YES We show the termination of the TRS R: |0|(|#|()) -> |#|() +(x,|#|()) -> x +(|#|(),x) -> x +(|0|(x),|0|(y)) -> |0|(+(x,y)) +(|0|(x),|1|(y)) -> |1|(+(x,y)) +(|1|(x),|0|(y)) -> |1|(+(x,y)) +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) +(x,+(y,z)) -> +(+(x,y),z) -(x,|#|()) -> x -(|#|(),x) -> |#|() -(|0|(x),|0|(y)) -> |0|(-(x,y)) -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) -(|1|(x),|0|(y)) -> |1|(-(x,y)) -(|1|(x),|1|(y)) -> |0|(-(x,y)) not(false()) -> true() not(true()) -> false() and(x,true()) -> x and(x,false()) -> false() if(true(),x,y) -> x if(false(),x,y) -> y ge(|0|(x),|0|(y)) -> ge(x,y) ge(|0|(x),|1|(y)) -> not(ge(y,x)) ge(|1|(x),|0|(y)) -> ge(x,y) ge(|1|(x),|1|(y)) -> ge(x,y) ge(x,|#|()) -> true() ge(|#|(),|1|(x)) -> false() ge(|#|(),|0|(x)) -> ge(|#|(),x) val(l(x)) -> x val(n(x,y,z)) -> x min(l(x)) -> x min(n(x,y,z)) -> min(y) max(l(x)) -> x max(n(x,y,z)) -> max(z) bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) size(l(x)) -> |1|(|#|()) size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: +#(|0|(x),|0|(y)) -> |0|#(+(x,y)) p2: +#(|0|(x),|0|(y)) -> +#(x,y) p3: +#(|0|(x),|1|(y)) -> +#(x,y) p4: +#(|1|(x),|0|(y)) -> +#(x,y) p5: +#(|1|(x),|1|(y)) -> |0|#(+(+(x,y),|1|(|#|()))) p6: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) p7: +#(|1|(x),|1|(y)) -> +#(x,y) p8: +#(x,+(y,z)) -> +#(+(x,y),z) p9: +#(x,+(y,z)) -> +#(x,y) p10: -#(|0|(x),|0|(y)) -> |0|#(-(x,y)) p11: -#(|0|(x),|0|(y)) -> -#(x,y) p12: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) p13: -#(|0|(x),|1|(y)) -> -#(x,y) p14: -#(|1|(x),|0|(y)) -> -#(x,y) p15: -#(|1|(x),|1|(y)) -> |0|#(-(x,y)) p16: -#(|1|(x),|1|(y)) -> -#(x,y) p17: ge#(|0|(x),|0|(y)) -> ge#(x,y) p18: ge#(|0|(x),|1|(y)) -> not#(ge(y,x)) p19: ge#(|0|(x),|1|(y)) -> ge#(y,x) p20: ge#(|1|(x),|0|(y)) -> ge#(x,y) p21: ge#(|1|(x),|1|(y)) -> ge#(x,y) p22: ge#(|#|(),|0|(x)) -> ge#(|#|(),x) p23: min#(n(x,y,z)) -> min#(y) p24: max#(n(x,y,z)) -> max#(z) p25: bs#(n(x,y,z)) -> and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) p26: bs#(n(x,y,z)) -> and#(ge(x,max(y)),ge(min(z),x)) p27: bs#(n(x,y,z)) -> ge#(x,max(y)) p28: bs#(n(x,y,z)) -> max#(y) p29: bs#(n(x,y,z)) -> ge#(min(z),x) p30: bs#(n(x,y,z)) -> min#(z) p31: bs#(n(x,y,z)) -> and#(bs(y),bs(z)) p32: bs#(n(x,y,z)) -> bs#(y) p33: bs#(n(x,y,z)) -> bs#(z) p34: size#(n(x,y,z)) -> +#(+(size(x),size(y)),|1|(|#|())) p35: size#(n(x,y,z)) -> +#(size(x),size(y)) p36: size#(n(x,y,z)) -> size#(x) p37: size#(n(x,y,z)) -> size#(y) p38: wb#(n(x,y,z)) -> and#(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) p39: wb#(n(x,y,z)) -> if#(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))) p40: wb#(n(x,y,z)) -> ge#(size(y),size(z)) p41: wb#(n(x,y,z)) -> size#(y) p42: wb#(n(x,y,z)) -> size#(z) p43: wb#(n(x,y,z)) -> ge#(|1|(|#|()),-(size(y),size(z))) p44: wb#(n(x,y,z)) -> -#(size(y),size(z)) p45: wb#(n(x,y,z)) -> ge#(|1|(|#|()),-(size(z),size(y))) p46: wb#(n(x,y,z)) -> -#(size(z),size(y)) p47: wb#(n(x,y,z)) -> and#(wb(y),wb(z)) p48: wb#(n(x,y,z)) -> wb#(y) p49: wb#(n(x,y,z)) -> wb#(z) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p48, p49} {p36, p37} {p2, p3, p4, p6, p7, p8, p9} {p11, p12, p13, p14, p16} {p32, p33} {p17, p19, p20, p21} {p22} {p23} {p24} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: wb#(n(x,y,z)) -> wb#(z) p2: wb#(n(x,y,z)) -> wb#(y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r4, r5, r6, r7, r9, r10, r11, r12, r13, r14, r15, r16, r18, r25, r26, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: wb#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + x3 + 29 |0|_A(x1) = 5 |#|_A() = 3 +_A(x1,x2) = 6 |1|_A(x1) = 4 -_A(x1,x2) = x1 + x2 + 6 not_A(x1) = x1 + 10 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 5 min_A(x1) = x1 + 4 size_A(x1) = x1 + 6 base order 2: precedence: min > + > wb# = n = not = val = l > - = ge = size > |1| = and > |0| > |#| > true > false partial status: pi(wb#) = [1] pi(n) = [3] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: wb#(n(x,y,z)) -> wb#(y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: wb#(n(x,y,z)) -> wb#(y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r9, r10, r12, r13, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: wb#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + x3 + 35 |0|_A(x1) = x1 + 11 |#|_A() = 3 +_A(x1,x2) = x2 + 6 |1|_A(x1) = 5 -_A(x1,x2) = x1 + x2 + 12 not_A(x1) = x1 + 17 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 6 min_A(x1) = x1 + 4 size_A(x1) = x1 + 5 base order 2: precedence: not > ge = l > n = |0| = |1| = - = if = val = size > wb# = + = min > and > |#| > true > false partial status: pi(wb#) = [] pi(n) = [2] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: size#(n(x,y,z)) -> size#(y) p2: size#(n(x,y,z)) -> size#(x) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r9, r10, r12, r13, r15, r16, r18, r25, r26, r27, r28, r29, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: size#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + 7 |0|_A(x1) = x1 + 11 |#|_A() = 3 +_A(x1,x2) = 6 |1|_A(x1) = 5 -_A(x1,x2) = x1 + x2 + 12 not_A(x1) = x1 + 17 false_A() = 1 true_A() = 2 and_A(x1,x2) = 3 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 6 size_A(x1) = x1 + 5 base order 2: precedence: not > |0| > ge > size# = l > n = |1| = - = val = size > + > |#| = and > true > false partial status: pi(size#) = [1] pi(n) = [1, 2] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [2] pi(val) = [1] pi(l) = [1] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: size#(n(x,y,z)) -> size#(x) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: size#(n(x,y,z)) -> size#(x) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r9, r10, r12, r13, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: size#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + x3 + 35 |0|_A(x1) = x1 + 11 |#|_A() = 3 +_A(x1,x2) = x2 + 6 |1|_A(x1) = 5 -_A(x1,x2) = x1 + x2 + 12 not_A(x1) = x1 + 17 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 6 min_A(x1) = x1 + 4 size_A(x1) = x1 + 5 base order 2: precedence: not > ge = l > n = |0| = |1| = - = val = size > size# = + = if = min > and > |#| > true > false partial status: pi(size#) = [] pi(n) = [2] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: +#(|0|(x),|0|(y)) -> +#(x,y) p2: +#(x,+(y,z)) -> +#(x,y) p3: +#(x,+(y,z)) -> +#(+(x,y),z) p4: +#(|1|(x),|1|(y)) -> +#(x,y) p5: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) p6: +#(|0|(x),|1|(y)) -> +#(x,y) p7: +#(|1|(x),|0|(y)) -> +#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r10, r15, r16, r18, r21, r22, r23, r24, r25, r26, r27, r28, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: +#_A(x1,x2) = x2 + 4 |0|_A(x1) = x1 + 12 +_A(x1,x2) = x1 + x2 + 8 |1|_A(x1) = x1 + 5 |#|_A() = 0 -_A(x1,x2) = 