0 QTRS
↳1 AAECC Innermost (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 QReductionProof (⇔)
↳11 QDP
↳12 QDPSizeChangeProof (⇔)
↳13 TRUE
↳14 QDP
↳15 UsableRulesProof (⇔)
↳16 QDP
↳17 QReductionProof (⇔)
↳18 QDP
↳19 QDPSizeChangeProof (⇔)
↳20 TRUE
↳21 QDP
↳22 UsableRulesProof (⇔)
↳23 QDP
↳24 QReductionProof (⇔)
↳25 QDP
↳26 Instantiation (⇔)
↳27 QDP
↳28 NonInfProof (⇔)
↳29 QDP
↳30 PisEmptyProof (⇔)
↳31 TRUE
f(true, x, y, z) → f(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, y, s(z))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, y, s(z))
f(true, x, y, z) → f(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, y, s(z))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, s(y), z)
F(true, x, y, z) → GT(x, plus(y, z))
F(true, x, y, z) → PLUS(y, z)
F(true, x, y, z) → F(gt(x, plus(y, z)), x, y, s(z))
PLUS(n, s(m)) → PLUS(n, m)
GT(s(u), s(v)) → GT(u, v)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, y, s(z))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
GT(s(u), s(v)) → GT(u, v)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, y, s(z))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
GT(s(u), s(v)) → GT(u, v)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
GT(s(u), s(v)) → GT(u, v)
From the DPs we obtained the following set of size-change graphs:
PLUS(n, s(m)) → PLUS(n, m)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, y, s(z))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
PLUS(n, s(m)) → PLUS(n, m)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
PLUS(n, s(m)) → PLUS(n, m)
From the DPs we obtained the following set of size-change graphs:
F(true, x, y, z) → F(gt(x, plus(y, z)), x, y, s(z))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, s(y), z)
f(true, x, y, z) → f(gt(x, plus(y, z)), x, y, s(z))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, y, s(z))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, s(y), z)
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
f(true, x0, x1, x2)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
f(true, x0, x1, x2)
F(true, x, y, z) → F(gt(x, plus(y, z)), x, y, s(z))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, s(y), z)
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
F(true, z0, z1, s(z2)) → F(gt(z0, plus(z1, s(z2))), z0, z1, s(s(z2)))
F(true, z0, s(z1), z2) → F(gt(z0, plus(s(z1), z2)), z0, s(z1), s(z2))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, s(y), z)
F(true, z0, z1, s(z2)) → F(gt(z0, plus(z1, s(z2))), z0, z1, s(s(z2)))
F(true, z0, s(z1), z2) → F(gt(z0, plus(s(z1), z2)), z0, s(z1), s(z2))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
