476.56/157.82
476.56/157.82
(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
ge(x, 0) → true
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ge(0, s(y)) → false
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ge(s(x), s(y)) → ge(x, y)
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minus(x, 0) → x
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minus(0, y) → 0
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minus(s(x), s(y)) → minus(x, y)
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id_inc(x) → x
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id_inc(x) → s(x)
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quot(x, y) → div(x, y, 0)
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div(x, y, z) → if(ge(y, s(0)), ge(x, y), x, y, z)
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if(false, b, x, y, z) → div_by_zero
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if(true, false, x, y, z) → z
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if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
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(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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QUOT(z0, z1) → c8(DIV(z0, z1, 0))
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DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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QUOT(z0, z1) → c8(DIV(z0, z1, 0))
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DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))
K tuples:none
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, QUOT, DIV, IF
Compound Symbols:
c2, c5, c8, c9, c12
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(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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QUOT(z0, z1) → c8(DIV(z0, z1, 0))
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DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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QUOT(z0, z1) → c8(DIV(z0, z1, 0))
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DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, QUOT, DIV, IF
Compound Symbols:
c2, c5, c8, c9, c12
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(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
QUOT(z0, z1) → c8(DIV(z0, z1, 0))
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c12
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(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
DIV(
z0,
z1,
z2) →
c9(
IF(
ge(
z1,
s(
0)),
ge(
z0,
z1),
z0,
z1,
z2),
GE(
z1,
s(
0)),
GE(
z0,
z1)) by
DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
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DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 0))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
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DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 0))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
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DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 0))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:none
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, IF, DIV
Compound Symbols:
c2, c5, c12, c9
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(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing tuple parts
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(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
K tuples:none
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, IF, DIV
Compound Symbols:
c2, c5, c12, c9, c9, c9
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(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
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(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
K tuples:none
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, IF, DIV
Compound Symbols:
c2, c5, c12, c9, c9, c9
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(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
We considered the (Usable) Rules:
ge(0, s(z0)) → false
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ge(z0, 0) → true
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
476.98/157.95
minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
And the Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.95
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
476.98/157.95
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.95
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.95
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.95
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.95
DIV(x0, 0, x2) → c9
The order we found is given by the following interpretation:
Polynomial interpretation :
476.98/157.95
POL(0) = 0
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POL(DIV(x1, x2, x3)) = x2
476.98/157.95
POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x4
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POL(MINUS(x1, x2)) = 0
476.98/157.95
POL(c12(x1, x2)) = x1 + x2
476.98/157.95
POL(c2(x1)) = x1
476.98/157.95
POL(c5(x1)) = x1
476.98/157.95
POL(c9) = 0
476.98/157.95
POL(c9(x1)) = x1
476.98/157.95
POL(c9(x1, x2, x3)) = x1 + x2 + x3
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POL(false) = [3]
476.98/157.95
POL(ge(x1, x2)) = 0
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POL(id_inc(x1)) = x1
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POL(minus(x1, x2)) = x2
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POL(s(x1)) = [2]
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POL(true) = 0
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(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
476.98/157.95
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.95
minus(z0, 0) → z0
476.98/157.95
minus(0, z0) → 0
476.98/157.95
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.95
id_inc(z0) → z0
476.98/157.95
id_inc(z0) → s(z0)
476.98/157.95
quot(z0, z1) → div(z0, z1, 0)
476.98/157.95
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
476.98/157.95
if(false, z0, z1, z2, z3) → div_by_zero
476.98/157.95
if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.95
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
476.98/157.95
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.95
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.95
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.95
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.95
DIV(x0, 0, x2) → c9
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.95
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
476.98/157.95
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.95
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.95
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.95
DIV(x0, 0, x2) → c9
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, IF, DIV
Compound Symbols:
c2, c5, c12, c9, c9, c9
476.98/157.95
476.98/157.95
(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DIV(x0, 0, x2) → c9
We considered the (Usable) Rules:
ge(0, s(z0)) → false
476.98/157.95
ge(z0, 0) → true
476.98/157.95
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.95
minus(z0, 0) → z0
476.98/157.95
minus(0, z0) → 0
476.98/157.95
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.95
id_inc(z0) → z0
476.98/157.95
id_inc(z0) → s(z0)
And the Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.95
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
476.98/157.95
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.95
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.95
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.95
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.95
DIV(x0, 0, x2) → c9
The order we found is given by the following interpretation:
Polynomial interpretation :
476.98/157.95
POL(0) = [4]
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POL(DIV(x1, x2, x3)) = [4]x2
476.98/157.95
POL(GE(x1, x2)) = 0
476.98/157.95
POL(IF(x1, x2, x3, x4, x5)) = [4]x4
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POL(MINUS(x1, x2)) = 0
476.98/157.95
POL(c12(x1, x2)) = x1 + x2
476.98/157.95
POL(c2(x1)) = x1
476.98/157.95
POL(c5(x1)) = x1
476.98/157.95
POL(c9) = 0
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POL(c9(x1)) = x1
476.98/157.95
POL(c9(x1, x2, x3)) = x1 + x2 + x3
476.98/157.95
POL(false) = [5]
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POL(ge(x1, x2)) = [5]x2
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POL(id_inc(x1)) = x1
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POL(minus(x1, x2)) = x2
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POL(s(x1)) = [2]
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POL(true) = 0
476.98/157.95
476.98/157.95
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
476.98/157.95
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.95
minus(z0, 0) → z0
476.98/157.95
minus(0, z0) → 0
476.98/157.95
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.95
id_inc(z0) → z0
476.98/157.95
id_inc(z0) → s(z0)
476.98/157.95
quot(z0, z1) → div(z0, z1, 0)
476.98/157.95
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
476.98/157.95
if(false, z0, z1, z2, z3) → div_by_zero
476.98/157.95
if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
DIV(x0, 0, x2) → c9
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.99
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(x0, 0, x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, IF, DIV
Compound Symbols:
c2, c5, c12, c9, c9, c9
476.98/157.99
476.98/157.99
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
z0,
z1,
z2) →
c12(
DIV(
minus(
z0,
z1),
z1,
id_inc(
z2)),
MINUS(
z0,
z1)) by