6 not_A(x1) = x1 false_A() = 1 true_A() = 1 and_A(x1,x2) = 34 ge_A(x1,x2) = 3 val_A(x1) = x1 l_A(x1) = x1 + 6 size_A(x1) = 6 base order 2: precedence: ge > and = val > not > true > l = size > |#| > + > |0| > +# = - > |1| > false partial status: pi(+#) = [2] pi(|0|) = [] pi(+) = [1, 2] pi(|1|) = [] pi(|#|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(size) = [] The following rules are removed: p7 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: +#(|0|(x),|0|(y)) -> +#(x,y) p2: +#(x,+(y,z)) -> +#(x,y) p3: +#(x,+(y,z)) -> +#(+(x,y),z) p4: +#(|1|(x),|1|(y)) -> +#(x,y) p5: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) p6: +#(|0|(x),|1|(y)) -> +#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: +#(|0|(x),|0|(y)) -> +#(x,y) p2: +#(|0|(x),|1|(y)) -> +#(x,y) p3: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) p4: +#(|1|(x),|1|(y)) -> +#(x,y) p5: +#(x,+(y,z)) -> +#(+(x,y),z) p6: +#(x,+(y,z)) -> +#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: +#_A(x1,x2) = x1 + x2 + 18 |0|_A(x1) = x1 + 6 |1|_A(x1) = x1 + 17 +_A(x1,x2) = x1 + x2 + 7 |#|_A() = 3 -_A(x1,x2) = x1 + x2 + 36 not_A(x1) = x1 + 16 false_A() = 1 true_A() = 2 and_A(x1,x2) = 38 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 2 val_A(x1) = x1 l_A(x1) = x1 + 21 n_A(x1,x2,x3) = x1 + x2 + 37 min_A(x1) = x1 + 36 base order 2: precedence: + = - = not = and = if > |0| = ge = n > val = l > +# = |1| = min > |#| > true > false partial status: pi(+#) = [] pi(|0|) = [1] pi(|1|) = [] pi(+) = [] pi(|#|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [2] pi(min) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: +#(|0|(x),|1|(y)) -> +#(x,y) p2: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) p3: +#(|1|(x),|1|(y)) -> +#(x,y) p4: +#(x,+(y,z)) -> +#(+(x,y),z) p5: +#(x,+(y,z)) -> +#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: +#(|0|(x),|1|(y)) -> +#(x,y) p2: +#(x,+(y,z)) -> +#(x,y) p3: +#(x,+(y,z)) -> +#(+(x,y),z) p4: +#(|1|(x),|1|(y)) -> +#(x,y) p5: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: +#_A(x1,x2) = x1 + x2 + 16 |0|_A(x1) = x1 + 10 |1|_A(x1) = x1 + 15 +_A(x1,x2) = x1 + x2 + 1 |#|_A() = 3 -_A(x1,x2) = x1 + x2 + 6 not_A(x1) = x1 + 14 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 2 val_A(x1) = x1 l_A(x1) = x1 + 19 n_A(x1,x2,x3) = x1 + x2 + 31 min_A(x1) = x1 + 30 base order 2: precedence: l > true > +# = not > |0| = |1| = - = if = min > val = n > + > ge > |#| > and > false partial status: pi(+#) = [] pi(|0|) = [] pi(|1|) = [] pi(+) = [] pi(|#|) = [] pi(-) = [1] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [2] pi(min) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: +#(x,+(y,z)) -> +#(x,y) p2: +#(x,+(y,z)) -> +#(+(x,y),z) p3: +#(|1|(x),|1|(y)) -> +#(x,y) p4: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: +#(x,+(y,z)) -> +#(x,y) p2: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) p3: +#(|1|(x),|1|(y)) -> +#(x,y) p4: +#(x,+(y,z)) -> +#(+(x,y),z) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r9, r10, r15, r16, r18, r25, r26, r27, r28, r29, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: +#_A(x1,x2) = x2 +_A(x1,x2) = x1 + x2 + 7 |1|_A(x1) = x1 + 14 |#|_A() = 0 |0|_A(x1) = x1 + 6 -_A(x1,x2) = x1 + x2 + 30 not_A(x1) = x1 + 2 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 ge_A(x1,x2) = x2 + 3 val_A(x1) = x1 l_A(x1) = x1 + 15 n_A(x1,x2,x3) = x1 + x2 + 37 