(1) (F(gt(x0, plus(x1, x2)), x0, x1, s(x2))=F(true, x3, x4, x5) ⇒ F(true, x3, x4, x5)≥F(gt(x3, plus(x4, x5)), x3, x4, s(x5)))
(2) (plus(x1, x2)=x24∧gt(x0, x24)=true ⇒ F(true, x0, x1, s(x2))≥F(gt(x0, plus(x1, s(x2))), x0, x1, s(s(x2))))
(3) (true=true∧plus(x1, x2)=0 ⇒ F(true, s(x26), x1, s(x2))≥F(gt(s(x26), plus(x1, s(x2))), s(x26), x1, s(s(x2))))
(4) (gt(x28, x27)=true∧plus(x1, x2)=s(x27)∧(∀x29,x30:gt(x28, x27)=true∧plus(x29, x30)=x27 ⇒ F(true, x28, x29, s(x30))≥F(gt(x28, plus(x29, s(x30))), x28, x29, s(s(x30)))) ⇒ F(true, s(x28), x1, s(x2))≥F(gt(s(x28), plus(x1, s(x2))), s(x28), x1, s(s(x2))))
(5) (plus(x1, x2)=0 ⇒ F(true, s(x26), x1, s(x2))≥F(gt(s(x26), plus(x1, s(x2))), s(x26), x1, s(s(x2))))
(6) (x34=s(x27)∧gt(x28, x27)=true∧(∀x29,x30:gt(x28, x27)=true∧plus(x29, x30)=x27 ⇒ F(true, x28, x29, s(x30))≥F(gt(x28, plus(x29, s(x30))), x28, x29, s(s(x30)))) ⇒ F(true, s(x28), x34, s(0))≥F(gt(s(x28), plus(x34, s(0))), s(x28), x34, s(s(0))))
(7) (s(plus(x36, x35))=s(x27)∧gt(x28, x27)=true∧(∀x29,x30:gt(x28, x27)=true∧plus(x29, x30)=x27 ⇒ F(true, x28, x29, s(x30))≥F(gt(x28, plus(x29, s(x30))), x28, x29, s(s(x30))))∧(∀x37,x38,x39,x40:plus(x36, x35)=s(x37)∧gt(x38, x37)=true∧(∀x39,x40:gt(x38, x37)=true∧plus(x39, x40)=x37 ⇒ F(true, x38, x39, s(x40))≥F(gt(x38, plus(x39, s(x40))), x38, x39, s(s(x40)))) ⇒ F(true, s(x38), x36, s(x35))≥F(gt(s(x38), plus(x36, s(x35))), s(x38), x36, s(s(x35)))) ⇒ F(true, s(x28), x36, s(s(x35)))≥F(gt(s(x28), plus(x36, s(s(x35)))), s(x28), x36, s(s(s(x35)))))
(8) (x31=0 ⇒ F(true, s(x26), x31, s(0))≥F(gt(s(x26), plus(x31, s(0))), s(x26), x31, s(s(0))))
(9) (F(true, s(x26), 0, s(0))≥F(gt(s(x26), plus(0, s(0))), s(x26), 0, s(s(0))))
(10) (gt(x28, x27)=true ⇒ F(true, s(x28), s(x27), s(0))≥F(gt(s(x28), plus(s(x27), s(0))), s(x28), s(x27), s(s(0))))
(11) (plus(x36, x35)=x27∧gt(x28, x27)=true∧(∀x29,x30:gt(x28, x27)=true∧plus(x29, x30)=x27 ⇒ F(true, x28, x29, s(x30))≥F(gt(x28, plus(x29, s(x30))), x28, x29, s(s(x30))))∧(∀x37,x38,x39,x40:plus(x36, x35)=s(x37)∧gt(x38, x37)=true∧(∀x39,x40:gt(x38, x37)=true∧plus(x39, x40)=x37 ⇒ F(true, x38, x39, s(x40))≥F(gt(x38, plus(x39, s(x40))), x38, x39, s(s(x40)))) ⇒ F(true, s(x38), x36, s(x35))≥F(gt(s(x38), plus(x36, s(x35))), s(x38), x36, s(s(x35)))) ⇒ F(true, s(x28), x36, s(s(x35)))≥F(gt(s(x28), plus(x36, s(s(x35)))), s(x28), x36, s(s(s(x35)))))
(12) (true=true ⇒ F(true, s(s(x42)), s(0), s(0))≥F(gt(s(s(x42)), plus(s(0), s(0))), s(s(x42)), s(0), s(s(0))))
