/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 212 ms] (2) CpxRelTRS (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (4) CdtProblem (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (12) CdtProblem (13) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 112 ms] (18) CdtProblem (19) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CdtProblem (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 551 ms] (22) CdtProblem (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 286 ms] (24) CdtProblem (25) CdtKnowledgeProof [FINISHED, 0 ms] (26) BOUNDS(1, 1) (27) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (28) TRS for Loop Detection (29) DecreasingLoopProof [LOWER BOUND(ID), 20 ms] (30) BEST (31) proven lower bound (32) LowerBoundPropagationProof [FINISHED, 0 ms] (33) BOUNDS(n^1, INF) (34) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 flatten(@t) -> flatten#1(@t) flatten#1(leaf) -> nil flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) flattensort(@t) -> insertionsort(flatten(@t)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST ---------------------------------------- (1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 flatten(@t) -> flatten#1(@t) flatten#1(leaf) -> nil flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) flattensort(@t) -> insertionsort(flatten(@t)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Tuples: #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) FLATTENSORT(z0) -> c22(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c25 INSERT#2(#false, z0, z1, z2) -> c26 INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c30 S tuples: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) FLATTENSORT(z0) -> c22(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c25 INSERT#2(#false, z0, z1, z2) -> c26 INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c30 K tuples:none Defined Rule Symbols: #less_2, append_2, append#1_2, flatten_1, flatten#1_1, flattensort_1, insert_2, insert#1_2, insert#2_4, insertionsort_1, insertionsort#1_1, #cklt_1, #compare_2 Defined Pair Symbols: #CKLT_1, #COMPARE_2, #LESS_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, FLATTENSORT_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8_1, c9, c10, c11, c12_1, c13, c14_1, c15_2, c16_1, c17_1, c18, c19_1, c20, c21_4, c22_2, c23_1, c24_2, c25, c26, c27_1, c28_1, c29_2, c30 ---------------------------------------- (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 17 trailing nodes: #COMPARE(#0, #0) -> c3 #CKLT(#GT) -> c1 #COMPARE(#neg(z0), #0) -> c7 INSERT#1(nil, z0) -> c25 #COMPARE(#0, #neg(z0)) -> c4 APPEND#1(nil, z0) -> c18 INSERT#2(#false, z0, z1, z2) -> c26 #CKLT(#LT) -> c2 #COMPARE(#s(z0), #0) -> c13 FLATTEN#1(leaf) -> c20 INSERTIONSORT#1(nil) -> c30 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#0, #pos(z0)) -> c5 #CKLT(#EQ) -> c #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #0) -> c10 ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) FLATTENSORT(z0) -> c22(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) S tuples: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) FLATTENSORT(z0) -> c22(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) K tuples:none Defined Rule Symbols: #less_2, append_2, append#1_2, flatten_1, flatten#1_1, flattensort_1, insert_2, insert#1_2, insert#2_4, insertionsort_1, insertionsort#1_1, #cklt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, #LESS_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, FLATTENSORT_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1 Compound Symbols: c8_1, c12_1, c14_1, c15_2, c16_1, c17_1, c19_1, c21_4, c22_2, c23_1, c24_2, c27_1, c28_1, c29_2 ---------------------------------------- (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) FLATTENSORT(z0) -> c22(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) S tuples: APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) FLATTENSORT(z0) -> c22(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) K tuples:none Defined Rule Symbols: #less_2, append_2, append#1_2, flatten_1, flatten#1_1, flattensort_1, insert_2, insert#1_2, insert#2_4, insertionsort_1, insertionsort#1_1, #cklt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, FLATTENSORT_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c22_2, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1 ---------------------------------------- (9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) FLATTENSORT(z0) -> c(FLATTEN(z0)) S tuples: APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) FLATTENSORT(z0) -> c(FLATTEN(z0)) K tuples:none Defined Rule Symbols: #less_2, append_2, append#1_2, flatten_1, flatten#1_1, flattensort_1, insert_2, insert#1_2, insert#2_4, insertionsort_1, insertionsort#1_1, #cklt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (11) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FLATTENSORT(z0) -> c(FLATTEN(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) S tuples: APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) K tuples:none Defined Rule Symbols: #less_2, append_2, append#1_2, flatten_1, flatten#1_1, flattensort_1, insert_2, insert#1_2, insert#2_4, insertionsort_1, insertionsort#1_1, #cklt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (13) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) S tuples: APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) K tuples: FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) Defined Rule Symbols: #less_2, append_2, append#1_2, flatten_1, flatten#1_1, flattensort_1, insert_2, insert#1_2, insert#2_4, insertionsort_1, insertionsort#1_1, #cklt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: flattensort(z0) -> insertionsort(flatten(z0)) ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: append(z0, z1) -> append#1(z0, z1) flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 #less(z0, z1) -> #cklt(#compare(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) S tuples: APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) K tuples: FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) Defined Rule Symbols: append_2, flatten_1, flatten#1_1, append#1_2, #less_2, #cklt_1, #compare_2, insertionsort_1, insertionsort#1_1, insert_2, insert#1_2, insert#2_4 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) We considered the (Usable) Rules: flatten#1(leaf) -> nil flatten(z0) -> flatten#1(z0) append(z0, z1) -> append#1(z0, z1) append#1(nil, z0) -> z0 flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) And the Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(#0) = [1] POL(#COMPARE(x_1, x_2)) = 0 POL(#EQ) = 0 POL(#GT) = 0 POL(#LESS(x_1, x_2)) = 0 POL(#LT) = 0 POL(#cklt(x_1)) = [1] + x_1 POL(#compare(x_1, x_2)) = 0 POL(#false) = [1] POL(#less(x_1, x_2)) = [1] + x_1 + x_2 POL(#neg(x_1)) = [1] + x_1 POL(#pos(x_1)) = [1] + x_1 POL(#s(x_1)) = [1] + x_1 POL(#true) = [1] POL(::(x_1, x_2)) = [1] + x_1 + x_2 POL(APPEND(x_1, x_2)) = 0 POL(APPEND#1(x_1, x_2)) = 0 POL(FLATTEN(x_1)) = x_1 POL(FLATTEN#1(x_1)) = x_1 POL(FLATTENSORT(x_1)) = [1] + x_1 POL(INSERT(x_1, x_2)) = x_1 POL(INSERT#1(x_1, x_2)) = x_2 POL(INSERT#2(x_1, x_2, x_3, x_4)) = x_2 POL(INSERTIONSORT(x_1)) = [1] + x_1 POL(INSERTIONSORT#1(x_1)) = [1] + x_1 POL(append(x_1, x_2)) = x_1 + x_2 POL(append#1(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c23(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(flatten(x_1)) = x_1 POL(flatten#1(x_1)) = x_1 POL(insert(x_1, x_2)) = [1] + x_1 + x_2 POL(insert#1(x_1, x_2)) = [1] + x_1 + x_2 POL(insert#2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(insertionsort(x_1)) = [1] + x_1 POL(insertionsort#1(x_1)) = [1] + x_1 POL(leaf) = [1] POL(nil) = [1] POL(node(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: append(z0, z1) -> append#1(z0, z1) flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 #less(z0, z1) -> #cklt(#compare(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) S tuples: APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) K tuples: FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) Defined Rule Symbols: append_2, flatten_1, flatten#1_1, append#1_2, #less_2, #cklt_1, #compare_2, insertionsort_1, insertionsort#1_1, insert_2, insert#1_2, insert#2_4 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (19) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FLATTEN(z0) -> c19(FLATTEN#1(z0)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: append(z0, z1) -> append#1(z0, z1) flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 #less(z0, z1) -> #cklt(#compare(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) S tuples: APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) K tuples: FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) Defined Rule Symbols: append_2, flatten_1, flatten#1_1, append#1_2, #less_2, #cklt_1, #compare_2, insertionsort_1, insertionsort#1_1, insert_2, insert#1_2, insert#2_4 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) We considered the (Usable) Rules: flatten#1(leaf) -> nil flatten(z0) -> flatten#1(z0) append(z0, z1) -> append#1(z0, z1) append#1(nil, z0) -> z0 flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) And the Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(#0) = [2] POL(#COMPARE(x_1, x_2)) = 0 POL(#EQ) = [2] POL(#GT) = [1] POL(#LESS(x_1, x_2)) = 0 POL(#LT) = [1] POL(#cklt(x_1)) = [1] POL(#compare(x_1, x_2)) = 0 POL(#false) = [1] POL(#less(x_1, x_2)) = 0 POL(#neg(x_1)) = 0 POL(#pos(x_1)) = 0 POL(#s(x_1)) = 0 POL(#true) = 0 POL(::(x_1, x_2)) = [1] + x_1 + x_2 POL(APPEND(x_1, x_2)) = [1] + [2]x_1 + [2]x_1*x_2 POL(APPEND#1(x_1, x_2)) = [2]x_1 + [2]x_1*x_2 POL(FLATTEN(x_1)) = x_1 + x_1^2 POL(FLATTEN#1(x_1)) = x_1 + x_1^2 POL(FLATTENSORT(x_1)) = [2] + [2]x_1 + [2]x_1^2 POL(INSERT(x_1, x_2)) = 0 POL(INSERT#1(x_1, x_2)) = 0 POL(INSERT#2(x_1, x_2, x_3, x_4)) = 0 POL(INSERTIONSORT(x_1)) = x_1 POL(INSERTIONSORT#1(x_1)) = x_1 POL(append(x_1, x_2)) = x_1 + x_2 POL(append#1(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c23(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(flatten(x_1)) = x_1 POL(flatten#1(x_1)) = x_1 POL(insert(x_1, x_2)) = [1] + [2]x_1 + x_1^2 POL(insert#1(x_1, x_2)) = [1] + x_2 + x_2^2 POL(insert#2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 + x_4^2 + x_3*x_4 + x_2*x_4 + x_3^2 + x_2*x_3 + x_2^2 POL(insertionsort(x_1)) = 0 POL(insertionsort#1(x_1)) = [1] POL(leaf) = 0 POL(nil) = 0 POL(node(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: append(z0, z1) -> append#1(z0, z1) flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 #less(z0, z1) -> #cklt(#compare(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) S tuples: INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) K tuples: FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) Defined Rule Symbols: append_2, flatten_1, flatten#1_1, append#1_2, #less_2, #cklt_1, #compare_2, insertionsort_1, insertionsort#1_1, insert_2, insert#1_2, insert#2_4 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) We considered the (Usable) Rules: flatten#1(leaf) -> nil flatten(z0) -> flatten#1(z0) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) #cklt(#GT) -> #false append(z0, z1) -> append#1(z0, z1) append#1(nil, z0) -> z0 flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) insertionsort#1(nil) -> nil insert(z0, z1) -> insert#1(z1, z0) #cklt(#EQ) -> #false insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insert#1(nil, z0) -> ::(z0, nil) insertionsort(z0) -> insertionsort#1(z0) #less(z0, z1) -> #cklt(#compare(z0, z1)) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) #cklt(#LT) -> #true And the Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(#0) = [2] POL(#COMPARE(x_1, x_2)) = 0 POL(#EQ) = 0 POL(#GT) = 0 POL(#LESS(x_1, x_2)) = 0 POL(#LT) = 0 