(0) Obligation:
Clauses:p(a, b).
p(b, c).
tc(X, X).
tc(X, Y) :- ','(p(X, Z), tc(Z, Y)).
Queries:
tc(g,a).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:tc21(c).
tc10(b).
tc10(T15) :- tc21(T15).
tc1(T4, T4).
tc1(a, T9) :- tc10(T9).
tc1(b, T9) :- tc21(T9).
tc1(a, T28) :- tc10(T28).
tc1(b, T28) :- tc21(T28).
Queries:
tc1(g,a).
(3) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:tc1_in: (b,f)
tc10_in: (f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
tc1_in_ga(T4, T4) → tc1_out_ga(T4, T4)
tc1_in_ga(a, T9) → U2_ga(T9, tc10_in_a(T9))
tc10_in_a(b) → tc10_out_a(b)
tc10_in_a(T15) → U1_a(T15, tc21_in_a(T15))
tc21_in_a(c) → tc21_out_a(c)
U1_a(T15, tc21_out_a(T15)) → tc10_out_a(T15)
U2_ga(T9, tc10_out_a(T9)) → tc1_out_ga(a, T9)
tc1_in_ga(b, T9) → U3_ga(T9, tc21_in_a(T9))
U3_ga(T9, tc21_out_a(T9)) → tc1_out_ga(b, T9)
tc1_in_ga(a, T28) → U4_ga(T28, tc10_in_a(T28))
U4_ga(T28, tc10_out_a(T28)) → tc1_out_ga(a, T28)
tc1_in_ga(b, T28) → U5_ga(T28, tc21_in_a(T28))
U5_ga(T28, tc21_out_a(T28)) → tc1_out_ga(b, T28)
The argument filtering Pi contains the following mapping:
tc1_in_ga(x1, x2) = tc1_in_ga(x1)
tc1_out_ga(x1, x2) = tc1_out_ga(x2)
a = a
U2_ga(x1, x2) = U2_ga(x2)
tc10_in_a(x1) = tc10_in_a
tc10_out_a(x1) = tc10_out_a(x1)
U1_a(x1, x2) = U1_a(x2)
tc21_in_a(x1) = tc21_in_a
tc21_out_a(x1) = tc21_out_a(x1)
b = b
U3_ga(x1, x2) = U3_ga(x2)
U4_ga(x1, x2) = U4_ga(x2)
U5_ga(x1, x2) = U5_ga(x2)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
tc1_in_ga(T4, T4) → tc1_out_ga(T4, T4)
tc1_in_ga(a, T9) → U2_ga(T9, tc10_in_a(T9))
tc10_in_a(b) → tc10_out_a(b)
tc10_in_a(T15) → U1_a(T15, tc21_in_a(T15))
tc21_in_a(c) → tc21_out_a(c)
U1_a(T15, tc21_out_a(T15)) → tc10_out_a(T15)
U2_ga(T9, tc10_out_a(T9)) → tc1_out_ga(a, T9)
tc1_in_ga(b, T9) → U3_ga(T9, tc21_in_a(T9))
U3_ga(T9, tc21_out_a(T9)) → tc1_out_ga(b, T9)
tc1_in_ga(a, T28) → U4_ga(T28, tc10_in_a(T28))
U4_ga(T28, tc10_out_a(T28)) → tc1_out_ga(a, T28)
tc1_in_ga(b, T28) → U5_ga(T28, tc21_in_a(T28))
U5_ga(T28, tc21_out_a(T28)) → tc1_out_ga(b, T28)
The argument filtering Pi contains the following mapping:
tc1_in_ga(x1, x2) = tc1_in_ga(x1)
tc1_out_ga(x1, x2) = tc1_out_ga(x2)
a = a
U2_ga(x1, x2) = U2_ga(x2)
tc10_in_a(x1) = tc10_in_a
tc10_out_a(x1) = tc10_out_a(x1)
U1_a(x1, x2) = U1_a(x2)
tc21_in_a(x1) = tc21_in_a
tc21_out_a(x1) = tc21_out_a(x1)
b = b
U3_ga(x1, x2) = U3_ga(x2)
U4_ga(x1, x2) = U4_ga(x2)
U5_ga(x1, x2) = U5_ga(x2)
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:Pi DP problem:
The TRS P consists of the following rules:
TC1_IN_GA(a, T9) → U2_GA(T9, tc10_in_a(T9))
TC1_IN_GA(a, T9) → TC10_IN_A(T9)
TC10_IN_A(T15) → U1_A(T15, tc21_in_a(T15))
TC10_IN_A(T15) → TC21_IN_A(T15)
TC1_IN_GA(b, T9) → U3_GA(T9, tc21_in_a(T9))
TC1_IN_GA(b, T9) → TC21_IN_A(T9)
TC1_IN_GA(a, T28) → U4_GA(T28, tc10_in_a(T28))
TC1_IN_GA(b, T28) → U5_GA(T28, tc21_in_a(T28))
The TRS R consists of the following rules:
tc1_in_ga(T4, T4) → tc1_out_ga(T4, T4)
tc1_in_ga(a, T9) → U2_ga(T9, tc10_in_a(T9))
tc10_in_a(b) → tc10_out_a(b)
tc10_in_a(T15) → U1_a(T15, tc21_in_a(T15))
tc21_in_a(c) → tc21_out_a(c)
U1_a(T15, tc21_out_a(T15)) → tc10_out_a(T15)
U2_ga(T9, tc10_out_a(T9)) → tc1_out_ga(a, T9)
tc1_in_ga(b, T9) → U3_ga(T9, tc21_in_a(T9))
U3_ga(T9, tc21_out_a(T9)) → tc1_out_ga(b, T9)
tc1_in_ga(a, T28) → U4_ga(T28, tc10_in_a(T28))
U4_ga(T28, tc10_out_a(T28)) → tc1_out_ga(a, T28)
tc1_in_ga(b, T28) → U5_ga(T28, tc21_in_a(T28))
U5_ga(T28, tc21_out_a(T28)) → tc1_out_ga(b, T28)
The argument filtering Pi contains the following mapping:
tc1_in_ga(x1, x2) = tc1_in_ga(x1)
tc1_out_ga(x1, x2) = tc1_out_ga(x2)
a = a
U2_ga(x1, x2) = U2_ga(x2)
tc10_in_a(x1) = tc10_in_a
tc10_out_a(x1) = tc10_out_a(x1)
U1_a(x1, x2) = U1_a(x2)
tc21_in_a(x1) = tc21_in_a
tc21_out_a(x1) = tc21_out_a(x1)
b = b
U3_ga(x1, x2) = U3_ga(x2)
U4_ga(x1, x2) = U4_ga(x2)
U5_ga(x1, x2) = U5_ga(x2)
TC1_IN_GA(x1, x2) = TC1_IN_GA(x1)
U2_GA(x1, x2) = U2_GA(x2)
TC10_IN_A(x1) = TC10_IN_A
U1_A(x1, x2) = U1_A(x2)
TC21_IN_A(x1) = TC21_IN_A
U3_GA(x1, x2) = U3_GA(x2)
U4_GA(x1, x2) = U4_GA(x2)
U5_GA(x1, x2) = U5_GA(x2)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
TC1_IN_GA(a, T9) → U2_GA(T9, tc10_in_a(T9))
TC1_IN_GA(a, T9) → TC10_IN_A(T9)
TC10_IN_A(T15) → U1_A(T15, tc21_in_a(T15))
TC10_IN_A(T15) → TC21_IN_A(T15)
TC1_IN_GA(b, T9) → U3_GA(T9, tc21_in_a(T9))
TC1_IN_GA(b, T9) → TC21_IN_A(T9)
TC1_IN_GA(a, T28) → U4_GA(T28, tc10_in_a(T28))
TC1_IN_GA(b, T28) → U5_GA(T28, tc21_in_a(T28))
The TRS R consists of the following rules:
tc1_in_ga(T4, T4) → tc1_out_ga(T4, T4)
tc1_in_ga(a, T9) → U2_ga(T9, tc10_in_a(T9))
tc10_in_a(b) → tc10_out_a(b)
tc10_in_a(T15) → U1_a(T15, tc21_in_a(T15))
tc21_in_a(c) → tc21_out_a(c)
U1_a(T15, tc21_out_a(T15)) → tc10_out_a(T15)
U2_ga(T9, tc10_out_a(T9)) → tc1_out_ga(a, T9)
tc1_in_ga(b, T9) → U3_ga(T9, tc21_in_a(T9))
U3_ga(T9, tc21_out_a(T9)) → tc1_out_ga(b, T9)
tc1_in_ga(a, T28) → U4_ga(T28, tc10_in_a(T28))
U4_ga(T28, tc10_out_a(T28)) → tc1_out_ga(a, T28)
tc1_in_ga(b, T28) → U5_ga(T28, tc21_in_a(T28))
U5_ga(T28, tc21_out_a(T28)) → tc1_out_ga(b, T28)
The argument filtering Pi contains the following mapping:
tc1_in_ga(x1, x2) = tc1_in_ga(x1)
tc1_out_ga(x1, x2) = tc1_out_ga(x2)
a = a
U2_ga(x1, x2) = U2_ga(x2)
tc10_in_a(x1) = tc10_in_a
tc10_out_a(x1) = tc10_out_a(x1)
U1_a(x1, x2) = U1_a(x2)
tc21_in_a(x1) = tc21_in_a
tc21_out_a(x1) = tc21_out_a(x1)
b = b
U3_ga(x1, x2) = U3_ga(x2)
U4_ga(x1, x2) = U4_ga(x2)
U5_ga(x1, x2) = U5_ga(x2)
TC1_IN_GA(x1, x2) = TC1_IN_GA(x1)
U2_GA(x1, x2) = U2_GA(x2)
TC10_IN_A(x1) = TC10_IN_A
U1_A(x1, x2) = U1_A(x2)
TC21_IN_A(x1) = TC21_IN_A
U3_GA(x1, x2) = U3_GA(x2)
U4_GA(x1, x2) = U4_GA(x2)
U5_GA(x1, x2) = U5_GA(x2)
We have to consider all (P,R,Pi)-chains