(13) (gt(x44, x43)=true∧(gt(x44, x43)=true ⇒ F(true, s(x44), s(x43), s(0))≥F(gt(s(x44), plus(s(x43), s(0))), s(x44), s(x43), s(s(0)))) ⇒ F(true, s(s(x44)), s(s(x43)), s(0))≥F(gt(s(s(x44)), plus(s(s(x43)), s(0))), s(s(x44)), s(s(x43)), s(s(0))))
(14) (F(true, s(s(x42)), s(0), s(0))≥F(gt(s(s(x42)), plus(s(0), s(0))), s(s(x42)), s(0), s(s(0))))
(15) (F(true, s(x44), s(x43), s(0))≥F(gt(s(x44), plus(s(x43), s(0))), s(x44), s(x43), s(s(0))) ⇒ F(true, s(s(x44)), s(s(x43)), s(0))≥F(gt(s(s(x44)), plus(s(s(x43)), s(0))), s(s(x44)), s(s(x43)), s(s(0))))
(16) (F(true, x28, x36, s(x35))≥F(gt(x28, plus(x36, s(x35))), x28, x36, s(s(x35)))∧(∀x37,x38,x39,x40:plus(x36, x35)=s(x37)∧gt(x38, x37)=true∧(∀x39,x40:gt(x38, x37)=true∧plus(x39, x40)=x37 ⇒ F(true, x38, x39, s(x40))≥F(gt(x38, plus(x39, s(x40))), x38, x39, s(s(x40)))) ⇒ F(true, s(x38), x36, s(x35))≥F(gt(s(x38), plus(x36, s(x35))), s(x38), x36, s(s(x35)))) ⇒ F(true, s(x28), x36, s(s(x35)))≥F(gt(s(x28), plus(x36, s(s(x35)))), s(x28), x36, s(s(s(x35)))))
(17) (F(true, x28, x36, s(x35))≥F(gt(x28, plus(x36, s(x35))), x28, x36, s(s(x35))) ⇒ F(true, s(x28), x36, s(s(x35)))≥F(gt(s(x28), plus(x36, s(s(x35)))), s(x28), x36, s(s(s(x35)))))
(18) (F(gt(x6, plus(x7, x8)), x6, s(x7), x8)=F(true, x9, x10, x11) ⇒ F(true, x9, x10, x11)≥F(gt(x9, plus(x10, x11)), x9, x10, s(x11)))
(19) (plus(x7, x8)=x45∧gt(x6, x45)=true ⇒ F(true, x6, s(x7), x8)≥F(gt(x6, plus(s(x7), x8)), x6, s(x7), s(x8)))
(20) (true=true∧plus(x7, x8)=0 ⇒ F(true, s(x47), s(x7), x8)≥F(gt(s(x47), plus(s(x7), x8)), s(x47), s(x7), s(x8)))
(21) (gt(x49, x48)=true∧plus(x7, x8)=s(x48)∧(∀x50,x51:gt(x49, x48)=true∧plus(x50, x51)=x48 ⇒ F(true, x49, s(x50), x51)≥F(gt(x49, plus(s(x50), x51)), x49, s(x50), s(x51))) ⇒ F(true, s(x49), s(x7), x8)≥F(gt(s(x49), plus(s(x7), x8)), s(x49), s(x7), s(x8)))
(22) (plus(x7, x8)=0 ⇒ F(true, s(x47), s(x7), x8)≥F(gt(s(x47), plus(s(x7), x8)), s(x47), s(x7), s(x8)))
(23) (x55=s(x48)∧gt(x49, x48)=true∧(∀x50,x51:gt(x49, x48)=true∧plus(x50, x51)=x48 ⇒ F(true, x49, s(x50), x51)≥F(gt(x49, plus(s(x50), x51)), x49, s(x50), s(x51))) ⇒ F(true, s(x49), s(x55), 0)≥F(gt(s(x49), plus(s(x55), 0)), s(x49), s(x55), s(0)))
(24) (s(plus(x57, x56))=s(x48)∧gt(x49, x48)=true∧(∀x50,x51:gt(x49, x48)=true∧plus(x50, x51)=x48 ⇒ F(true, x49, s(x50), x51)≥F(gt(x49, plus(s(x50), x51)), x49, s(x50), s(x51)))∧(∀x58,x59,x60,x61:plus(x57, x56)=s(x58)∧gt(x59, x58)=true∧(∀x60,x61:gt(x59, x58)=true∧plus(x60, x61)=x58 ⇒ F(true, x59, s(x60), x61)≥F(gt(x59, plus(s(x60), x61)), x59, s(x60), s(x61))) ⇒ F(true, s(x59), s(x57), x56)≥F(gt(s(x59), plus(s(x57), x56)), s(x59), s(x57), s(x56))) ⇒ F(true, s(x49), s(x57), s(x56))≥F(gt(s(x49), plus(s(x57), s(x56))), s(x49), s(x57), s(s(x56))))