POL(#cklt(x_1)) = [2] POL(#compare(x_1, x_2)) = 0 POL(#false) = [2] POL(#less(x_1, x_2)) = [2] POL(#neg(x_1)) = 0 POL(#pos(x_1)) = 0 POL(#s(x_1)) = 0 POL(#true) = [2] POL(::(x_1, x_2)) = [2] + x_2 POL(APPEND(x_1, x_2)) = 0 POL(APPEND#1(x_1, x_2)) = 0 POL(FLATTEN(x_1)) = 0 POL(FLATTEN#1(x_1)) = 0 POL(FLATTENSORT(x_1)) = [2] + [2]x_1^2 POL(INSERT(x_1, x_2)) = [2]x_2 POL(INSERT#1(x_1, x_2)) = [2]x_1 POL(INSERT#2(x_1, x_2, x_3, x_4)) = [2]x_1 + [2]x_4 POL(INSERTIONSORT(x_1)) = x_1^2 POL(INSERTIONSORT#1(x_1)) = x_1^2 POL(append(x_1, x_2)) = x_1 + x_2 POL(append#1(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c23(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(flatten(x_1)) = x_1 POL(flatten#1(x_1)) = x_1 POL(insert(x_1, x_2)) = [2] + x_2 POL(insert#1(x_1, x_2)) = [2] + x_1 POL(insert#2(x_1, x_2, x_3, x_4)) = x_4 + x_1^2 POL(insertionsort(x_1)) = [2] + x_1 POL(insertionsort#1(x_1)) = [2] + x_1 POL(leaf) = 0 POL(nil) = 0 POL(node(x_1, x_2, x_3)) = x_1 + x_2 + x_3 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: append(z0, z1) -> append#1(z0, z1) flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 #less(z0, z1) -> #cklt(#compare(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) S tuples: INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) K tuples: FLATTENSORT(z0) -> c(INSERTIONSORT(flatten(z0))) FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1), FLATTEN(z2)) INSERTIONSORT#1(::(z0, z1)) -> c29(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) FLATTEN(z0) -> c19(FLATTEN#1(z0)) INSERTIONSORT(z0) -> c28(INSERTIONSORT#1(z0)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) Defined Rule Symbols: append_2, flatten_1, flatten#1_1, append#1_2, #less_2, #cklt_1, #compare_2, insertionsort_1, insertionsort#1_1, insert_2, insert#1_2, insert#2_4 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1, #LESS_2, FLATTENSORT_1 Compound Symbols: c8_1, c12_1, c14_1, c16_1, c17_1, c19_1, c21_4, c23_1, c24_2, c27_1, c28_1, c29_2, c15_1, c_1 ---------------------------------------- (25) CdtKnowledgeProof (FINISHED) The following tuples could be moved from S to K by knowledge propagation: INSERT(z0, z1) -> c23(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) INSERT#1(::(z0, z1), z2) -> c24(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#2(#true, z0, z1, z2) -> c27(INSERT(z0, z2)) #LESS(z0, z1) -> c15(#COMPARE(z0, z1)) Now S is empty ---------------------------------------- (26) BOUNDS(1, 1) ---------------------------------------- (27) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (28) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 flatten(@t) -> flatten#1(@t) flatten#1(leaf) -> nil flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) flattensort(@t) -> insertionsort(flatten(@t)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST ---------------------------------------- (29) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence insertionsort(::(@x1_0, @xs2_0)) ->^+ insert(@x1_0, insertionsort(@xs2_0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [@xs2_0 / ::(@x1_0, @xs2_0)]. The result substitution is [ ]. ---------------------------------------- (30) Complex Obligation (BEST) ---------------------------------------- (31) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 flatten(@t) -> flatten#1(@t) flatten#1(leaf) -> nil flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) flattensort(@t) -> insertionsort(flatten(@t)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST ---------------------------------------- (32) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (33) BOUNDS(n^1, INF) ---------------------------------------- (34) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 flatten(@t) -> flatten#1(@t) flatten#1(leaf) -> nil flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) flattensort(@t) -> insertionsort(flatten(@t)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST