Left Termination of the query pattern
div_in_3(g, g, a)
w.r.t. the given Prolog program could not be shown:
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
Clauses:
le(0, Y, true).
le(s(X), 0, false).
le(s(X), s(Y), B) :- le(X, Y, B).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
div(X, s(Y), Z) :- ','(le(s(Y), X, B), if(B, X, s(Y), Z)).
if(false, X, s(Y), 0).
if(true, X, s(Y), s(Z)) :- ','(minus(X, Y, U), div(U, s(Y), Z)).
Queries:
div(g,g,a).
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
div_in: (b,b,f)
le_in: (b,b,f)
if_in: (b,b,b,f)
minus_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x4)
div_out_gga(x1, x2, x3) = div_out_gga(x3)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PrologToPiTRSProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x4)
div_out_gga(x1, x2, x3) = div_out_gga(x3)
Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
DIV_IN_GGA(X, s(Y), Z) → LE_IN_GGA(s(Y), X, B)
LE_IN_GGA(s(X), s(Y), B) → U1_GGA(X, Y, B, le_in_gga(X, Y, B))
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_GGA(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
IF_IN_GGGA(true, X, s(Y), s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U2_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → U6_GGGA(X, Y, Z, div_in_gga(U, s(Y), Z))
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x4)
div_out_gga(x1, x2, x3) = div_out_gga(x3)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
DIV_IN_GGA(X, s(Y), Z) → LE_IN_GGA(s(Y), X, B)
LE_IN_GGA(s(X), s(Y), B) → U1_GGA(X, Y, B, le_in_gga(X, Y, B))
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_GGA(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
IF_IN_GGGA(true, X, s(Y), s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U2_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → U6_GGGA(X, Y, Z, div_in_gga(U, s(Y), Z))
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x4)
div_out_gga(x1, x2, x3) = div_out_gga(x3)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 6 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x4)
div_out_gga(x1, x2, x3) = div_out_gga(x3)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x4)
div_out_gga(x1, x2, x3) = div_out_gga(x3)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
LE_IN_GGA(s(X), s(Y)) → LE_IN_GGA(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- LE_IN_GGA(s(X), s(Y)) → LE_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x4)
div_out_gga(x1, x2, x3) = div_out_gga(x3)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x2, x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
The TRS R consists of the following rules:
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
false = false
true = true
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x2, x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(X, s(Y)) → U3_GGA(X, Y, le_in_gga(s(Y), X))
U5_GGGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, s(Y))
IF_IN_GGGA(true, X, s(Y)) → U5_GGGA(Y, minus_in_gga(X, Y))
U3_GGA(X, Y, le_out_gga(B)) → IF_IN_GGGA(B, X, s(Y))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule IF_IN_GGGA(true, X, s(Y)) → U5_GGGA(Y, minus_in_gga(X, Y)) at position [1] we obtained the following new rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
DIV_IN_GGA(X, s(Y)) → U3_GGA(X, Y, le_in_gga(s(Y), X))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U5_GGGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, s(Y))
U3_GGA(X, Y, le_out_gga(B)) → IF_IN_GGGA(B, X, s(Y))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule DIV_IN_GGA(X, s(Y)) → U3_GGA(X, Y, le_in_gga(s(Y), X)) at position [2] we obtained the following new rules:
DIV_IN_GGA(0, s(x0)) → U3_GGA(0, x0, le_out_gga(false))
DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(le_in_gga(x0, x1)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(le_in_gga(x0, x1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U5_GGGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, s(Y))
DIV_IN_GGA(0, s(x0)) → U3_GGA(0, x0, le_out_gga(false))
U3_GGA(X, Y, le_out_gga(B)) → IF_IN_GGGA(B, X, s(Y))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U5_GGGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, s(Y)) we obtained the following new rules:
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(le_in_gga(x0, x1)))
DIV_IN_GGA(0, s(x0)) → U3_GGA(0, x0, le_out_gga(false))
U3_GGA(X, Y, le_out_gga(B)) → IF_IN_GGGA(B, X, s(Y))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGA(X, Y, le_out_gga(B)) → IF_IN_GGGA(B, X, s(Y)) we obtained the following new rules:
U3_GGA(0, z0, le_out_gga(false)) → IF_IN_GGGA(false, 0, s(z0))
U3_GGA(s(z0), z1, le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(z1))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U3_GGA(0, z0, le_out_gga(false)) → IF_IN_GGGA(false, 0, s(z0))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(le_in_gga(x0, x1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
DIV_IN_GGA(0, s(x0)) → U3_GGA(0, x0, le_out_gga(false))