(25) (x52=0 ⇒ F(true, s(x47), s(x52), 0)≥F(gt(s(x47), plus(s(x52), 0)), s(x47), s(x52), s(0)))
(26) (F(true, s(x47), s(0), 0)≥F(gt(s(x47), plus(s(0), 0)), s(x47), s(0), s(0)))
(27) (gt(x49, x48)=true ⇒ F(true, s(x49), s(s(x48)), 0)≥F(gt(s(x49), plus(s(s(x48)), 0)), s(x49), s(s(x48)), s(0)))
(28) (plus(x57, x56)=x48∧gt(x49, x48)=true∧(∀x50,x51:gt(x49, x48)=true∧plus(x50, x51)=x48 ⇒ F(true, x49, s(x50), x51)≥F(gt(x49, plus(s(x50), x51)), x49, s(x50), s(x51)))∧(∀x58,x59,x60,x61:plus(x57, x56)=s(x58)∧gt(x59, x58)=true∧(∀x60,x61:gt(x59, x58)=true∧plus(x60, x61)=x58 ⇒ F(true, x59, s(x60), x61)≥F(gt(x59, plus(s(x60), x61)), x59, s(x60), s(x61))) ⇒ F(true, s(x59), s(x57), x56)≥F(gt(s(x59), plus(s(x57), x56)), s(x59), s(x57), s(x56))) ⇒ F(true, s(x49), s(x57), s(x56))≥F(gt(s(x49), plus(s(x57), s(x56))), s(x49), s(x57), s(s(x56))))
(29) (true=true ⇒ F(true, s(s(x63)), s(s(0)), 0)≥F(gt(s(s(x63)), plus(s(s(0)), 0)), s(s(x63)), s(s(0)), s(0)))
(30) (gt(x65, x64)=true∧(gt(x65, x64)=true ⇒ F(true, s(x65), s(s(x64)), 0)≥F(gt(s(x65), plus(s(s(x64)), 0)), s(x65), s(s(x64)), s(0))) ⇒ F(true, s(s(x65)), s(s(s(x64))), 0)≥F(gt(s(s(x65)), plus(s(s(s(x64))), 0)), s(s(x65)), s(s(s(x64))), s(0)))
(31) (F(true, s(s(x63)), s(s(0)), 0)≥F(gt(s(s(x63)), plus(s(s(0)), 0)), s(s(x63)), s(s(0)), s(0)))
(32) (F(true, s(x65), s(s(x64)), 0)≥F(gt(s(x65), plus(s(s(x64)), 0)), s(x65), s(s(x64)), s(0)) ⇒ F(true, s(s(x65)), s(s(s(x64))), 0)≥F(gt(s(s(x65)), plus(s(s(s(x64))), 0)), s(s(x65)), s(s(s(x64))), s(0)))
(33) (F(true, x49, s(x57), x56)≥F(gt(x49, plus(s(x57), x56)), x49, s(x57), s(x56))∧(∀x58,x59,x60,x61:plus(x57, x56)=s(x58)∧gt(x59, x58)=true∧(∀x60,x61:gt(x59, x58)=true∧plus(x60, x61)=x58 ⇒ F(true, x59, s(x60), x61)≥F(gt(x59, plus(s(x60), x61)), x59, s(x60), s(x61))) ⇒ F(true, s(x59), s(x57), x56)≥F(gt(s(x59), plus(s(x57), x56)), s(x59), s(x57), s(x56))) ⇒ F(true, s(x49), s(x57), s(x56))≥F(gt(s(x49), plus(s(x57), s(x56))), s(x49), s(x57), s(s(x56))))
(34) (F(true, x49, s(x57), x56)≥F(gt(x49, plus(s(x57), x56)), x49, s(x57), s(x56)) ⇒ F(true, s(x49), s(x57), s(x56))≥F(gt(s(x49), plus(s(x57), s(x56))), s(x49), s(x57), s(s(x56))))
(35) (F(gt(x12, plus(x13, x14)), x12, x13, s(x14))=F(true, x15, x16, x17) ⇒ F(true, x15, x16, x17)≥F(gt(x15, plus(x16, x17)), x15, s(x16), x17))
(36) (plus(x13, x14)=x66∧gt(x12, x66)=true ⇒ F(true, x12, x13, s(x14))≥F(gt(x12, plus(x13, s(x14))), x12, s(x13), s(x14)))
(37) (true=true∧plus(x13, x14)=0 ⇒ F(true, s(x68), x13, s(x14))≥F(gt(s(x68), plus(x13, s(x14))), s(x68), s(x13), s(x14)))