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)), MINUS(z0, 0))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)), MINUS(0, z0))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
476.98/157.99
476.98/157.99
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
476.98/157.99
ge(0, s(z0)) → false
476.98/157.99
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
476.98/157.99
minus(0, z0) → 0
476.98/157.99
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.99
id_inc(z0) → z0
476.98/157.99
id_inc(z0) → s(z0)
476.98/157.99
quot(z0, z1) → div(z0, z1, 0)
476.98/157.99
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
476.98/157.99
if(false, z0, z1, z2, z3) → div_by_zero
476.98/157.99
if(true, false, z0, z1, z2) → z2
476.98/157.99
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.99
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
DIV(x0, 0, x2) → c9
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)), MINUS(z0, 0))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)), MINUS(0, z0))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.99
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)), MINUS(z0, 0))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)), MINUS(0, z0))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(x0, 0, x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12
476.98/157.99
476.98/157.99
(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
476.98/157.99
476.98/157.99
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
476.98/157.99
ge(0, s(z0)) → false
476.98/157.99
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
476.98/157.99
minus(0, z0) → 0
476.98/157.99
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.99
id_inc(z0) → z0
476.98/157.99
id_inc(z0) → s(z0)
476.98/157.99
quot(z0, z1) → div(z0, z1, 0)
476.98/157.99
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
476.98/157.99
if(false, z0, z1, z2, z3) → div_by_zero
476.98/157.99
if(true, false, z0, z1, z2) → z2
476.98/157.99
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.99
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
DIV(x0, 0, x2) → c9
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.99
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12
476.98/157.99
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(21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
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(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
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minus(0, z0) → 0
476.98/157.99
minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
476.98/157.99
id_inc(z0) → s(z0)
476.98/157.99
quot(z0, z1) → div(z0, z1, 0)
476.98/157.99
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
DIV(x0, 0, x2) → c9
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(x0, 0, x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12
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(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
We considered the (Usable) Rules:
id_inc(z0) → z0
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id_inc(z0) → s(z0)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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ge(0, s(z0)) → false
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ge(z0, 0) → true
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ge(s(z0), s(z1)) → ge(z0, z1)
And the Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
DIV(x0, 0, x2) → c9
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(DIV(x1, x2, x3)) = x1
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POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x3
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POL(MINUS(x1, x2)) = 0
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POL(c12(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2, x3)) = x1 + x2 + x3
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POL(false) = [3]
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POL(ge(x1, x2)) = 0
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POL(id_inc(x1)) = [5] + x1
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POL(minus(x1, x2)) = x1
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POL(s(x1)) = [4] + x1
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POL(true) = 0
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(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
476.98/157.99
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
476.98/157.99
minus(0, z0) → 0
476.98/157.99
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.99
id_inc(z0) → z0
476.98/157.99
id_inc(z0) → s(z0)
476.98/157.99
quot(z0, z1) → div(z0, z1, 0)
476.98/157.99
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
476.98/157.99
if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
DIV(x0, 0, x2) → c9
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12
476.98/157.99
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(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
DIV(
s(
z0),
s(
z1),
x2) →
c9(
IF(
ge(
s(
z1),
s(
0)),
ge(
z0,
z1),
s(
z0),
s(
z1),
x2),
GE(
s(
z1),
s(
0)),
GE(
s(
z0),
s(
z1))) by
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
476.98/157.99
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c9(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0)), x2), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
476.98/157.99
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
476.98/157.99
476.98/157.99
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
476.98/157.99
ge(0, s(z0)) → false
476.98/157.99
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
476.98/157.99
minus(0, z0) → 0
476.98/157.99
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.99
id_inc(z0) → z0
476.98/157.99
id_inc(z0) → s(z0)
476.98/157.99
quot(z0, z1) → div(z0, z1, 0)
476.98/157.99
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
476.98/157.99
if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.99
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
476.98/157.99
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c9(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0)), x2), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
476.98/157.99
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
476.98/157.99
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
476.98/157.99
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
476.98/157.99
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c9(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0)), x2), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
476.98/157.99
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12
476.98/157.99
476.98/157.99
(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
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(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c9(GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c9(GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12, c9
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(29) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
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(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12, c
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(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
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(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12, c
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(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
We considered the (Usable) Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :
476.98/157.99
POL(0) = 0
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POL(DIV(x1, x2, x3)) = [1]
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POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x2 + [2]x4
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POL(MINUS(x1, x2)) = 0
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POL(c(x1)) = x1
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POL(c12(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2, x3)) = x1 + x2 + x3
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POL(false) = [1]
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POL(ge(x1, x2)) = [1]
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POL(id_inc(x1)) = [2] + x1
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POL(minus(x1, x2)) = 0
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POL(s(x1)) = 0
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POL(true) = [1]
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(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
476.98/157.99
ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
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minus(0, z0) → 0
476.98/157.99
minus(s(z0), s(z1)) → minus(z0, z1)
476.98/157.99
id_inc(z0) → z0
476.98/157.99
id_inc(z0) → s(z0)
476.98/157.99
quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
476.98/157.99
if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
476.98/157.99
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c9, c12, c12, c
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(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
DIV(
x0,
s(
z0),
x2) →
c9(
IF(
ge(
z0,
0),
ge(
x0,
s(
z0)),
x0,
s(
z0),
x2),
GE(
s(
z0),
s(
0)),
GE(
x0,
s(
z0))) by
DIV(0, s(z0), x2) → c9(IF(ge(z0, 0), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
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DIV(s(z0), s(z1), x2) → c9(IF(ge(z1, 0), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
476.98/157.99
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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DIV(0, s(z0), x2) → c9(IF(ge(z0, 0), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
476.98/157.99
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(IF(ge(z0, 0), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
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476.98/157.99
(37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
476.98/157.99
476.98/157.99
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
476.98/157.99
ge(s(z0), s(z1)) → ge(z0, z1)
476.98/157.99
minus(z0, 0) → z0
476.98/157.99
minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
476.98/157.99
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
476.98/157.99
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
476.98/157.99
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
476.98/157.99
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
476.98/157.99
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
476.98/157.99
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
476.98/157.99
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
476.98/157.99
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
476.98/157.99
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
476.98/157.99
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
476.98/157.99
DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
476.98/157.99
476.98/157.99
(39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
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(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
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(41) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
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(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
DIV(
z0,
0,
x2) →
c9(
IF(
ge(
0,
s(
0)),
true,
z0,
0,
x2)) by
DIV(x0, 0, x1) → c9(IF(false, true, x0, 0, x1))
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(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(x0, 0, x1) → c9(IF(false, true, x0, 0, x1))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
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DIV(x0, 0, x1) → c9(IF(false, true, x0, 0, x1))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
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(45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
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(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
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IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
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IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
476.98/157.99
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
476.98/157.99
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
476.98/157.99
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
476.98/157.99
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
476.98/157.99
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.01
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.01
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.01
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.01
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.01
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.01
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.01
DIV(x0, 0, x1) → c9
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.01
DIV(x0, 0, x2) → c9
477.34/158.01
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
477.34/158.01
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
477.34/158.01
477.34/158.01
(47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.01
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.01
477.34/158.01
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.01
ge(0, s(z0)) → false
477.34/158.01
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.01
minus(z0, 0) → z0
477.34/158.01
minus(0, z0) → 0
477.34/158.01
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.01
id_inc(z0) → z0
477.34/158.01
id_inc(z0) → s(z0)
477.34/158.01
quot(z0, z1) → div(z0, z1, 0)
477.34/158.01
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.01
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.01
if(true, false, z0, z1, z2) → z2
477.34/158.01
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.01
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.01
DIV(x0, 0, x2) → c9
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.01
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
477.34/158.01
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.01
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.01
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.01
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.01
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.01
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.01
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.01
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.01
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.01
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.01
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.01
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.01
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.01
DIV(x0, 0, x1) → c9
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.01
DIV(x0, 0, x2) → c9
477.34/158.01
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
477.34/158.01
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
477.34/158.01
477.34/158.01
(49) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.01
DIV(x0, 0, x2) → c9
477.34/158.01
DIV(x0, 0, x1) → c9
477.34/158.01
DIV(x0, 0, x2) → c9
477.34/158.01
DIV(x0, 0, x1) → c9
477.34/158.01
477.34/158.01
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.01
ge(0, s(z0)) → false
477.34/158.01
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.01
minus(z0, 0) → z0
477.34/158.01
minus(0, z0) → 0
477.34/158.01
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.01
id_inc(z0) → z0
477.34/158.01
id_inc(z0) → s(z0)
477.34/158.01
quot(z0, z1) → div(z0, z1, 0)
477.34/158.01
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.01
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.01
if(true, false, z0, z1, z2) → z2
477.34/158.01
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.01
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.01
DIV(x0, 0, x2) → c9
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.01
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
477.34/158.01
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.01
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.01
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.01
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.01
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.01
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.01
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.01
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.01
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.02
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.02
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.02
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.02
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.02
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.02
DIV(x0, 0, x2) → c9
477.34/158.02
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
477.34/158.02
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.02
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
477.34/158.02
477.34/158.02
(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
s(
z0),
s(
z1),
x2) →
c12(
DIV(
minus(
z0,
z1),
s(
z1),
id_inc(
x2)),
MINUS(
s(
z0),
s(
z1))) by
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.02
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.02
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.02
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.02
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.02
477.34/158.02
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.02
ge(0, s(z0)) → false
477.34/158.02
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.02
minus(z0, 0) → z0
477.34/158.02
minus(0, z0) → 0
477.34/158.02
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.02
id_inc(z0) → z0
477.34/158.02
id_inc(z0) → s(z0)
477.34/158.02
quot(z0, z1) → div(z0, z1, 0)
477.34/158.02
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.02
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.02
if(true, false, z0, z1, z2) → z2
477.34/158.02
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.02
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.02
DIV(x0, 0, x2) → c9
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.02
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.02
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.02
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.02
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.02
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.02
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.02
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.02
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.02
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.02
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.02
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.02
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.02
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.02
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.02
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.02
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.02
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.02
DIV(x0, 0, x2) → c9
477.34/158.02
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
477.34/158.02
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.02
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
477.34/158.02
477.34/158.02
(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.02
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.02
477.34/158.02
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.02
ge(0, s(z0)) → false
477.34/158.02
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.02
minus(z0, 0) → z0
477.34/158.02
minus(0, z0) → 0
477.34/158.02
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.02
id_inc(z0) → z0
477.34/158.02
id_inc(z0) → s(z0)
477.34/158.02
quot(z0, z1) → div(z0, z1, 0)
477.34/158.02
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.02
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.02
if(true, false, z0, z1, z2) → z2
477.34/158.02
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.02
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.02
DIV(x0, 0, x2) → c9
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.02
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
477.34/158.02
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.02
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.02
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.02
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.02
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.02
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.02
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.02
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.02
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.02
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.02
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.02
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.02
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.02
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.02
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.02
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.02
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.03
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.03
DIV(x0, 0, x2) → c9
477.34/158.03
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.03
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
477.34/158.03
477.34/158.03
(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
z0,
0,
x2) →
c12(
DIV(
z0,
0,
id_inc(
x2))) by
IF(true, true, x0, 0, z0) → c12(DIV(x0, 0, z0))
477.34/158.03
IF(true, true, x0, 0, z0) → c12(DIV(x0, 0, s(z0)))
477.34/158.03
477.34/158.03
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.03
ge(0, s(z0)) → false
477.34/158.03
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.03
minus(z0, 0) → z0
477.34/158.03
minus(0, z0) → 0
477.34/158.03
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.03
id_inc(z0) → z0
477.34/158.03
id_inc(z0) → s(z0)
477.34/158.03
quot(z0, z1) → div(z0, z1, 0)
477.34/158.03
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.03
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.03
if(true, false, z0, z1, z2) → z2
477.34/158.03
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.03
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.03
DIV(x0, 0, x2) → c9
477.34/158.03
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.03
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.03
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.03
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.03
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.03
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.03
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.03
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.03
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.03
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.03
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.03
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.03
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.03
IF(true, true, x0, 0, z0) → c12(DIV(x0, 0, z0))
477.34/158.03
IF(true, true, x0, 0, z0) → c12(DIV(x0, 0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.03
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.03
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.03
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.03
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.03
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.03
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.03
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.03
DIV(x0, 0, x2) → c9
477.34/158.03
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.03
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
477.34/158.03
477.34/158.03
(57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.03
IF(true, true, x0, 0, z0) → c12(DIV(x0, 0, s(z0)))
477.34/158.03
IF(true, true, x0, 0, z0) → c12(DIV(x0, 0, z0))
477.34/158.03
477.34/158.03
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.03
ge(0, s(z0)) → false
477.34/158.03
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.03
minus(z0, 0) → z0
477.34/158.03
minus(0, z0) → 0
477.34/158.03
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.03
id_inc(z0) → z0
477.34/158.03
id_inc(z0) → s(z0)
477.34/158.03
quot(z0, z1) → div(z0, z1, 0)
477.34/158.03
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.03
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.03
if(true, false, z0, z1, z2) → z2
477.34/158.03
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.03
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.03
DIV(x0, 0, x2) → c9
477.34/158.03
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.03
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.03
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.03
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.03
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.03
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.03
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.03
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.03
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.03
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.03
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.03
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.03
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.07
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c12, c9, c
477.34/158.07
477.34/158.07
(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
0,
z0,
x2) →
c12(
DIV(
0,
z0,
id_inc(
x2))) by
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
477.34/158.07
477.34/158.07
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.07
ge(0, s(z0)) → false
477.34/158.07
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.07
minus(z0, 0) → z0
477.34/158.07
minus(0, z0) → 0
477.34/158.07
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.07
id_inc(z0) → z0
477.34/158.07
id_inc(z0) → s(z0)
477.34/158.07
quot(z0, z1) → div(z0, z1, 0)
477.34/158.07
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.07
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.07
if(true, false, z0, z1, z2) → z2
477.34/158.07
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.07
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.07
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.07
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.07
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c12
477.34/158.07
477.34/158.07
(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.07
477.34/158.07
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.07
ge(0, s(z0)) → false
477.34/158.07
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.07
minus(z0, 0) → z0
477.34/158.07
minus(0, z0) → 0
477.34/158.07
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.07
id_inc(z0) → z0
477.34/158.07
id_inc(z0) → s(z0)
477.34/158.07
quot(z0, z1) → div(z0, z1, 0)
477.34/158.07
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.07
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.07
if(true, false, z0, z1, z2) → z2
477.34/158.07
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.07
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.07
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.07
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.07
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c12
477.34/158.07
477.34/158.07
(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
DIV(
x0,
s(
z0),
x2) →
c9(
IF(
true,
ge(
x0,
s(
z0)),
x0,
s(
z0),
x2),
GE(
s(
z0),
s(
0)),
GE(
x0,
s(
z0))) by
DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
477.34/158.07
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.07
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.07
477.34/158.07
(64) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.07
ge(0, s(z0)) → false
477.34/158.07
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.07
minus(z0, 0) → z0
477.34/158.07
minus(0, z0) → 0
477.34/158.07
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.07
id_inc(z0) → z0
477.34/158.07
id_inc(z0) → s(z0)
477.34/158.07
quot(z0, z1) → div(z0, z1, 0)
477.34/158.07
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.07
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.07
if(true, false, z0, z1, z2) → z2
477.34/158.07
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.07
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.07
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.07
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
477.34/158.07
DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
477.34/158.07
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.07
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.07
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
477.34/158.07
DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
477.34/158.07
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.07
