0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 QReductionProof (⇔)
↳11 QDP
↳12 QDPSizeChangeProof (⇔)
↳13 TRUE
↳14 QDP
↳15 UsableRulesProof (⇔)
↳16 QDP
↳17 QReductionProof (⇔)
↳18 QDP
↳19 QDPSizeChangeProof (⇔)
↳20 TRUE
↳21 QDP
↳22 UsableRulesProof (⇔)
↳23 QDP
↳24 QReductionProof (⇔)
↳25 QDP
↳26 QDPSizeChangeProof (⇔)
↳27 TRUE
↳28 QDP
↳29 UsableRulesProof (⇔)
↳30 QDP
↳31 QReductionProof (⇔)
↳32 QDP
↳33 Instantiation (⇔)
↳34 QDP
↳35 Narrowing (⇔)
↳36 QDP
↳37 DependencyGraphProof (⇔)
↳38 QDP
↳39 Narrowing (⇔)
↳40 QDP
↳41 QDPOrderProof (⇔)
↳42 QDP
↳43 DependencyGraphProof (⇔)
↳44 TRUE
division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
DIVISION(x, y) → DIV(x, y, 0)
DIV(x, y, z) → IF(lt(x, y), x, y, inc(z))
DIV(x, y, z) → LT(x, y)
DIV(x, y, z) → INC(z)
IF(false, x, s(y), z) → DIV(minus(x, s(y)), s(y), z)
IF(false, x, s(y), z) → MINUS(x, s(y))
MINUS(s(x), s(y)) → MINUS(x, y)
LT(s(x), s(y)) → LT(x, y)
INC(s(x)) → INC(x)
division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
INC(s(x)) → INC(x)
division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
INC(s(x)) → INC(x)
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
INC(s(x)) → INC(x)
From the DPs we obtained the following set of size-change graphs:
LT(s(x), s(y)) → LT(x, y)
division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
LT(s(x), s(y)) → LT(x, y)
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
LT(s(x), s(y)) → LT(x, y)
From the DPs we obtained the following set of size-change graphs:
MINUS(s(x), s(y)) → MINUS(x, y)
division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
MINUS(s(x), s(y)) → MINUS(x, y)
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
MINUS(s(x), s(y)) → MINUS(x, y)
From the DPs we obtained the following set of size-change graphs:
IF(false, x, s(y), z) → DIV(minus(x, s(y)), s(y), z)
DIV(x, y, z) → IF(lt(x, y), x, y, inc(z))
division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
IF(false, x, s(y), z) → DIV(minus(x, s(y)), s(y), z)
DIV(x, y, z) → IF(lt(x, y), x, y, inc(z))
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
division(x0, x1)
div(x0, x1, x2)
if(true, x0, x1, x2)
if(false, x0, s(x1), x2)
IF(false, x, s(y), z) → DIV(minus(x, s(y)), s(y), z)
DIV(x, y, z) → IF(lt(x, y), x, y, inc(z))
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
DIV(y_0, s(z1), z2) → IF(lt(y_0, s(z1)), y_0, s(z1), inc(z2))
IF(false, x, s(y), z) → DIV(minus(x, s(y)), s(y), z)
DIV(y_0, s(z1), z2) → IF(lt(y_0, s(z1)), y_0, s(z1), inc(z2))
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
DIV(x0, 0, y2) → IF(false, x0, 0, inc(y2))
DIV(0, s(x0), y2) → IF(true, 0, s(x0), inc(y2))
DIV(s(x0), s(x1), y2) → IF(lt(x0, x1), s(x0), s(x1), inc(y2))
IF(false, x, s(y), z) → DIV(minus(x, s(y)), s(y), z)
DIV(x0, 0, y2) → IF(false, x0, 0, inc(y2))
DIV(0, s(x0), y2) → IF(true, 0, s(x0), inc(y2))
DIV(s(x0), s(x1), y2) → IF(lt(x0, x1), s(x0), s(x1), inc(y2))
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
DIV(s(x0), s(x1), y2) → IF(lt(x0, x1), s(x0), s(x1), inc(y2))
IF(false, x, s(y), z) → DIV(minus(x, s(y)), s(y), z)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
IF(false, s(x0), s(x1), y2) → DIV(minus(x0, x1), s(x1), y2)
DIV(s(x0), s(x1), y2) → IF(lt(x0, x1), s(x0), s(x1), inc(y2))
IF(false, s(x0), s(x1), y2) → DIV(minus(x0, x1), s(x1), y2)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
IF(false, s(x0), s(x1), y2) → DIV(minus(x0, x1), s(x1), y2)
POL(0) = 0
POL(DIV(x1, x2, x3)) = 1 + x1
POL(IF(x1, x2, x3, x4)) = x1 + x2
POL(false) = 1
POL(inc(x1)) = 0
POL(lt(x1, x2)) = 1
POL(minus(x1, x2)) = x1
POL(s(x1)) = 1 + x1
POL(true) = 0
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
DIV(s(x0), s(x1), y2) → IF(lt(x0, x1), s(x0), s(x1), inc(y2))
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x0, 0)
minus(s(x0), s(x1))
lt(x0, 0)
lt(0, s(x0))
lt(s(x0), s(x1))
inc(0)
inc(s(x0))