U3_GGA(s(z0), z1, le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(z1))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(le_in_gga(x0, x1)))
U3_GGA(s(z0), z1, le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(z1))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(le_in_gga(x0, x1))) we obtained the following new rules:
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, U1_gga(le_in_gga(0, x0)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, U1_gga(le_in_gga(0, x0)))
U3_GGA(s(z0), z1, le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(z1))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, U1_gga(le_in_gga(0, x0))) at position [2,0] we obtained the following new rules:
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, U1_gga(le_out_gga(true)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, U1_gga(le_out_gga(true)))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U3_GGA(s(z0), z1, le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(z1))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, U1_gga(le_out_gga(true))) at position [2] we obtained the following new rules:
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U3_GGA(s(z0), z1, le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(z1))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGA(s(z0), z1, le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(z1)) we obtained the following new rules:
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
U3_GGA(s(z0), s(z1), le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(s(z1)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U3_GGA(s(z0), s(z1), le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(s(z1)))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U3_GGA(s(z0), s(z1), le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(s(z1)))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U5_GGGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(s(z1))) we obtained the following new rules:
U5_GGGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(s(x0)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U3_GGA(s(z0), s(z1), le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(s(z1)))
U5_GGGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(s(x0)))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U3_GGA(s(z0), s(z1), le_out_gga(x2)) → IF_IN_GGGA(x2, s(z0), s(s(z1))) we obtained the following new rules:
U3_GGA(s(x0), s(x1), le_out_gga(true)) → IF_IN_GGGA(true, s(x0), s(s(x1)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
U3_GGA(s(x0), s(x1), le_out_gga(true)) → IF_IN_GGGA(true, s(x0), s(s(x1)))
U5_GGGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(s(x0)))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x1), U2_gga(minus_in_gga(x0, x1)))
The remaining pairs can at least be oriented weakly.
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
U3_GGA(s(x0), s(x1), le_out_gga(true)) → IF_IN_GGGA(true, s(x0), s(s(x1)))
U5_GGGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(s(x0)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( minus_out_gga(x1) ) = | | + | | · | x1 |
| M( minus_in_gga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( le_in_gga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( le_out_gga(x1) ) = | | + | | · | x1 |
Tuple symbols:
| M( U5_GGGA(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
| M( DIV_IN_GGA(x1, x2) ) = | 1 | + | | · | x1 | + | | · | x2 |
| M( IF_IN_GGGA(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U3_GGA(x1, ..., x3) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(s(z0))) → U3_GGA(s(x0), s(z0), U1_gga(le_in_gga(s(z0), x0)))
U3_GGA(s(x0), s(x1), le_out_gga(true)) → IF_IN_GGGA(true, s(x0), s(s(x1)))
U5_GGGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(s(x0)))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U2_gga(minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(false)
le_in_gga(s(X), s(Y)) → U1_gga(le_in_gga(X, Y))
U2_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(le_out_gga(B)) → le_out_gga(B)
le_in_gga(0, Y) → le_out_gga(true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
R is empty.
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0)
U1_gga(x0)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(0, minus_out_gga(x0)) we obtained the following new rules:
IF_IN_GGGA(true, s(z0), s(0)) → U5_GGGA(0, minus_out_gga(s(z0)))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0))
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
IF_IN_GGGA(true, s(z0), s(0)) → U5_GGGA(0, minus_out_gga(s(z0)))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U5_GGGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, s(0)) we obtained the following new rules:
U5_GGGA(0, minus_out_gga(s(z0))) → DIV_IN_GGA(s(z0), s(0))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ NonTerminationProof
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
IF_IN_GGGA(true, s(z0), s(0)) → U5_GGGA(0, minus_out_gga(s(z0)))
