YES Problem: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 Proof: Matrix Interpretation Processor: dim=1 interpretation: [sixtimes](x0) = x0, [thrice](x0) = x0, [p](x0) = x0, [half](x0) = x0, [0](x0) = 4x0 + 2, [s](x0) = x0 orientation: thrice(0(x1)) = 4x1 + 2 >= 4x1 + 2 = p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) = x1 >= x1 = p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) = 4x1 + 2 >= 4x1 + 2 = p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) = x1 >= x1 = p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) = x1 >= x1 = p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) = 4x1 + 2 >= 4x1 + 2 = p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) = x1 >= x1 = p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) = x1 >= x1 = p(x1) p(s(x1)) = x1 >= x1 = x1 p(0(x1)) = 4x1 + 2 >= 4x1 + 2 = 0(s(s(s(s(x1))))) 0(x1) = 4x1 + 2 >= x1 = x1 problem: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) Matrix Interpretation Processor: dim=1 interpretation: [sixtimes](x0) = 2x0, [thrice](x0) = 4x0, [p](x0) = x0, [half](x0) = 2x0, [0](x0) = 6x0 + 4, [s](x0) = x0 orientation: thrice(0(x1)) = 24x1 + 16 >= 6x1 + 4 = p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) = 4x1 >= 4x1 = p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) = 12x1 + 8 >= 6x1 + 4 = p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) = 2x1 >= 2x1 = p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) = 2x1 >= 2x1 = p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) = 12x1 + 8 >= 6x1 + 4 = p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) = 2x1 >= 2x1 = p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) = x1 >= x1 = p(x1) p(s(x1)) = x1 >= x1 = x1 p(0(x1)) = 6x1 + 4 >= 6x1 + 4 = 0(s(s(s(s(x1))))) problem: thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) Matrix Interpretation Processor: dim=1 interpretation: [sixtimes](x0) = x0, [thrice](x0) = 4x0 + 3, [p](x0) = x0, [half](x0) = 4x0 + 2, [0](x0) = x0 + 2, [s](x0) = x0 orientation: thrice(s(x1)) = 4x1 + 3 >= 4x1 + 2 = p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(s(x1)) = 4x1 + 2 >= 4x1 + 2 = p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) = 4x1 + 2 >= 4x1 + 2 = p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(s(x1)) = x1 >= x1 = p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) = x1 >= x1 = p(x1) p(s(x1)) = x1 >= x1 = x1 p(0(x1)) = x1 + 2 >= x1 + 2 = 0(s(s(s(s(x1))))) problem: half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) Bounds Processor: bound: 1 enrichment: match automaton: final states: {46,2,45,21,19,1} transitions: p0(16) -> 17* p0(44) -> 21* p0(34) -> 35* p0(12) -> 13* p0(20) -> 19* p0(35) -> 36* p0(27) -> 28* p0(15) -> 16* p0(7) -> 8* p0(32) -> 33* p0(23) -> 24* p0(6) -> 7* p0(11) -> 12* p0(3) -> 4* p0(43) -> 44* p0(25) -> 26* p0(24) -> 25* p0(18) -> 1* p0(2) -> 45* f60() -> 2* p1(56) -> 57* p1(74) -> 75* p1(62) -> 63* p1(54) -> 55* p1(48) -> 49* p1(70) -> 71* p1(64) -> 65* 00(47) -> 46* s0(22) -> 23* s0(16) -> 20* s0(39) -> 40* s0(4) -> 5* s0(2) -> 3* s0(10) -> 11* s0(17) -> 18* s0(5) -> 6* s0(31) -> 32* s0(13) -> 14* s0(9) -> 10* s0(40) -> 41* s0(3) -> 22* s0(29) -> 30* s0(36) -> 37* s0(38) -> 39* s0(37) -> 38* s0(30) -> 31* s0(14) -> 15* s0(41) -> 42* s0(42) -> 43* s0(26) -> 27* s0(23) -> 47* s0(33) -> 34* sixtimes0(28) -> 29* half0(8) -> 9* 19 -> 71* 17 -> 1* 46 -> 45* 42 -> 44,48 30 -> 55,36 63 -> 8* 14 -> 16,70 16 -> 19* 4 -> 63* 2 -> 45,75,4 75 -> 26* 22 -> 24,56 21 -> 29* 26 -> 28* 13 -> 71,17,1 71 -> 17* 1 -> 9,65,71 49 -> 21* 31 -> 33* 55 -> 36* 65 -> 13* 5 -> 7,62 9 -> 65* 57 -> 25* 3 -> 57,25,74 41 -> 49* 10 -> 12,64 33 -> 35,54 problem: Qed