(38) (gt(x70, x69)=true∧plus(x13, x14)=s(x69)∧(∀x71,x72:gt(x70, x69)=true∧plus(x71, x72)=x69 ⇒ F(true, x70, x71, s(x72))≥F(gt(x70, plus(x71, s(x72))), x70, s(x71), s(x72))) ⇒ F(true, s(x70), x13, s(x14))≥F(gt(s(x70), plus(x13, s(x14))), s(x70), s(x13), s(x14)))
(39) (plus(x13, x14)=0 ⇒ F(true, s(x68), x13, s(x14))≥F(gt(s(x68), plus(x13, s(x14))), s(x68), s(x13), s(x14)))
(40) (x76=s(x69)∧gt(x70, x69)=true∧(∀x71,x72:gt(x70, x69)=true∧plus(x71, x72)=x69 ⇒ F(true, x70, x71, s(x72))≥F(gt(x70, plus(x71, s(x72))), x70, s(x71), s(x72))) ⇒ F(true, s(x70), x76, s(0))≥F(gt(s(x70), plus(x76, s(0))), s(x70), s(x76), s(0)))
(41) (s(plus(x78, x77))=s(x69)∧gt(x70, x69)=true∧(∀x71,x72:gt(x70, x69)=true∧plus(x71, x72)=x69 ⇒ F(true, x70, x71, s(x72))≥F(gt(x70, plus(x71, s(x72))), x70, s(x71), s(x72)))∧(∀x79,x80,x81,x82:plus(x78, x77)=s(x79)∧gt(x80, x79)=true∧(∀x81,x82:gt(x80, x79)=true∧plus(x81, x82)=x79 ⇒ F(true, x80, x81, s(x82))≥F(gt(x80, plus(x81, s(x82))), x80, s(x81), s(x82))) ⇒ F(true, s(x80), x78, s(x77))≥F(gt(s(x80), plus(x78, s(x77))), s(x80), s(x78), s(x77))) ⇒ F(true, s(x70), x78, s(s(x77)))≥F(gt(s(x70), plus(x78, s(s(x77)))), s(x70), s(x78), s(s(x77))))
(42) (x73=0 ⇒ F(true, s(x68), x73, s(0))≥F(gt(s(x68), plus(x73, s(0))), s(x68), s(x73), s(0)))
(43) (F(true, s(x68), 0, s(0))≥F(gt(s(x68), plus(0, s(0))), s(x68), s(0), s(0)))
(44) (gt(x70, x69)=true ⇒ F(true, s(x70), s(x69), s(0))≥F(gt(s(x70), plus(s(x69), s(0))), s(x70), s(s(x69)), s(0)))
(45) (plus(x78, x77)=x69∧gt(x70, x69)=true∧(∀x71,x72:gt(x70, x69)=true∧plus(x71, x72)=x69 ⇒ F(true, x70, x71, s(x72))≥F(gt(x70, plus(x71, s(x72))), x70, s(x71), s(x72)))∧(∀x79,x80,x81,x82:plus(x78, x77)=s(x79)∧gt(x80, x79)=true∧(∀x81,x82:gt(x80, x79)=true∧plus(x81, x82)=x79 ⇒ F(true, x80, x81, s(x82))≥F(gt(x80, plus(x81, s(x82))), x80, s(x81), s(x82))) ⇒ F(true, s(x80), x78, s(x77))≥F(gt(s(x80), plus(x78, s(x77))), s(x80), s(x78), s(x77))) ⇒ F(true, s(x70), x78, s(s(x77)))≥F(gt(s(x70), plus(x78, s(s(x77)))), s(x70), s(x78), s(s(x77))))
(46) (true=true ⇒ F(true, s(s(x84)), s(0), s(0))≥F(gt(s(s(x84)), plus(s(0), s(0))), s(s(x84)), s(s(0)), s(0)))
(47) (gt(x86, x85)=true∧(gt(x86, x85)=true ⇒ F(true, s(x86), s(x85), s(0))≥F(gt(s(x86), plus(s(x85), s(0))), s(x86), s(s(x85)), s(0))) ⇒ F(true, s(s(x86)), s(s(x85)), s(0))≥F(gt(s(s(x86)), plus(s(s(x85)), s(0))), s(s(x86)), s(s(s(x85))), s(0)))
(48) (F(true, s(s(x84)), s(0), s(0))≥F(gt(s(s(x84)), plus(s(0), s(0))), s(s(x84)), s(s(0)), s(0)))
(49) (F(true, s(x86), s(x85), s(0))≥F(gt(s(x86), plus(s(x85), s(0))), s(x86), s(s(x85)), s(0)) ⇒ F(true, s(s(x86)), s(s(x85)), s(0))≥F(gt(s(s(x86)), plus(s(s(x85)), s(0))), s(s(x86)), s(s(s(x85))), s(0)))
(50) (F(true, x70, x78, s(x77))≥F(gt(x70, plus(x78, s(x77))), x70, s(x78), s(x77))∧(∀x79,x80,x81,x82:plus(x78, x77)=s(x79)∧gt(x80, x79)=true∧(∀x81,x82:gt(x80, x79)=true∧plus(x81, x82)=x79 ⇒ F(true, x80, x81, s(x82))≥F(gt(x80, plus(x81, s(x82))), x80, s(x81), s(x82))) ⇒ F(true, s(x80), x78, s(x77))≥F(gt(s(x80), plus(x78, s(x77))), s(x80), s(x78), s(x77))) ⇒ F(true, s(x70), x78, s(s(x77)))≥F(gt(s(x70), plus(x78, s(s(x77)))), s(x70), s(x78), s(s(x77))))
(51) (F(true, x70, x78, s(x77))≥F(gt(x70, plus(x78, s(x77))), x70, s(x78), s(x77)) ⇒ F(true, s(x70), x78, s(s(x77)))≥F(gt(s(x70), plus(x78, s(s(x77)))), s(x70), s(x78), s(s(x77))))
(52) (F(gt(x18, plus(x19, x20)), x18, s(x19), x20)=F(true, x21, x22, x23) ⇒ F(true, x21, x22, x23)≥F(gt(x21, plus(x22, x23)), x21, s(x22), x23))
(53) (plus(x19, x20)=x87∧gt(x18, x87)=true ⇒ F(true, x18, s(x19), x20)≥F(gt(x18, plus(s(x19), x20)), x18, s(s(x19)), x20))
(54) (true=true∧plus(x19, x20)=0 ⇒ F(true, s(x89), s(x19), x20)≥F(gt(s(x89), plus(s(x19), x20)), s(x89), s(s(x19)), x20))
(55) (gt(x91, x90)=true∧plus(x19, x20)=s(x90)∧(∀x92,x93:gt(x91, x90)=true∧plus(x92, x93)=x90 ⇒ F(true, x91, s(x92), x93)≥F(gt(x91, plus(s(x92), x93)), x91, s(s(x92)), x93)) ⇒ F(true, s(x91), s(x19), x20)≥F(gt(s(x91), plus(s(x19), x20)), s(x91), s(s(x19)), x20))
(56) (plus(x19, x20)=0 ⇒ F(true, s(x89), s(x19), x20)≥F(gt(s(x89), plus(s(x19), x20)), s(x89), s(s(x19)), x20))
(57) (x97=s(x90)∧gt(x91, x90)=true∧(∀x92,x93:gt(x91, x90)=true∧plus(x92, x93)=x90 ⇒ F(true, x91, s(x92), x93)≥F(gt(x91, plus(s(x92), x93)), x91, s(s(x92)), x93)) ⇒ F(true, s(x91), s(x97), 0)≥F(gt(s(x91), plus(s(x97), 0)), s(x91), s(s(x97)), 0))
(58) (s(plus(x99, x98))=s(x90)∧gt(x91, x90)=true∧(∀x92,x93:gt(x91, x90)=true∧plus(x92, x93)=x90 ⇒ F(true, x91, s(x92), x93)≥F(gt(x91, plus(s(x92), x93)), x91, s(s(x92)), x93))∧(∀x100,x101,x102,x103:plus(x99, x98)=s(x100)∧gt(x101, x100)=true∧(∀x102,x103:gt(x101, x100)=true∧plus(x102, x103)=x100 ⇒ F(true, x101, s(x102), x103)≥F(gt(x101, plus(s(x102), x103)), x101, s(s(x102)), x103)) ⇒ F(true, s(x101), s(x99), x98)≥F(gt(s(x101), plus(s(x99), x98)), s(x101), s(s(x99)), x98)) ⇒ F(true, s(x91), s(x99), s(x98))≥F(gt(s(x91), plus(s(x99), s(x98))), s(x91), s(s(x99)), s(x98)))
(59) (x94=0 ⇒ F(true, s(x89), s(x94), 0)≥F(gt(s(x89), plus(s(x94), 0)), s(x89), s(s(x94)), 0))
(60) (F(true, s(x89), s(0), 0)≥F(gt(s(x89), plus(s(0), 0)), s(x89), s(s(0)), 0))
(61) (gt(x91, x90)=true ⇒ F(true, s(x91), s(s(x90)), 0)≥F(gt(s(x91), plus(s(s(x90)), 0)), s(x91), s(s(s(x90))), 0))
