0 Prolog
↳1 PrologToPiTRSProof (⇒, 0 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 11 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
↳7 PiDP
↳8 UsableRulesProof (⇔, 1 ms)
↳9 PiDP
↳10 PiDPToQDPProof (⇒, 0 ms)
↳11 QDP
↳12 QDPSizeChangeProof (⇔, 0 ms)
↳13 YES
↳14 PiDP
↳15 UsableRulesProof (⇔, 0 ms)
↳16 PiDP
↳17 PiDPToQDPProof (⇒, 0 ms)
↳18 QDP
↳19 QDPSizeChangeProof (⇔, 0 ms)
↳20 YES
↳21 PiDP
↳22 UsableRulesProof (⇔, 0 ms)
↳23 PiDP
↳24 PiDPToQDPProof (⇒, 4 ms)
↳25 QDP
↳26 MRRProof (⇔, 35 ms)
↳27 QDP
↳28 PisEmptyProof (⇔, 0 ms)
↳29 YES
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
APP_IN_GGA(.(Xs), Ys) → APP_IN_GGA(Xs, Ys)
From the DPs we obtained the following set of size-change graphs:
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
APP_IN_AAG(.(Zs)) → APP_IN_AAG(Zs)
From the DPs we obtained the following set of size-change graphs:
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_GA(app_out_aag(X1s, .(X2s))) → U2_GA(app_in_gga(X1s, X2s))
U2_GA(app_out_gga(Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U1_GA(app_in_aag(Xs))
app_in_gga(.(Xs), Ys) → U4_gga(app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
app_in_aag(.(Zs)) → U4_aag(app_in_aag(Zs))
app_in_aag(Ys) → app_out_aag([], Ys)
U4_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))
U4_aag(app_out_aag(Xs, Ys)) → app_out_aag(.(Xs), Ys)
app_in_gga(x0, x1)
app_in_aag(x0)
U4_gga(x0)
U4_aag(x0)
U1_GA(app_out_aag(X1s, .(X2s))) → U2_GA(app_in_gga(X1s, X2s))
U2_GA(app_out_gga(Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U1_GA(app_in_aag(Xs))
app_in_gga(.(Xs), Ys) → U4_gga(app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
app_in_aag(.(Zs)) → U4_aag(app_in_aag(Zs))
app_in_aag(Ys) → app_out_aag([], Ys)
U4_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))
U4_aag(app_out_aag(Xs, Ys)) → app_out_aag(.(Xs), Ys)
U1GA1 > appinaag1 > .1 > appingga2 > PERMINGA1 > U4gga1 > U4aag1 > U2GA1 > appoutaag2 > appoutgga1 > []
[]=2
._1=3
U4_gga_1=3
app_out_gga_1=7
app_in_aag_1=4
U4_aag_1=3
U1_GA_1=2
U2_GA_1=1
PERM_IN_GA_1=7
app_in_gga_2=5
app_out_aag_2=1
app_in_gga(x0, x1)
app_in_aag(x0)
U4_gga(x0)
U4_aag(x0)