U5_GGGA(0, minus_out_gga(s(z0))) → DIV_IN_GGA(s(z0), s(0))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
The TRS P consists of the following rules:
DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
IF_IN_GGGA(true, s(z0), s(0)) → U5_GGGA(0, minus_out_gga(s(z0)))
U5_GGGA(0, minus_out_gga(s(z0))) → DIV_IN_GGA(s(z0), s(0))
The TRS R consists of the following rules:none
s = U3_GGA(s(z0), 0, le_out_gga(true)) evaluates to t =U3_GGA(s(z0), 0, le_out_gga(true))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
U3_GGA(s(z0), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0), s(0))
with rule U3_GGA(s(z0'), 0, le_out_gga(true)) → IF_IN_GGGA(true, s(z0'), s(0)) at position [] and matcher [z0' / z0]
IF_IN_GGGA(true, s(z0), s(0)) → U5_GGGA(0, minus_out_gga(s(z0)))
with rule IF_IN_GGGA(true, s(z0'), s(0)) → U5_GGGA(0, minus_out_gga(s(z0'))) at position [] and matcher [z0' / z0]
U5_GGGA(0, minus_out_gga(s(z0))) → DIV_IN_GGA(s(z0), s(0))
with rule U5_GGGA(0, minus_out_gga(s(z0'))) → DIV_IN_GGA(s(z0'), s(0)) at position [] and matcher [z0' / z0]
DIV_IN_GGA(s(z0), s(0)) → U3_GGA(s(z0), 0, le_out_gga(true))
with rule DIV_IN_GGA(s(x0), s(0)) → U3_GGA(s(x0), 0, le_out_gga(true))
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
div_in: (b,b,f)
le_in: (b,b,f)
if_in: (b,b,b,f)
minus_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x1, x2, x3, x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x1, x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x1, x2, x4)
div_out_gga(x1, x2, x3) = div_out_gga(x1, x2, x3)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x1, x2, x3, x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x1, x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x1, x2, x4)
div_out_gga(x1, x2, x3) = div_out_gga(x1, x2, x3)
Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
DIV_IN_GGA(X, s(Y), Z) → LE_IN_GGA(s(Y), X, B)
LE_IN_GGA(s(X), s(Y), B) → U1_GGA(X, Y, B, le_in_gga(X, Y, B))
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_GGA(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
IF_IN_GGGA(true, X, s(Y), s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U2_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → U6_GGGA(X, Y, Z, div_in_gga(U, s(Y), Z))
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x1, x2, x3, x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x1, x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x1, x2, x4)
div_out_gga(x1, x2, x3) = div_out_gga(x1, x2, x3)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
Pi DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
DIV_IN_GGA(X, s(Y), Z) → LE_IN_GGA(s(Y), X, B)
LE_IN_GGA(s(X), s(Y), B) → U1_GGA(X, Y, B, le_in_gga(X, Y, B))
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_GGA(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
IF_IN_GGGA(true, X, s(Y), s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U2_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → U6_GGGA(X, Y, Z, div_in_gga(U, s(Y), Z))
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x1, x2, x3, x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x1, x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x1, x2, x4)
div_out_gga(x1, x2, x3) = div_out_gga(x1, x2, x3)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
U6_GGGA(x1, x2, x3, x4) = U6_GGGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 6 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x1, x2, x3, x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x1, x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x1, x2, x4)
div_out_gga(x1, x2, x3) = div_out_gga(x1, x2, x3)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
MINUS_IN_GGA(x1, x2, x3) = MINUS_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x1, x2, x3, x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x1, x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x1, x2, x4)
div_out_gga(x1, x2, x3) = div_out_gga(x1, x2, x3)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
LE_IN_GGA(s(X), s(Y), B) → LE_IN_GGA(X, Y, B)
R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE_IN_GGA(x1, x2, x3) = LE_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
LE_IN_GGA(s(X), s(Y)) → LE_IN_GGA(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- LE_IN_GGA(s(X), s(Y)) → LE_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
Pi DP problem:
The TRS P consists of the following rules:
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
The TRS R consists of the following rules:
div_in_gga(X, s(Y), Z) → U3_gga(X, Y, Z, le_in_gga(s(Y), X, B))
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
U3_gga(X, Y, Z, le_out_gga(s(Y), X, B)) → U4_gga(X, Y, Z, if_in_ggga(B, X, s(Y), Z))
if_in_ggga(false, X, s(Y), 0) → if_out_ggga(false, X, s(Y), 0)
if_in_ggga(true, X, s(Y), s(Z)) → U5_ggga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U5_ggga(X, Y, Z, minus_out_gga(X, Y, U)) → U6_ggga(X, Y, Z, div_in_gga(U, s(Y), Z))