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c12
477.34/158.07
477.34/158.07
(65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
477.34/158.07
477.34/158.07
(66) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.07
ge(0, s(z0)) → false
477.34/158.07
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.07
minus(z0, 0) → z0
477.34/158.07
minus(0, z0) → 0
477.34/158.07
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.07
id_inc(z0) → z0
477.34/158.07
id_inc(z0) → s(z0)
477.34/158.07
quot(z0, z1) → div(z0, z1, 0)
477.34/158.07
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.07
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.07
if(true, false, z0, z1, z2) → z2
477.34/158.07
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.07
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.07
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.07
DIV(x0, 0, x2) → c9
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.07
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.07
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.07
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.07
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.07
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.07
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.07
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c12
477.34/158.08
477.34/158.08
(67) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
477.34/158.08
477.34/158.08
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c12, c1
477.34/158.08
477.34/158.08
(69) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 2 leading nodes:
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
477.34/158.08
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.08
477.34/158.08
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
477.34/158.08
477.34/158.08
(71) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
477.34/158.08
(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
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(73) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
We considered the (Usable) Rules:
id_inc(z0) → z0
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id_inc(z0) → s(z0)
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ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = [1]
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POL(DIV(x1, x2, x3)) = x2
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POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x4
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POL(MINUS(x1, x2)) = 0
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POL(c(x1)) = x1
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POL(c1(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2, x3)) = x1 + x2 + x3
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POL(false) = [4]
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POL(ge(x1, x2)) = [2] + [4]x2
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POL(id_inc(x1)) = [4] + x1
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POL(minus(x1, x2)) = 0
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POL(s(x1)) = [2] + x1
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POL(true) = 0
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(74) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
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(75) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1)))) by DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
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(76) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
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DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
477.34/158.08
477.34/158.08
(77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.08
477.34/158.08
(78) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
477.34/158.08
477.34/158.08
(79) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0))) by DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
477.34/158.08
(80) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
477.34/158.08
477.34/158.08
(81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.08
477.34/158.08
(82) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
477.34/158.08
477.34/158.08
(83) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0))) by DIV(s(z0), s(z1), z2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), z2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
477.34/158.08
(84) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c9, c, c1
477.34/158.08
477.34/158.08
(85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.08
477.34/158.08
(86) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(ge(z1, 0), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c, c9, c1
477.34/158.08
477.34/158.08
(87) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace DIV(s(z0), s(z1), x2) → c9(IF(ge(z1, 0), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1))) by DIV(s(z0), s(z1), z2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), z2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
477.34/158.08
(88) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c, c9, c1
477.34/158.08
477.34/158.08
(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.08
477.34/158.08
(90) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c, c9, c1
477.34/158.08
477.34/158.08
(91) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1)))) by DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.08
477.34/158.08
(92) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c, c9, c1
477.34/158.08
477.34/158.08
(93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
DIV(x0, 0, x2) → c9
477.34/158.08
477.34/158.08
(94) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.08
ge(0, s(z0)) → false
477.34/158.08
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.08
minus(z0, 0) → z0
477.34/158.08
minus(0, z0) → 0
477.34/158.08
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.08
id_inc(z0) → z0
477.34/158.08
id_inc(z0) → s(z0)
477.34/158.08
quot(z0, z1) → div(z0, z1, 0)
477.34/158.08
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.08
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.08
if(true, false, z0, z1, z2) → z2
477.34/158.08
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.08
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.08
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.08
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.08
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
S tuples:
GE(s(z0), s(z1)) → c2(GE(z0, z1))
477.34/158.08
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.08
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.08
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.08
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.08
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.08
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.09
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
GE, MINUS, DIV, IF
Compound Symbols:
c2, c5, c9, c9, c12, c, c9, c1
477.34/158.09
477.34/158.09
(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
GE(
s(
z0),
s(
z1)) →
c2(
GE(
z0,
z1)) by
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
477.34/158.09
(96) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1), GE(s(0), s(0)), GE(s(z0), s(0)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
K tuples:
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
477.34/158.09
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c9, c12, c, c9, c1, c2
477.34/158.09
477.34/158.09
(97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 12 trailing tuple parts
477.34/158.09
477.34/158.09
(98) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
477.34/158.09
(100) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(101) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace DIV(s(z0), s(0), z1) → c9(IF(ge(0, 0), true, s(z0), s(0), z1)) by DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
477.34/158.09
(102) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
477.34/158.09
(104) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(105) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
IF(
true,
true,
x0,
x1,
z0) →
c12(
DIV(
minus(
x0,
x1),
x1,
z0),
MINUS(
x0,
x1)) by
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), x2), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
477.34/158.09
477.34/158.09
(106) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), x2), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), x2), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
477.34/158.09
(108) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), x2), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), x2), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
s(
x0),
s(
x1),
x2) →
c12(
DIV(
minus(
s(
x0),
s(
x1)),
s(
x1),
x2),
MINUS(
s(
x0),
s(
x1))) by
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), x2), MINUS(s(z0), s(z1)))
477.34/158.09
477.34/158.09
(110) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), x2), MINUS(s(z0), s(z1)))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
477.34/158.09
(112) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), x2), MINUS(s(z0), s(z1)))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(113) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
We considered the (Usable) Rules:
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
ge(z0, 0) → true
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ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
id_inc(z0) → z0
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id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
477.34/158.09
POL(0) = 0
477.34/158.09