size_A(x1) = x1 + 4 base order 2: precedence: not = l > |1| = size > |#| = - > |0| > ge > val = n > + > +# > true = and > false partial status: pi(+#) = [] pi(+) = [] pi(|1|) = [] pi(|#|) = [] pi(|0|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [2] pi(size) = [] The following rules are removed: p3 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: +#(x,+(y,z)) -> +#(x,y) p2: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) p3: +#(x,+(y,z)) -> +#(+(x,y),z) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p3} {p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: +#(x,+(y,z)) -> +#(x,y) p2: +#(x,+(y,z)) -> +#(+(x,y),z) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r18, r19, r25, r26, r28, r29 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: +#_A(x1,x2) = x1 + x2 + 7 +_A(x1,x2) = x1 + x2 + 6 |0|_A(x1) = 4 |#|_A() = 3 |1|_A(x1) = 5 -_A(x1,x2) = x1 + x2 + 19 not_A(x1) = x1 + 10 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 19 val_A(x1) = x1 l_A(x1) = x1 + 6 n_A(x1,x2,x3) = x1 + x2 + 32 base order 2: precedence: - = not = if > +# = ge = val = l = n > + > |1| = and > |0| > |#| > true > false partial status: pi(+#) = [] pi(+) = [] pi(|0|) = [] pi(|#|) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [2] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: +#(x,+(y,z)) -> +#(+(x,y),z) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: +#(x,+(y,z)) -> +#(+(x,y),z) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r18, r19, r25, r26, r28, r29, r30, r31 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: +#_A(x1,x2) = x1 + x2 + 1 +_A(x1,x2) = x1 + x2 + 7 |0|_A(x1) = 4 |#|_A() = 3 |1|_A(x1) = 6 -_A(x1,x2) = x1 + x2 + 5 not_A(x1) = x1 + 11 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 22 val_A(x1) = x1 l_A(x1) = x1 + 7 n_A(x1,x2,x3) = x1 + x2 + 37 min_A(x1) = x1 + 14 base order 2: precedence: - = not = if = min > +# = ge = val = l = n > + > |0| > |1| > |#| = and > true > false partial status: pi(+#) = [2] pi(+) = [] pi(|0|) = [] pi(|#|) = [] pi(|1|) = [] pi(-) = [1] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [2] pi(min) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|())) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: +#_A(x1,x2) = x1 + x2 |1|_A(x1) = x1 + 22 +_A(x1,x2) = x1 + x2 + 10 |#|_A() = 2 |0|_A(x1) = x1 + 9 -_A(x1,x2) = x1 + x2 + 23 not_A(x1) = x1 + 3 false_A() = 1 true_A() = 3 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x2 + 5 val_A(x1) = x1 + 1 l_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + 95 min_A(x1) = x1 base order 2: precedence: min > +# = + = |#| = - = n > |1| = |0| = not = val > true = if = ge > l > and > false partial status: pi(+#) = [1] pi(|1|) = [] pi(+) = [] pi(|#|) = [] pi(|0|) = [1] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [2] pi(val) = [1] pi(l) = [1] pi(n) = [2] pi(min) = [1] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: -#(|0|(x),|0|(y)) -> -#(x,y) p2: -#(|1|(x),|1|(y)) -> -#(x,y) p3: -#(|1|(x),|0|(y)) -> -#(x,y) p4: -#(|0|(x),|1|(y)) -> -#(x,y) p5: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r9, r10, r11, r12, r13, r14, r15, r16, r18, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: -#_A(x1,x2) = x1 + x2 + 18 |0|_A(x1) = x1 + 17 |1|_A(x1) = x1 + 11 -_A(x1,x2) = x1 + x2 + 2 |#|_A() = 3 +_A(x1,x2) = x1 + x2 + 7 not_A(x1) = x1 + 12 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 ge_A(x1,x2) = x2 + 4 val_A(x1) = x1 l_A(x1) = x1 + 15 n_A(x1,x2,x3) = x1 + x2 + x3 + 61 min_A(x1) = x1 + 4 size_A(x1) = x1 + 16 base order 2: precedence: ge > n > - = min > -# = size > |0| = + > not = val = l > |1| = and > |#| > true > false partial status: pi(-#) = [] pi(|0|) = [] pi(|1|) = [] pi(-) = [] pi(|#|) = [] pi(+) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [3] pi(min) = [] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: -#(|1|(x),|1|(y)) -> -#(x,y) p2: -#(|1|(x),|0|(y)) -> -#(x,y) p3: -#(|0|(x),|1|(y)) -> -#(x,y) p4: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: -#(|1|(x),|1|(y)) -> -#(x,y) p2: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) p3: -#(|0|(x),|1|(y)) -> -#(x,y) p4: -#(|1|(x),|0|(y)) -> -#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r3, r9, r10, r11, r12, r13, r14, r15, r16, r18, r25, r26, r27, r28, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: -#_A(x1,x2) = x1 + x2 + 14 |1|_A(x1) = x1 + 7 |0|_A(x1) = x1 + 13 -_A(x1,x2) = x1 + x2 + 2 |#|_A() = 3 +_A(x1,x2) = x2 + 5 not_A(x1) = x1 + 5 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 ge_A(x1,x2) = x1 + x2 + 9 val_A(x1) = x1 l_A(x1) = x1 + 11 min_A(x1) = x1 + 10 n_A(x1,x2,x3) = x2 + 18 size_A(x1) = x1 + 12 base order 2: precedence: -# > - = ge = val = l = min > not = n = size > + > |1| = |0| = and > |#| > true > false partial status: pi(-#) = [] pi(|1|) = [] pi(|0|) = [1] pi(-) = [] pi(|#|) = [] pi(+) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [2] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(n) = [2] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) p2: -#(|0|(x),|1|(y)) -> -#(x,y) p3: -#(|1|(x),|0|(y)) -> -#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p2, p3} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) p2: -#(|0|(x),|1|(y)) -> -#(x,y) p3: -#(|1|(x),|0|(y)) -> -#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r9, r10, r11, r12, r13, r14, r15, r16, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: -#_A(x1,x2) = x1 + 7 |0|_A(x1) = x1 + 8 |1|_A(x1) = x1 + 8 -_A(x1,x2) = x1 |#|_A() = 0 not_A(x1) = 3 false_A() = 1 true_A() = 2 and_A(x1,x2) = x2 + 7 if_A(x1,x2,x3) = x2 + x3 + 1 ge_A(x1,x2) = 4 val_A(x1) = x1 l_A(x1) = x1 + 9 n_A(x1,x2,x3) = x1 + x2 + x3 + 10 min_A(x1) = x1 + 9 max_A(x1) = x1 + 9 size_A(x1) = 9 base order 2: precedence: -# = |0| = - = l > |1| = and = ge = val = n = min = max > not > true = if = size > false > |#| partial status: pi(-#) = [1] pi(|0|) = [1] pi(|1|) = [] pi(-) = [1] pi(|#|) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [3] pi(min) = [] pi(max) = [] pi(size) = [] The following rules are removed: p2 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) p2: -#(|1|(x),|0|(y)) -> -#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p2} {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: -#(|1|(x),|0|(y)) -> -#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r3, r9, r10, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: -#_A(x1,x2) = x1 + x2 + 14 |1|_A(x1) = x1 + 7 |0|_A(x1) = x1 + 13 |#|_A() = 3 +_A(x1,x2) = x2 + 5 -_A(x1,x2) = x1 + x2 + 2 not_A(x1) = x1 + 5 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 9 val_A(x1) = x1 l_A(x1) = x1 + 11 n_A(x1,x2,x3) = x1 + x2 + 18 min_A(x1) = x1 + 10 size_A(x1) = x1 + 12 base order 2: precedence: - = if = ge = size > n = min > -# = + = not = val = l > |1| = |0| = and > |#| > true > false partial status: pi(-#) = [2] pi(|1|) = [] pi(|0|) = [1] pi(|#|) = [] pi(+) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(n) = [2] pi(min) = [] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|())) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r3, r9, r10, r11, r12, r13, r14, r15, r16, r18, r25, r26, r27, r28, r29, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: -#_A(x1,x2) = x1 + 3 |0|_A(x1) = x1 + 4 |1|_A(x1) = x1 + 4 -_A(x1,x2) = x1 |#|_A() = 2 +_A(x1,x2) = x2 + 5 not_A(x1) = x1 + 5 false_A() = 1 true_A() = 3 and_A(x1,x2) = 2 ge_A(x1,x2) = x1 + 13 val_A(x1) = x1 l_A(x1) = x1 + 7 n_A(x1,x2,x3) = x1 + 12 size_A(x1) = x1 + 13 base order 2: precedence: + = not = l > - = true = size > ge = n > val > -# = |0| = |1| = |#| = and > false partial status: pi(-#) = [1] pi(|0|) = [1] pi(|1|) = [] pi(-) = [1] pi(|#|) = [] pi(+) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(n) = [1] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: bs#(n(x,y,z)) -> bs#(z) p2: bs#(n(x,y,z)) -> bs#(y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r9, r10, r12, r13, r15, r16, r18, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: bs#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + x3 + 32 |0|_A(x1) = x1 + 9 |#|_A() = 3 +_A(x1,x2) = 5 |1|_A(x1) = 4 -_A(x1,x2) = x1 + x2 + 10 not_A(x1) = x1 + 14 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 ge_A(x1,x2) = x1 + x2 + 5 val_A(x1) = x1 l_A(x1) = x1 + 5 min_A(x1) = x1 + 4 size_A(x1) = x1 + 6 base order 2: precedence: bs# = n = not = val = l = min > |0| = - = size > ge > + = |1| = and > |#| > true > false partial status: pi(bs#) = [1] pi(n) = [3] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: bs#(n(x,y,z)) -> bs#(y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: bs#(n(x,y,z)) -> bs#(y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r9, r10, r12, r13, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: bs#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + x3 + 35 |0|_A(x1) = x1 + 11 |#|_A() = 3 +_A(x1,x2) = x2 + 6 |1|_A(x1) = 5 -_A(x1,x2) = x1 + x2 + 12 not_A(x1) = x1 + 17 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 6 min_A(x1) = x1 + 4 size_A(x1) = x1 + 5 base order 2: precedence: not > ge = l > n = |0| = |1| = - = val = size > bs# = + = if = min > and > |#| > true > false partial status: pi(bs#) = [] pi(n) = [2] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|0|(x),|0|(y)) -> ge#(x,y) p2: ge#(|1|(x),|1|(y)) -> ge#(x,y) p3: ge#(|1|(x),|0|(y)) -> ge#(x,y) p4: ge#(|0|(x),|1|(y)) -> ge#(y,x) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r3, r9, r10, r15, r16, r18, r19, r25, r26, r27, r28, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: ge#_A(x1,x2) = x1 + x2 + 14 |0|_A(x1) = x1 + 13 |1|_A(x1) = x1 + 7 |#|_A() = 3 +_A(x1,x2) = x2 + 5 -_A(x1,x2) = x1 + x2 + 1 not_A(x1) = x1 + 5 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 9 val_A(x1) = x1 l_A(x1) = x1 + 11 min_A(x1) = x1 + 10 n_A(x1,x2,x3) = x1 + x2 + 18 size_A(x1) = x1 + 12 base order 2: precedence: - = if = min = size > ge = n > ge# = + = not = val = l > |0| = |1| = and > |#| > true > false partial status: pi(ge#) = [] pi(|0|) = [] pi(|1|) = [] pi(|#|) = [] pi(+) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(n) = [1, 2] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|1|(x),|1|(y)) -> ge#(x,y) p2: ge#(|1|(x),|0|(y)) -> ge#(x,y) p3: ge#(|0|(x),|1|(y)) -> ge#(y,x) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p2, p3} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|1|(x),|1|(y)) -> ge#(x,y) p2: ge#(|0|(x),|1|(y)) -> ge#(y,x) p3: ge#(|1|(x),|0|(y)) -> ge#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r3, r9, r10, r15, r16, r18, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: ge#_A(x1,x2) = x1 + x2 + 6 |1|_A(x1) = x1 + 3 |0|_A(x1) = x1 + 5 |#|_A() = 15 +_A(x1,x2) = x2 + 1 -_A(x1,x2) = x1 + x2 + 6 not_A(x1) = x1 + 2 false_A() = 1 true_A() = 2 and_A(x1,x2) = 4 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 19 n_A(x1,x2,x3) = x1 + x2 + x3 + 71 min_A(x1) = x1 + 70 max_A(x1) = x1 + 66 size_A(x1) = x1 + 20 base order 2: precedence: ge > max > n > and = l = min = size > ge# = |#| = not = val > |1| = |0| = + = true > - = false partial status: pi(ge#) = [1] pi(|1|) = [] pi(|0|) = [1] pi(|#|) = [] pi(+) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [3] pi(min) = [] pi(max) = [] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|0|(x),|1|(y)) -> ge#(y,x) p2: ge#(|1|(x),|0|(y)) -> ge#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|0|(x),|1|(y)) -> ge#(y,x) p2: ge#(|1|(x),|0|(y)) -> ge#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r9, r10, r15, r16, r18, r19, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: ge#_A(x1,x2) = x1 + x2 + 10 |0|_A(x1) = x1 + 11 |1|_A(x1) = x1 + 9 |#|_A() = 3 +_A(x1,x2) = x1 + 4 -_A(x1,x2) = x1 + x2 + 7 not_A(x1) = x1 + 5 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 7 val_A(x1) = x1 l_A(x1) = x1 + 13 n_A(x1,x2,x3) = x1 + x2 + x3 + 56 min_A(x1) = x1 + 4 max_A(x1) = x1 + 4 size_A(x1) = x1 + 14 base order 2: precedence: ge# = not > |0| > + = max > ge = val = n > - = if = min = size > |1| > and = l > |#| > true > false partial status: pi(ge#) = [1] pi(|0|) = [1] pi(|1|) = [1] pi(|#|) = [] pi(+) = [1] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [3] pi(min) = [] pi(max) = [] pi(size) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|1|(x),|0|(y)) -> ge#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|1|(x),|0|(y)) -> ge#(x,y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r2, r3, r9, r10, r15, r16, r18, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: ge#_A(x1,x2) = x1 + x2 + 3 |1|_A(x1) = x1 + 4 |0|_A(x1) = x1 + 2 |#|_A() = 10 +_A(x1,x2) = x1 + x2 + 7 -_A(x1,x2) = x1 + x2 + 3 not_A(x1) = x1 + 2 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 ge_A(x1,x2) = x1 + x2 + 30 val_A(x1) = x1 l_A(x1) = x1 + 15 n_A(x1,x2,x3) = x1 + x2 + 61 min_A(x1) = x1 + 32 size_A(x1) = x1 + 16 base order 2: precedence: size > + = - = ge > not = n = min > val = l > |#| = and > ge# = |1| = |0| = true > false partial status: pi(ge#) = [] pi(|1|) = [] pi(|0|) = [] pi(|#|) = [] pi(+) = [1] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [2] pi(min) = [] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: ge#(|#|(),|0|(x)) -> ge#(|#|(),x) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r10, r15, r16, r18, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: ge#_A(x1,x2) = x1 + x2 + 1 |#|_A() = 3 |0|_A(x1) = x1 + 9 +_A(x1,x2) = x2 + 8 |1|_A(x1) = 16 -_A(x1,x2) = 6 not_A(x1) = x1 + 26 false_A() = 1 true_A() = 2 and_A(x1,x2) = 4 ge_A(x1,x2) = x1 + x2 + 2 val_A(x1) = x1 l_A(x1) = x1 + 17 n_A(x1,x2,x3) = x1 + x2 + x3 + 25 min_A(x1) = x1 + 4 size_A(x1) = x1 + 11 base order 2: precedence: + = not = l > |1| = min > |0| = ge = val = n > ge# = - = size > |#| = and > true > false partial status: pi(ge#) = [] pi(|#|) = [] pi(|0|) = [1] pi(+) = [2] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(ge) = [] pi(val) = [1] pi(l) = [1] pi(n) = [3] pi(min) = [1] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: min#(n(x,y,z)) -> min#(y) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r9, r10, r12, r13, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: min#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + x3 + 35 |0|_A(x1) = x1 + 11 |#|_A() = 3 +_A(x1,x2) = x2 + 6 |1|_A(x1) = 5 -_A(x1,x2) = x1 + x2 + 12 not_A(x1) = x1 + 17 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 6 min_A(x1) = x1 + 4 size_A(x1) = x1 + 5 base order 2: precedence: not > ge = l > n = |0| = |1| = - = val = size > min# = + = if = min > and > |#| > true > false partial status: pi(min#) = [] pi(n) = [2] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: max#(n(x,y,z)) -> max#(z) and R consists of: r1: |0|(|#|()) -> |#|() r2: +(x,|#|()) -> x r3: +(|#|(),x) -> x r4: +(|0|(x),|0|(y)) -> |0|(+(x,y)) r5: +(|0|(x),|1|(y)) -> |1|(+(x,y)) r6: +(|1|(x),|0|(y)) -> |1|(+(x,y)) r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|()))) r8: +(x,+(y,z)) -> +(+(x,y),z) r9: -(x,|#|()) -> x r10: -(|#|(),x) -> |#|() r11: -(|0|(x),|0|(y)) -> |0|(-(x,y)) r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|()))) r13: -(|1|(x),|0|(y)) -> |1|(-(x,y)) r14: -(|1|(x),|1|(y)) -> |0|(-(x,y)) r15: not(false()) -> true() r16: not(true()) -> false() r17: and(x,true()) -> x r18: and(x,false()) -> false() r19: if(true(),x,y) -> x r20: if(false(),x,y) -> y r21: ge(|0|(x),|0|(y)) -> ge(x,y) r22: ge(|0|(x),|1|(y)) -> not(ge(y,x)) r23: ge(|1|(x),|0|(y)) -> ge(x,y) r24: ge(|1|(x),|1|(y)) -> ge(x,y) r25: ge(x,|#|()) -> true() r26: ge(|#|(),|1|(x)) -> false() r27: ge(|#|(),|0|(x)) -> ge(|#|(),x) r28: val(l(x)) -> x r29: val(n(x,y,z)) -> x r30: min(l(x)) -> x r31: min(n(x,y,z)) -> min(y) r32: max(l(x)) -> x r33: max(n(x,y,z)) -> max(z) r34: bs(l(x)) -> true() r35: bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) r36: size(l(x)) -> |1|(|#|()) r37: size(n(x,y,z)) -> +(+(size(x),size(y)),|1|(|#|())) r38: wb(l(x)) -> true() r39: wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(|1|(|#|()),-(size(y),size(z))),ge(|1|(|#|()),-(size(z),size(y)))),and(wb(y),wb(z))) The set of usable rules consists of r1, r5, r6, r9, r10, r12, r13, r15, r16, r18, r19, r25, r26, r27, r28, r29, r30, r31, r36 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: max#_A(x1) = x1 n_A(x1,x2,x3) = x1 + x2 + x3 + 35 |0|_A(x1) = x1 + 11 |#|_A() = 3 +_A(x1,x2) = x2 + 6 |1|_A(x1) = 5 -_A(x1,x2) = x1 + x2 + 12 not_A(x1) = x1 + 17 false_A() = 1 true_A() = 2 and_A(x1,x2) = 2 if_A(x1,x2,x3) = x2 + 1 ge_A(x1,x2) = x1 + x2 + 6 val_A(x1) = x1 l_A(x1) = x1 + 6 min_A(x1) = x1 + 4 size_A(x1) = x1 + 5 base order 2: precedence: not > ge = l > n = |0| = |1| = - = val = size > max# = + = if = min > and > |#| > true > false partial status: pi(max#) = [] pi(n) = [2] pi(|0|) = [] pi(|#|) = [] pi(+) = [] pi(|1|) = [] pi(-) = [] pi(not) = [] pi(false) = [] pi(true) = [] pi(and) = [] pi(if) = [] pi(ge) = [1] pi(val) = [1] pi(l) = [1] pi(min) = [] pi(size) = [] The following rules are removed: p1 Then no dependency pair remains.