U6_ggga(X, Y, Z, div_out_gga(U, s(Y), Z)) → if_out_ggga(true, X, s(Y), s(Z))
U4_gga(X, Y, Z, if_out_ggga(B, X, s(Y), Z)) → div_out_gga(X, s(Y), Z)
The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3) = div_in_gga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
if_in_ggga(x1, x2, x3, x4) = if_in_ggga(x1, x2, x3)
false = false
if_out_ggga(x1, x2, x3, x4) = if_out_ggga(x1, x2, x3, x4)
true = true
U5_ggga(x1, x2, x3, x4) = U5_ggga(x1, x2, x4)
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U6_ggga(x1, x2, x3, x4) = U6_ggga(x1, x2, x4)
div_out_gga(x1, x2, x3) = div_out_gga(x1, x2, x3)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
Pi DP problem:
The TRS P consists of the following rules:
U5_GGGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y), Z)
IF_IN_GGGA(true, X, s(Y), s(Z)) → U5_GGGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, s(Y), Z) → U3_GGA(X, Y, Z, le_in_gga(s(Y), X, B))
U3_GGA(X, Y, Z, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y), Z)
The TRS R consists of the following rules:
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U2_gga(X, Y, Z, minus_in_gga(X, Y, Z))
le_in_gga(s(X), 0, false) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y), B) → U1_gga(X, Y, B, le_in_gga(X, Y, B))
U2_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U1_gga(X, Y, B, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
le_in_gga(0, Y, true) → le_out_gga(0, Y, true)
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
le_in_gga(x1, x2, x3) = le_in_gga(x1, x2)
0 = 0
le_out_gga(x1, x2, x3) = le_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
false = false
true = true
minus_in_gga(x1, x2, x3) = minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3) = minus_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
U5_GGGA(x1, x2, x3, x4) = U5_GGGA(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2)
IF_IN_GGGA(x1, x2, x3, x4) = IF_IN_GGGA(x1, x2, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
DIV_IN_GGA(X, s(Y)) → U3_GGA(X, Y, le_in_gga(s(Y), X))
IF_IN_GGGA(true, X, s(Y)) → U5_GGGA(X, Y, minus_in_gga(X, Y))
U5_GGGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y))
U3_GGA(X, Y, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U2_gga(X, Y, minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y)) → U1_gga(X, Y, le_in_gga(X, Y))
U2_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U1_gga(X, Y, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
le_in_gga(0, Y) → le_out_gga(0, Y, true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0, x1, x2)
U1_gga(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule IF_IN_GGGA(true, X, s(Y)) → U5_GGGA(X, Y, minus_in_gga(X, Y)) at position [2] we obtained the following new rules:
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x0), s(x1), U2_gga(x0, x1, minus_in_gga(x0, x1)))
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(x0, 0, minus_out_gga(x0, 0, x0))
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x0), s(x1), U2_gga(x0, x1, minus_in_gga(x0, x1)))
DIV_IN_GGA(X, s(Y)) → U3_GGA(X, Y, le_in_gga(s(Y), X))
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(x0, 0, minus_out_gga(x0, 0, x0))
U5_GGGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y))
U3_GGA(X, Y, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U2_gga(X, Y, minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y)) → U1_gga(X, Y, le_in_gga(X, Y))
U2_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U1_gga(X, Y, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
le_in_gga(0, Y) → le_out_gga(0, Y, true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0, x1, x2)
U1_gga(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule DIV_IN_GGA(X, s(Y)) → U3_GGA(X, Y, le_in_gga(s(Y), X)) at position [2] we obtained the following new rules:
DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(x0, x1, le_in_gga(x0, x1)))
DIV_IN_GGA(0, s(x0)) → U3_GGA(0, x0, le_out_gga(s(x0), 0, false))
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
IF_IN_GGGA(true, s(x0), s(s(x1))) → U5_GGGA(s(x0), s(x1), U2_gga(x0, x1, minus_in_gga(x0, x1)))
IF_IN_GGGA(true, x0, s(0)) → U5_GGGA(x0, 0, minus_out_gga(x0, 0, x0))
DIV_IN_GGA(0, s(x0)) → U3_GGA(0, x0, le_out_gga(s(x0), 0, false))
DIV_IN_GGA(s(x1), s(x0)) → U3_GGA(s(x1), x0, U1_gga(x0, x1, le_in_gga(x0, x1)))
U5_GGGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, s(Y))
U3_GGA(X, Y, le_out_gga(s(Y), X, B)) → IF_IN_GGGA(B, X, s(Y))
The TRS R consists of the following rules:
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U2_gga(X, Y, minus_in_gga(X, Y))
le_in_gga(s(X), 0) → le_out_gga(s(X), 0, false)
le_in_gga(s(X), s(Y)) → U1_gga(X, Y, le_in_gga(X, Y))
U2_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U1_gga(X, Y, le_out_gga(X, Y, B)) → le_out_gga(s(X), s(Y), B)
le_in_gga(0, Y) → le_out_gga(0, Y, true)
The set Q consists of the following terms:
minus_in_gga(x0, x1)
le_in_gga(x0, x1)
U2_gga(x0, x1, x2)
U1_gga(x0, x1, x2)
We have to consider all (P,Q,R)-chains.