POL(DIV(x1, x2, x3)) = x1
477.34/158.09
POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x1 + x3
477.34/158.09
POL(MINUS(x1, x2)) = 0
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POL(c) = 0
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POL(c1(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2)) = x1 + x2
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POL(false) = [3]
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POL(ge(x1, x2)) = 0
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POL(id_inc(x1)) = [2] + x1
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POL(minus(x1, x2)) = x1
477.34/158.09
POL(s(x1)) = [1] + x1
477.34/158.09
POL(true) = 0
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(114) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
s(
s(
x0)),
s(
s(
x1)),
x2) →
c12(
DIV(
minus(
s(
s(
x0)),
s(
s(
x1))),
s(
s(
x1)),
x2),
MINUS(
s(
s(
x0)),
s(
s(
x1)))) by
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
477.34/158.09
(116) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(x0, 0, x2) → c9
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
K tuples:
DIV(x0, 0, x2) → c9
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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DIV(s(0), s(s(z0)), x2) → c
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DIV(0, s(z0), x2) → c9
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DIV(x0, 0, x2) → c9
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DIV(x0, s(x1), x2) → c9
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(118) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
K tuples:
DIV(x0, 0, x2) → c9
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
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(119) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
We considered the (Usable) Rules:
minus(s(z0), s(z1)) → minus(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(DIV(x1, x2, x3)) = [4]x1
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POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x1 + [4]x3
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POL(MINUS(x1, x2)) = 0
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POL(c) = 0
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POL(c1(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2)) = x1 + x2
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POL(false) = [3]
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POL(ge(x1, x2)) = 0
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POL(id_inc(x1)) = x1
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POL(minus(x1, x2)) = x1
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POL(s(x1)) = [4] + x1
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POL(true) = 0
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(120) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), x1), MINUS(s(x0), s(0)))
K tuples:
DIV(x0, 0, x2) → c9
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
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(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
s(
x0),
s(
0),
x1) →
c12(
DIV(
minus(
s(
x0),
s(
0)),
s(
0),
x1),
MINUS(
s(
x0),
s(
0))) by
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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(122) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
K tuples:
DIV(x0, 0, x2) → c9
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
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(123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, 0, x2) → c9
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DIV(x0, s(x1), x2) → c9
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(124) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
K tuples:
DIV(x0, 0, x2) → c9
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
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(125) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
We considered the (Usable) Rules:
minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(DIV(x1, x2, x3)) = x1
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POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x1 + x3
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POL(MINUS(x1, x2)) = 0
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POL(c) = 0
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POL(c1(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2)) = x1 + x2
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POL(false) = [3]
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POL(ge(x1, x2)) = 0
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POL(id_inc(x1)) = x1
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POL(minus(x1, x2)) = x1
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POL(s(x1)) = [1] + x1
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POL(true) = 0
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(126) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
K tuples:
DIV(x0, 0, x2) → c9
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
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(127) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
DIV(
x0,
0,
x2) →
c9 by
DIV(y0, 0, s(x2)) → c9
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(128) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(x0, 0, x2) → c9
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IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
DIV(y0, 0, s(x2)) → c9
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
DIV(y0, 0, s(x2)) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(x0, 0, x1) → c9
477.34/158.09
DIV(y0, 0, s(x2)) → c9
477.34/158.09
477.34/158.09
(130) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(x0, 0, x1) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
DIV(y0, 0, s(x2)) → c9
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
DIV(y0, 0, s(x2)) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, IF, DIV, GE
Compound Symbols:
c5, c12, c9, c9, c1, c2, c, c9
477.34/158.09
477.34/158.09
(131) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
IF(
true,
true,
x0,
x1,
z0) →
c12(
DIV(
minus(
x0,
x1),
x1,
s(
z0)),
MINUS(
x0,
x1)) by
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), s(x2)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
477.34/158.09
(132) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(x0, 0, x1) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
DIV(y0, 0, s(x2)) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), s(x2)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), s(x2)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
DIV(y0, 0, s(x2)) → c9
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c9, c12, c1, c2, c, c9
477.34/158.09
477.34/158.09
(133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(x0, 0, x1) → c9
477.34/158.09
DIV(y0, 0, s(x2)) → c9
477.34/158.09
477.34/158.09
(134) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), s(x2)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(x1), x2) → c12(DIV(minus(s(x0), s(x1)), s(x1), s(x2)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.09
477.34/158.09
(135) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
s(
x0),
s(
x1),
x2) →
c12(
DIV(
minus(
s(
x0),
s(
x1)),
s(
x1),
s(
x2)),
MINUS(
s(
x0),
s(
x1))) by
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), s(x2)), MINUS(s(z0), s(z1)))
477.34/158.09
477.34/158.09
(136) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), s(x2)), MINUS(s(z0), s(z1)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.09
477.34/158.09
(137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
477.34/158.09
(138) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), s(x2)), MINUS(s(z0), s(z1)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.09
477.34/158.09
(139) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
We considered the (Usable) Rules:
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
477.34/158.09
POL(0) = 0
477.34/158.09
POL(DIV(x1, x2, x3)) = [1] + x1
477.34/158.09
POL(GE(x1, x2)) = 0
477.34/158.09
POL(IF(x1, x2, x3, x4, x5)) = x2 + x3
477.34/158.09
POL(MINUS(x1, x2)) = 0
477.34/158.09
POL(c) = 0
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POL(c1(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
477.34/158.09
POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2)) = x1 + x2
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POL(false) = [1]
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POL(ge(x1, x2)) = [1]
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POL(id_inc(x1)) = x1
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POL(minus(x1, x2)) = x1
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POL(s(x1)) = [1] + x1
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POL(true) = [1]
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(140) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.09
477.34/158.09
(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
s(
s(
x0)),
s(
s(
x1)),
x2) →
c12(
DIV(
minus(
s(
s(
x0)),
s(
s(
x1))),
s(
s(
x1)),
s(
x2)),
MINUS(
s(
s(
x0)),
s(
s(
x1)))) by
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
477.34/158.09
(142) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.09
477.34/158.09
(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
477.34/158.09
(144) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.09
if(true, false, z0, z1, z2) → z2
477.34/158.09
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.09
477.34/158.09
(145) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
We considered the (Usable) Rules:
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
477.34/158.09
POL(0) = 0
477.34/158.09
POL(DIV(x1, x2, x3)) = x1
477.34/158.09
POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x1 + x3
477.34/158.09
POL(MINUS(x1, x2)) = 0
477.34/158.09
POL(c) = 0
477.34/158.09
POL(c1(x1)) = x1
477.34/158.09
POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
477.34/158.09
POL(c5(x1)) = x1
477.34/158.09
POL(c9) = 0
477.34/158.09
POL(c9(x1)) = x1
477.34/158.09
POL(c9(x1, x2)) = x1 + x2
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POL(false) = [3]
477.34/158.09
POL(ge(x1, x2)) = 0
477.34/158.09
POL(id_inc(x1)) = x1
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POL(minus(x1, x2)) = x1
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POL(s(x1)) = [1] + x1
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POL(true) = 0
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477.34/158.09
(146) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
477.34/158.09
quot(z0, z1) → div(z0, z1, 0)
477.34/158.09
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.09
if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(x0, s(x1), x2) → c9
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.09
477.34/158.09
(147) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
We considered the (Usable) Rules:
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.09
minus(z0, 0) → z0
477.34/158.09
minus(0, z0) → 0
477.34/158.09
ge(z0, 0) → true
477.34/158.09
ge(0, s(z0)) → false
477.34/158.09
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.09
id_inc(z0) → z0
477.34/158.09
id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.09
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.09
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.09
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.09
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.09
DIV(0, s(z0), x2) → c9
477.34/158.09
DIV(s(0), s(s(z0)), x2) → c
477.34/158.09
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
477.34/158.09
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.09
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.09
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.09
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.09
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
477.34/158.09
POL(0) = 0
477.34/158.09
POL(DIV(x1, x2, x3)) = x2 + x1·x2
477.34/158.09
POL(GE(x1, x2)) = 0
477.34/158.09
POL(IF(x1, x2, x3, x4, x5)) = x2 + x3·x4
477.34/158.09
POL(MINUS(x1, x2)) = 0
477.34/158.09
POL(c) = 0
477.34/158.09
POL(c1(x1)) = x1
477.34/158.09
POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
477.34/158.09
POL(c5(x1)) = x1
477.34/158.09
POL(c9) = 0
477.34/158.09
POL(c9(x1)) = x1
477.34/158.09
POL(c9(x1, x2)) = x1 + x2
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POL(false) = [1]
477.34/158.09
POL(ge(x1, x2)) = [1]
477.34/158.09
POL(id_inc(x1)) = 0
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POL(minus(x1, x2)) = x1
477.34/158.10
POL(s(x1)) = [1] + x1
477.34/158.10
POL(true) = [1]
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477.34/158.10
(148) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.10
ge(0, s(z0)) → false
477.34/158.10
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.10
minus(z0, 0) → z0
477.34/158.10
minus(0, z0) → 0
477.34/158.10
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.10
id_inc(z0) → z0
477.34/158.10
id_inc(z0) → s(z0)
477.34/158.10
quot(z0, z1) → div(z0, z1, 0)
477.34/158.10
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.10
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.10
if(true, false, z0, z1, z2) → z2
477.34/158.10
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.10
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.10
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.10
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.10
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.10
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.10
DIV(0, s(z0), x2) → c9
477.34/158.10
DIV(s(0), s(s(z0)), x2) → c
477.34/158.10
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.10
DIV(x0, s(x1), x2) → c9
477.34/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.10
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.10
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
477.34/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.10
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.10
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.10
IF(true, true, s(x0), s(0), x1) → c12(DIV(minus(s(x0), s(0)), s(0), s(x1)), MINUS(s(x0), s(0)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.10
DIV(0, s(z0), x2) → c9
477.34/158.10
DIV(s(0), s(s(z0)), x2) → c
477.34/158.10
DIV(x0, s(x1), x2) → c9
477.34/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.10
477.34/158.10
(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
IF(
true,
true,
s(
x0),
s(
0),
x1) →
c12(
DIV(
minus(
s(
x0),
s(
0)),
s(
0),
s(
x1)),
MINUS(
s(
x0),
s(
0))) by
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
477.34/158.10
477.34/158.10
(150) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.10
ge(0, s(z0)) → false
477.34/158.10
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.10
minus(z0, 0) → z0
477.34/158.10
minus(0, z0) → 0
477.34/158.10
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.10
id_inc(z0) → z0
477.34/158.10
id_inc(z0) → s(z0)
477.34/158.10
quot(z0, z1) → div(z0, z1, 0)
477.34/158.10
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.34/158.10
if(false, z0, z1, z2, z3) → div_by_zero
477.34/158.10
if(true, false, z0, z1, z2) → z2
477.34/158.10
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.10
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.10
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
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(151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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DIV(s(0), s(s(z0)), x2) → c
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DIV(0, s(z0), x2) → c9
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DIV(x0, s(x1), x2) → c9
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(152) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
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(153) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
We considered the (Usable) Rules:
minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
477.34/158.10
POL(0) = 0
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POL(DIV(x1, x2, x3)) = [4]x1 + [2]x3
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POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = x1 + [4]x3 + [2]x5
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POL(MINUS(x1, x2)) = 0
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POL(c) = 0
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POL(c1(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2)) = x1 + x2
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POL(false) = [3]
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POL(ge(x1, x2)) = 0
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POL(id_inc(x1)) = [4] + x1
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POL(minus(x1, x2)) = x1
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POL(s(x1)) = [2] + x1
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POL(true) = 0
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(154) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.34/158.10
477.34/158.10
(155) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.10
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
We considered the (Usable) Rules:
minus(z0, 0) → z0
477.34/158.10
minus(0, z0) → 0
477.34/158.10
minus(s(z0), s(z1)) → minus(z0, z1)
477.34/158.10
ge(z0, 0) → true
477.34/158.10
ge(0, s(z0)) → false
477.34/158.10
ge(s(z0), s(z1)) → ge(z0, z1)
477.34/158.10
id_inc(z0) → z0
477.34/158.10
id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.34/158.10
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.34/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.34/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.34/158.10
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.34/158.10
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.34/158.10
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.34/158.10
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.34/158.10
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.34/158.10
DIV(0, s(z0), x2) → c9
477.34/158.10
DIV(s(0), s(s(z0)), x2) → c
477.34/158.10
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.34/158.10
DIV(x0, s(x1), x2) → c9
477.34/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.34/158.10
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.34/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.34/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.34/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.34/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
477.34/158.10
POL(0) = [1]
477.34/158.10
POL(DIV(x1, x2, x3)) = [5] + [4]x1 + [2]x3
477.34/158.10
POL(GE(x1, x2)) = 0
477.34/158.10
POL(IF(x1, x2, x3, x4, x5)) = [4]x2 + [4]x3 + [2]x5
477.34/158.10
POL(MINUS(x1, x2)) = 0
477.34/158.10
POL(c) = 0
477.34/158.10
POL(c1(x1)) = x1
477.34/158.10
POL(c12(x1, x2)) = x1 + x2
477.34/158.10
POL(c2(x1)) = x1
477.34/158.10
POL(c5(x1)) = x1
477.34/158.10
POL(c9) = 0
477.34/158.10
POL(c9(x1)) = x1
477.34/158.10
POL(c9(x1, x2)) = x1 + x2
477.34/158.10
POL(false) = [1]
477.34/158.10
POL(ge(x1, x2)) = [1]
477.34/158.10
POL(id_inc(x1)) = [4] + x1
477.34/158.10
POL(minus(x1, x2)) = x1
477.34/158.10
POL(s(x1)) = [4] + x1
477.34/158.10
POL(true) = [1]
477.34/158.10
477.34/158.10
(156) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.34/158.10
ge(0, s(z0)) → false
477.71/158.10
ge(s(z0), s(z1)) → ge(z0, z1)
477.71/158.10
minus(z0, 0) → z0
477.71/158.10