(62) (plus(x99, x98)=x90∧gt(x91, x90)=true∧(∀x92,x93:gt(x91, x90)=true∧plus(x92, x93)=x90 ⇒ F(true, x91, s(x92), x93)≥F(gt(x91, plus(s(x92), x93)), x91, s(s(x92)), x93))∧(∀x100,x101,x102,x103:plus(x99, x98)=s(x100)∧gt(x101, x100)=true∧(∀x102,x103:gt(x101, x100)=true∧plus(x102, x103)=x100 ⇒ F(true, x101, s(x102), x103)≥F(gt(x101, plus(s(x102), x103)), x101, s(s(x102)), x103)) ⇒ F(true, s(x101), s(x99), x98)≥F(gt(s(x101), plus(s(x99), x98)), s(x101), s(s(x99)), x98)) ⇒ F(true, s(x91), s(x99), s(x98))≥F(gt(s(x91), plus(s(x99), s(x98))), s(x91), s(s(x99)), s(x98)))
(63) (true=true ⇒ F(true, s(s(x105)), s(s(0)), 0)≥F(gt(s(s(x105)), plus(s(s(0)), 0)), s(s(x105)), s(s(s(0))), 0))
(64) (gt(x107, x106)=true∧(gt(x107, x106)=true ⇒ F(true, s(x107), s(s(x106)), 0)≥F(gt(s(x107), plus(s(s(x106)), 0)), s(x107), s(s(s(x106))), 0)) ⇒ F(true, s(s(x107)), s(s(s(x106))), 0)≥F(gt(s(s(x107)), plus(s(s(s(x106))), 0)), s(s(x107)), s(s(s(s(x106)))), 0))
(65) (F(true, s(s(x105)), s(s(0)), 0)≥F(gt(s(s(x105)), plus(s(s(0)), 0)), s(s(x105)), s(s(s(0))), 0))
(66) (F(true, s(x107), s(s(x106)), 0)≥F(gt(s(x107), plus(s(s(x106)), 0)), s(x107), s(s(s(x106))), 0) ⇒ F(true, s(s(x107)), s(s(s(x106))), 0)≥F(gt(s(s(x107)), plus(s(s(s(x106))), 0)), s(s(x107)), s(s(s(s(x106)))), 0))
(67) (F(true, x91, s(x99), x98)≥F(gt(x91, plus(s(x99), x98)), x91, s(s(x99)), x98)∧(∀x100,x101,x102,x103:plus(x99, x98)=s(x100)∧gt(x101, x100)=true∧(∀x102,x103:gt(x101, x100)=true∧plus(x102, x103)=x100 ⇒ F(true, x101, s(x102), x103)≥F(gt(x101, plus(s(x102), x103)), x101, s(s(x102)), x103)) ⇒ F(true, s(x101), s(x99), x98)≥F(gt(s(x101), plus(s(x99), x98)), s(x101), s(s(x99)), x98)) ⇒ F(true, s(x91), s(x99), s(x98))≥F(gt(s(x91), plus(s(x99), s(x98))), s(x91), s(s(x99)), s(x98)))
(68) (F(true, x91, s(x99), x98)≥F(gt(x91, plus(s(x99), x98)), x91, s(s(x99)), x98) ⇒ F(true, s(x91), s(x99), s(x98))≥F(gt(s(x91), plus(s(x99), s(x98))), s(x91), s(s(x99)), s(x98)))
POL(0) = 0
POL(F(x1, x2, x3, x4)) = -1 - x1 + x2 - x3 - x4
POL(c) = -1
POL(false) = 0
POL(gt(x1, x2)) = 0
POL(plus(x1, x2)) = 0
POL(s(x1)) = 1 + x1
POL(true) = 0
The following pairs are in Pbound:
F(true, x, y, z) → F(gt(x, plus(y, z)), x, y, s(z))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, s(y), z)
The following rules are usable:
F(true, x, y, z) → F(gt(x, plus(y, z)), x, y, s(z))
F(true, x, y, z) → F(gt(x, plus(y, z)), x, s(y), z)
false → gt(0, v)
gt(u, v) → gt(s(u), s(v))
true → gt(s(u), 0)
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
plus(x0, 0)
plus(x0, s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))