YES Problem: p(p(b(a(x0)),x1),p(x2,x3)) -> p(p(b(x2),a(a(b(x1)))),p(x3,x0)) Proof: DP Processor: DPs: p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(b(x2),a(a(b(x1)))) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) TRS: p(p(b(a(x0)),x1),p(x2,x3)) -> p(p(b(x2),a(a(b(x1)))),p(x3,x0)) EDG Processor: DPs: p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(b(x2),a(a(b(x1)))) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) TRS: p(p(b(a(x0)),x1),p(x2,x3)) -> p(p(b(x2),a(a(b(x1)))),p(x3,x0)) graph: p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) -> p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) -> p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(b(x2),a(a(b(x1)))) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) -> p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) -> p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) -> p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(b(x2),a(a(b(x1)))) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) -> p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) SCC Processor: #sccs: 1 #rules: 2 #arcs: 6/9 DPs: p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(x3,x0) TRS: p(p(b(a(x0)),x1),p(x2,x3)) -> p(p(b(x2),a(a(b(x1)))),p(x3,x0)) Matrix Interpretation Processor: dimension: 1 interpretation: [p#](x0, x1) = x0 + x1, [p](x0, x1) = x0 + x1 + 1, [b](x0) = x0, [a](x0) = x0 orientation: p#(p(b(a(x0)),x1),p(x2,x3)) = x0 + x1 + x2 + x3 + 2 >= x0 + x1 + x2 + x3 + 2 = p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) p#(p(b(a(x0)),x1),p(x2,x3)) = x0 + x1 + x2 + x3 + 2 >= x0 + x3 = p#(x3,x0) p(p(b(a(x0)),x1),p(x2,x3)) = x0 + x1 + x2 + x3 + 3 >= x0 + x1 + x2 + x3 + 3 = p(p(b(x2),a(a(b(x1)))),p(x3,x0)) problem: DPs: p#(p(b(a(x0)),x1),p(x2,x3)) -> p#(p(b(x2),a(a(b(x1)))),p(x3,x0)) TRS: p(p(b(a(x0)),x1),p(x2,x3)) -> p(p(b(x2),a(a(b(x1)))),p(x3,x0)) Bounds Processor: bound: 1 enrichment: match-dp automaton: final states: {1} transitions: f40() -> 2* p{#,0}(8,3) -> 1* p{#,0}(26,3) -> 1* p0(7,14) -> 8* p0(8,3) -> 3* p0(13,13) -> 21* p0(8,21) -> 3* p0(4,2) -> 3* p0(7,44) -> 26* p0(4,14) -> 8* p0(25,13) -> 3* p0(21,2) -> 3* p0(2,5) -> 3* p0(12,13) -> 3* p0(7,13) -> 3* p0(13,2) -> 21* p0(3,2) -> 3* p0(4,13) -> 3* p0(25,2) -> 3* p0(5,2) -> 3* p0(26,3) -> 3* p0(26,21) -> 3* p0(12,2) -> 3* p0(7,2) -> 3* p0(2,2) -> 3* p0(7,6) -> 8* p0(12,12) -> 3* b0(2) -> 7,4 b0(24) -> 42* b0(14) -> 4* b0(21) -> 7* b0(6) -> 12* b0(13) -> 7* b0(8) -> 7* a0(25) -> 13* a0(5) -> 6* a0(42) -> 43* a0(12) -> 13* a0(7) -> 13* a0(4) -> 5* a0(43) -> 44* a0(13) -> 14* p{#,1}(26,21) -> 1* p1(7,12) -> 21* p1(2,12) -> 21* p1(13,7) -> 21* p1(3,7) -> 21* p1(13,13) -> 21* p1(3,13) -> 21* p1(13,25) -> 21* p1(4,4) -> 21* p1(3,25) -> 21* p1(4,12) -> 21* p1(25,7) -> 21* p1(5,7) -> 21* p1(5,13) -> 21* p1(21,4) -> 21* p1(25,25) -> 21* p1(5,25) -> 21* p1(21,12) -> 21* p1(12,7) -> 21* p1(7,7) -> 21* p1(2,7) -> 21* p1(12,13) -> 21* p1(2,13) -> 21* p1(13,4) -> 21* p1(12,25) -> 21* p1(3,4) -> 21* p1(7,25) -> 21* p1(2,25) -> 21* p1(13,12) -> 21* p1(3,12) -> 21* p1(4,7) -> 21* p1(25,4) -> 21* p1(5,4) -> 21* p1(4,25) -> 21* p1(25,12) -> 21* p1(5,12) -> 21* p1(25,24) -> 26* p1(21,7) -> 21* p1(21,13) -> 21* p1(21,25) -> 21* p1(12,4) -> 21* p1(7,4) -> 21* p1(2,4) -> 21* p1(12,12) -> 21* b1(25) -> 25* b1(5) -> 25* b1(12) -> 25* b1(7) -> 25* b1(2) -> 25* b1(44) -> 22* b1(24) -> 22* b1(14) -> 22* b1(4) -> 25* b1(26) -> 25* b1(21) -> 25* b1(6) -> 22* b1(13) -> 25* b1(8) -> 25* b1(3) -> 25* a1(22) -> 23* a1(23) -> 24* problem: DPs: TRS: p(p(b(a(x0)),x1),p(x2,x3)) -> p(p(b(x2),a(a(b(x1)))),p(x3,x0)) Qed