minus(0, z0) → 0
477.71/158.10
minus(s(z0), s(z1)) → minus(z0, z1)
477.71/158.10
id_inc(z0) → z0
477.71/158.10
id_inc(z0) → s(z0)
477.71/158.10
quot(z0, z1) → div(z0, z1, 0)
477.71/158.10
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.71/158.10
if(false, z0, z1, z2, z3) → div_by_zero
477.71/158.10
if(true, false, z0, z1, z2) → z2
477.71/158.10
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.71/158.10
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.71/158.10
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.71/158.10
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.71/158.10
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.71/158.10
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.71/158.10
DIV(0, s(z0), x2) → c9
477.71/158.10
DIV(s(0), s(s(z0)), x2) → c
477.71/158.10
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.71/158.10
DIV(x0, s(x1), x2) → c9
477.71/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.71/158.10
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.71/158.10
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.71/158.10
DIV(0, s(z0), x2) → c9
477.71/158.10
DIV(s(0), s(s(z0)), x2) → c
477.71/158.10
DIV(x0, s(x1), x2) → c9
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
477.71/158.10
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.71/158.10
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
477.71/158.10
477.71/158.10
(157) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
We considered the (Usable) Rules:
minus(z0, 0) → z0
477.71/158.10
minus(0, z0) → 0
477.71/158.10
minus(s(z0), s(z1)) → minus(z0, z1)
477.71/158.10
ge(z0, 0) → true
477.71/158.10
ge(0, s(z0)) → false
477.71/158.10
ge(s(z0), s(z1)) → ge(z0, z1)
477.71/158.10
id_inc(z0) → z0
477.71/158.10
id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.71/158.10
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.71/158.10
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.71/158.10
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.71/158.10
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.71/158.10
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.71/158.10
DIV(0, s(z0), x2) → c9
477.71/158.10
DIV(s(0), s(s(z0)), x2) → c
477.71/158.10
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.71/158.10
DIV(x0, s(x1), x2) → c9
477.71/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.71/158.10
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
477.71/158.10
POL(0) = 0
477.71/158.10
POL(DIV(x1, x2, x3)) = [2]x1 + [2]x12
477.71/158.10
POL(GE(x1, x2)) = x1
477.71/158.10
POL(IF(x1, x2, x3, x4, x5)) = [1] + [3]x1 + [3]x1·x5 + [3]x12 + [2]x32
477.71/158.10
POL(MINUS(x1, x2)) = 0
477.71/158.10
POL(c) = 0
477.71/158.10
POL(c1(x1)) = x1
477.71/158.10
POL(c12(x1, x2)) = x1 + x2
477.71/158.10
POL(c2(x1)) = x1
477.71/158.10
POL(c5(x1)) = x1
477.71/158.10
POL(c9) = 0
477.71/158.10
POL(c9(x1)) = x1
477.71/158.10
POL(c9(x1, x2)) = x1 + x2
477.71/158.10
POL(false) = [3]
477.71/158.10
POL(ge(x1, x2)) = 0
477.71/158.10
POL(id_inc(x1)) = 0
477.71/158.10
POL(minus(x1, x2)) = x1
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POL(s(x1)) = [2] + x1
477.71/158.10
POL(true) = 0
477.71/158.10
477.71/158.10
(158) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
477.71/158.10
ge(0, s(z0)) → false
477.71/158.10
ge(s(z0), s(z1)) → ge(z0, z1)
477.71/158.10
minus(z0, 0) → z0
477.71/158.10
minus(0, z0) → 0
477.71/158.10
minus(s(z0), s(z1)) → minus(z0, z1)
477.71/158.10
id_inc(z0) → z0
477.71/158.10
id_inc(z0) → s(z0)
477.71/158.10
quot(z0, z1) → div(z0, z1, 0)
477.71/158.10
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
477.71/158.10
if(false, z0, z1, z2, z3) → div_by_zero
477.71/158.10
if(true, false, z0, z1, z2) → z2
477.71/158.10
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
477.71/158.10
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
477.71/158.10
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
477.71/158.10
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
477.71/158.10
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
477.71/158.10
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
477.71/158.10
DIV(0, s(z0), x2) → c9
477.71/158.10
DIV(s(0), s(s(z0)), x2) → c
477.71/158.10
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
477.71/158.10
DIV(x0, s(x1), x2) → c9
477.71/158.10
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
477.71/158.10
DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
S tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
477.71/158.10
DIV(0, s(z0), x2) → c9
477.71/158.10
DIV(s(0), s(s(z0)), x2) → c
477.71/158.10
DIV(x0, s(x1), x2) → c9
477.71/158.10
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
477.71/158.10
IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
477.71/158.10
IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
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(159) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
We considered the (Usable) Rules:
minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
And the Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(DIV(x1, x2, x3)) = [1] + x3 + [2]x1·x2
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POL(GE(x1, x2)) = 0
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POL(IF(x1, x2, x3, x4, x5)) = [3]x1 + x5 + [3]x1·x5 + [3]x12 + [2]x3·x4
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POL(MINUS(x1, x2)) = x2
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POL(c) = 0
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POL(c1(x1)) = x1
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POL(c12(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c5(x1)) = x1
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POL(c9) = 0
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POL(c9(x1)) = x1
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POL(c9(x1, x2)) = x1 + x2
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POL(false) = [3]
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POL(ge(x1, x2)) = 0
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POL(id_inc(x1)) = [3] + x1
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POL(minus(x1, x2)) = x1
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POL(s(x1)) = [2] + x1
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POL(true) = 0
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(160) Obligation:
Complexity Dependency Tuples Problem
Rules:
ge(z0, 0) → true
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ge(0, s(z0)) → false
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ge(s(z0), s(z1)) → ge(z0, z1)
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minus(z0, 0) → z0
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minus(0, z0) → 0
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minus(s(z0), s(z1)) → minus(z0, z1)
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id_inc(z0) → z0
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id_inc(z0) → s(z0)
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quot(z0, z1) → div(z0, z1, 0)
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div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
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if(false, z0, z1, z2, z3) → div_by_zero
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if(true, false, z0, z1, z2) → z2
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if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
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DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
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IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
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IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
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IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(x0, s(x1), x2) → c9
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
S tuples:none
K tuples:
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
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DIV(0, s(z0), x2) → c9
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DIV(s(0), s(s(z0)), x2) → c
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DIV(x0, s(x1), x2) → c9
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), x2), MINUS(s(s(x0)), s(s(x1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), x1), MINUS(s(z0), s(0)))
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IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
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IF(true, true, s(s(x0)), s(s(x1)), x2) → c12(DIV(minus(s(x0), s(x1)), s(s(x1)), s(x2)), MINUS(s(s(x0)), s(s(x1))))
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DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(true, ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
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IF(true, true, s(z0), s(0), x1) → c12(DIV(minus(z0, 0), s(0), s(x1)), MINUS(s(z0), s(0)))
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DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
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DIV(s(z0), s(0), z1) → c9(IF(true, true, s(z0), s(0), z1))
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GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
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MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:
ge, minus, id_inc, quot, div, if
Defined Pair Symbols:
MINUS, DIV, IF, GE
Compound Symbols:
c5, c9, c12, c1, c2, c9, c, c9
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(161) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(162) BOUNDS(O(1), O(1))
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