Left Termination of the query pattern len(b,f) w.r.t. the given Prolog program could successfully be proven:
↳ PROLOG
↳ PrologToPiTRSProof
len2({}0, 00).
len2(.2(underscore, Ts), s1(N)) :- len2(Ts, N).
With regard to the inferred argument filtering the predicates were used in the following modes:
len2: (b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
len_2_in_ga2([]_0, 0_0) -> len_2_out_ga2([]_0, 0_0)
len_2_in_ga2(._22(underscore, Ts), s_11(N)) -> if_len_2_in_1_ga4(underscore, Ts, N, len_2_in_ga2(Ts, N))
if_len_2_in_1_ga4(underscore, Ts, N, len_2_out_ga2(Ts, N)) -> len_2_out_ga2(._22(underscore, Ts), s_11(N))
The argument filtering Pi contains the following mapping:
len_2_in_ga2(x1, x2) = len_2_in_ga1(x1)
[]_0 = []_0
0_0 = 0_0
._22(x1, x2) = ._22(x1, x2)
s_11(x1) = s_11(x1)
len_2_out_ga2(x1, x2) = len_2_out_ga1(x2)
if_len_2_in_1_ga4(x1, x2, x3, x4) = if_len_2_in_1_ga1(x4)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
len_2_in_ga2([]_0, 0_0) -> len_2_out_ga2([]_0, 0_0)
len_2_in_ga2(._22(underscore, Ts), s_11(N)) -> if_len_2_in_1_ga4(underscore, Ts, N, len_2_in_ga2(Ts, N))
if_len_2_in_1_ga4(underscore, Ts, N, len_2_out_ga2(Ts, N)) -> len_2_out_ga2(._22(underscore, Ts), s_11(N))
The argument filtering Pi contains the following mapping:
len_2_in_ga2(x1, x2) = len_2_in_ga1(x1)
[]_0 = []_0
0_0 = 0_0
._22(x1, x2) = ._22(x1, x2)
s_11(x1) = s_11(x1)
len_2_out_ga2(x1, x2) = len_2_out_ga1(x2)
if_len_2_in_1_ga4(x1, x2, x3, x4) = if_len_2_in_1_ga1(x4)
Pi DP problem:
The TRS P consists of the following rules:
LEN_2_IN_GA2(._22(underscore, Ts), s_11(N)) -> IF_LEN_2_IN_1_GA4(underscore, Ts, N, len_2_in_ga2(Ts, N))
LEN_2_IN_GA2(._22(underscore, Ts), s_11(N)) -> LEN_2_IN_GA2(Ts, N)
The TRS R consists of the following rules:
len_2_in_ga2([]_0, 0_0) -> len_2_out_ga2([]_0, 0_0)
len_2_in_ga2(._22(underscore, Ts), s_11(N)) -> if_len_2_in_1_ga4(underscore, Ts, N, len_2_in_ga2(Ts, N))
if_len_2_in_1_ga4(underscore, Ts, N, len_2_out_ga2(Ts, N)) -> len_2_out_ga2(._22(underscore, Ts), s_11(N))
The argument filtering Pi contains the following mapping:
len_2_in_ga2(x1, x2) = len_2_in_ga1(x1)
[]_0 = []_0
0_0 = 0_0
._22(x1, x2) = ._22(x1, x2)
s_11(x1) = s_11(x1)
len_2_out_ga2(x1, x2) = len_2_out_ga1(x2)
if_len_2_in_1_ga4(x1, x2, x3, x4) = if_len_2_in_1_ga1(x4)
LEN_2_IN_GA2(x1, x2) = LEN_2_IN_GA1(x1)
IF_LEN_2_IN_1_GA4(x1, x2, x3, x4) = IF_LEN_2_IN_1_GA1(x4)
We have to consider all (P,R,Pi)-chains
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
Pi DP problem:
The TRS P consists of the following rules:
LEN_2_IN_GA2(._22(underscore, Ts), s_11(N)) -> IF_LEN_2_IN_1_GA4(underscore, Ts, N, len_2_in_ga2(Ts, N))
LEN_2_IN_GA2(._22(underscore, Ts), s_11(N)) -> LEN_2_IN_GA2(Ts, N)
The TRS R consists of the following rules:
len_2_in_ga2([]_0, 0_0) -> len_2_out_ga2([]_0, 0_0)
len_2_in_ga2(._22(underscore, Ts), s_11(N)) -> if_len_2_in_1_ga4(underscore, Ts, N, len_2_in_ga2(Ts, N))
if_len_2_in_1_ga4(underscore, Ts, N, len_2_out_ga2(Ts, N)) -> len_2_out_ga2(._22(underscore, Ts), s_11(N))
The argument filtering Pi contains the following mapping:
len_2_in_ga2(x1, x2) = len_2_in_ga1(x1)
[]_0 = []_0
0_0 = 0_0
._22(x1, x2) = ._22(x1, x2)
s_11(x1) = s_11(x1)
len_2_out_ga2(x1, x2) = len_2_out_ga1(x2)
if_len_2_in_1_ga4(x1, x2, x3, x4) = if_len_2_in_1_ga1(x4)
LEN_2_IN_GA2(x1, x2) = LEN_2_IN_GA1(x1)
IF_LEN_2_IN_1_GA4(x1, x2, x3, x4) = IF_LEN_2_IN_1_GA1(x4)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 1 SCC with 1 less node.
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PiDP
↳ UsableRulesProof
Pi DP problem:
The TRS P consists of the following rules:
LEN_2_IN_GA2(._22(underscore, Ts), s_11(N)) -> LEN_2_IN_GA2(Ts, N)
The TRS R consists of the following rules:
len_2_in_ga2([]_0, 0_0) -> len_2_out_ga2([]_0, 0_0)
len_2_in_ga2(._22(underscore, Ts), s_11(N)) -> if_len_2_in_1_ga4(underscore, Ts, N, len_2_in_ga2(Ts, N))
if_len_2_in_1_ga4(underscore, Ts, N, len_2_out_ga2(Ts, N)) -> len_2_out_ga2(._22(underscore, Ts), s_11(N))
The argument filtering Pi contains the following mapping:
len_2_in_ga2(x1, x2) = len_2_in_ga1(x1)
[]_0 = []_0
0_0 = 0_0
._22(x1, x2) = ._22(x1, x2)
s_11(x1) = s_11(x1)
len_2_out_ga2(x1, x2) = len_2_out_ga1(x2)
if_len_2_in_1_ga4(x1, x2, x3, x4) = if_len_2_in_1_ga1(x4)
LEN_2_IN_GA2(x1, x2) = LEN_2_IN_GA1(x1)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
Pi DP problem:
The TRS P consists of the following rules:
LEN_2_IN_GA2(._22(underscore, Ts), s_11(N)) -> LEN_2_IN_GA2(Ts, N)
R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2) = ._22(x1, x2)
s_11(x1) = s_11(x1)
LEN_2_IN_GA2(x1, x2) = LEN_2_IN_GA1(x1)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
LEN_2_IN_GA1(._22(underscore, Ts)) -> LEN_2_IN_GA1(Ts)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LEN_2_IN_GA1}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- LEN_2_IN_GA1(._22(underscore, Ts)) -> LEN_2_IN_GA1(Ts)
The graph contains the following edges 1 > 1