N,0) | → | a__U41(a__and(a__isNat(N),isNatKind(N)),N) | (22) |
a__plus(N,s(M)) | → | a__U51(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (23) |
a__x(N,0) | → | a__U61(a__and(a__isNat(N),isNatKind(N))) | (24) |
a__x(N,s(M)) | → | a__U71(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (25) |
prec(tt) | = | 13 | weight(tt) | = | 2 | ||||
prec(0) | = | 32 | weight(0) | = | 1 | ||||
prec(a__U13) | = | 12 | weight(a__U13) | = | 1 | ||||
prec(a__U33) | = | 35 | weight(a__U33) | = | 1 | ||||
prec(a__U61) | = | 33 | weight(a__U61) | = | 1 | ||||
prec(a__isNat) | = | 21 | weight(a__isNat) | = | 1 | ||||
prec(a__isNatKind) | = | 14 | weight(a__isNatKind) | = | 1 | ||||
prec(mark) | = | 37 | weight(mark) | = | 0 | ||||
prec(isNat) | = | 18 | weight(isNat) | = | 1 | ||||
prec(U13) | = | 11 | weight(U13) | = | 1 | ||||
prec(U22) | = | 1 | weight(U22) | = | 1 | ||||
prec(a__U22) | = | 27 | weight(a__U22) | = | 1 | ||||
prec(U33) | = | 28 | weight(U33) | = | 1 | ||||
prec(U61) | = | 30 | weight(U61) | = | 1 | ||||
prec(isNatKind) | = | 10 | weight(isNatKind) | = | 1 | ||||
prec(s) | = | 34 | weight(s) | = | 1 | ||||
prec(U11) | = | 0 | weight(U11) | = | 0 | ||||
prec(a__U11) | = | 36 | weight(a__U11) | = | 0 | ||||
prec(U12) | = | 19 | weight(U12) | = | 0 | ||||
prec(a__U12) | = | 24 | weight(a__U12) | = | 0 | ||||
prec(U21) | = | 25 | weight(U21) | = | 0 | ||||
prec(a__U21) | = | 26 | weight(a__U21) | = | 0 | ||||
prec(U31) | = | 2 | weight(U31) | = | 0 | ||||
prec(a__U31) | = | 3 | weight(a__U31) | = | 0 | ||||
prec(U32) | = | 15 | weight(U32) | = | 0 | ||||
prec(a__U32) | = | 16 | weight(a__U32) | = | 0 | ||||
prec(U41) | = | 4 | weight(U41) | = | 0 | ||||
prec(a__U41) | = | 5 | weight(a__U41) | = | 0 | ||||
prec(U51) | = | 6 | weight(U51) | = | 0 | ||||
prec(a__U51) | = | 29 | weight(a__U51) | = | 0 | ||||
prec(plus) | = | 17 | weight(plus) | = | 0 | ||||
prec(a__plus) | = | 20 | weight(a__plus) | = | 0 | ||||
prec(U71) | = | 7 | weight(U71) | = | 0 | ||||
prec(a__U71) | = | 8 | weight(a__U71) | = | 0 | ||||
prec(x) | = | 22 | weight(x) | = | 0 | ||||
prec(a__x) | = | 23 | weight(a__x) | = | 0 | ||||
prec(and) | = | 9 | weight(and) | = | 0 | ||||
prec(a__and) | = | 31 | weight(a__and) | = | 0 |
a__U13(tt) | → | tt | (3) |
a__U33(tt) | → | tt | (8) |
a__U61(tt) | → | 0 | (11) |
a__isNat(0) | → | tt | (14) |
a__isNatKind(0) | → | tt | (18) |
mark(U11(X1,X2,X3)) | → | a__U11(mark(X1),X2,X3) | (26) |
mark(U12(X1,X2)) | → | a__U12(mark(X1),X2) | (27) |
mark(isNat(X)) | → | a__isNat(X) | (28) |
mark(U13(X)) | → | a__U13(mark(X)) | (29) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (30) |
mark(U22(X)) | → | a__U22(mark(X)) | (31) |
mark(U31(X1,X2,X3)) | → | a__U31(mark(X1),X2,X3) | (32) |
mark(U32(X1,X2)) | → | a__U32(mark(X1),X2) | (33) |
mark(U33(X)) | → | a__U33(mark(X)) | (34) |
mark(U41(X1,X2)) | → | a__U41(mark(X1),X2) | (35) |
mark(U51(X1,X2,X3)) | → | a__U51(mark(X1),X2,X3) | (36) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (37) |
mark(U61(X)) | → | a__U61(mark(X)) | (38) |
mark(U71(X1,X2,X3)) | → | a__U71(mark(X1),X2,X3) | (39) |
mark(x(X1,X2)) | → | a__x(mark(X1),mark(X2)) | (40) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (41) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (42) |
mark(tt) | → | tt | (43) |
mark(s(X)) | → | s(mark(X)) | (44) |
mark(0) | → | 0 | (45) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (46) |
a__U12(X1,X2) | → | U12(X1,X2) | (47) |
a__isNat(X) | → | isNat(X) | (48) |
a__U13(X) | → | U13(X) | (49) |
a__U21(X1,X2) | → | U21(X1,X2) | (50) |
a__U22(X) | → | U22(X) | (51) |
a__U31(X1,X2,X3) | → | U31(X1,X2,X3) | (52) |
a__U32(X1,X2) | → | U32(X1,X2) | (53) |
a__U33(X) | → | U33(X) | (54) |
a__U41(X1,X2) | → | U41(X1,X2) | (55) |
a__U51(X1,X2,X3) | → | U51(X1,X2,X3) | (56) |
a__plus(X1,X2) | → | plus(X1,X2) | (57) |
a__U61(X) | → | U61(X) | (58) |
a__U71(X1,X2,X3) | → | U71(X1,X2,X3) | (59) |
a__x(X1,X2) | → | x(X1,X2) | (60) |
a__and(X1,X2) | → | and(X1,X2) | (61) |
a__isNatKind(X) | → | isNatKind(X) | (62) |
There are no rules in the TRS. Hence, it is terminating.