/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 221 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 288 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 322 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 869 ms] (20) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) The (relative) TRS S consists of the following rules: encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) The (relative) TRS S consists of the following rules: encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) The (relative) TRS S consists of the following rules: encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Types: isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encArg :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 hole_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if21_0 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0 :: Nat -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: concat, less_leaves, if1, if2, encArg They will be analysed ascendingly in the following order: concat < if2 concat < encArg less_leaves = if1 less_leaves = if2 less_leaves < encArg if1 = if2 if1 < encArg if2 < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Types: isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encArg :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 hole_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if21_0 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0 :: Nat -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 Generator Equations: gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0) <=> leaf gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(x, 1)) <=> cons(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(x)) The following defined symbols remain to be analysed: concat, less_leaves, if1, if2, encArg They will be analysed ascendingly in the following order: concat < if2 concat < encArg less_leaves = if1 less_leaves = if2 less_leaves < encArg if1 = if2 if1 < encArg if2 < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: concat(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n4_0), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(b)) -> gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n4_0, b)), rt in Omega(1 + n4_0) Induction Base: concat(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(b)) ->_R^Omega(1) gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(b) Induction Step: concat(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n4_0, 1)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(b)) ->_R^Omega(1) cons(leaf, concat(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n4_0), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(b))) ->_IH cons(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(b, c5_0))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Types: isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encArg :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 hole_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if21_0 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0 :: Nat -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 Generator Equations: gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0) <=> leaf gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(x, 1)) <=> cons(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(x)) The following defined symbols remain to be analysed: concat, less_leaves, if1, if2, encArg They will be analysed ascendingly in the following order: concat < if2 concat < encArg less_leaves = if1 less_leaves = if2 less_leaves < encArg if1 = if2 if1 < encArg if2 < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Types: isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encArg :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 hole_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if21_0 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0 :: Nat -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 Lemmas: concat(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n4_0), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(b)) -> gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n4_0, b)), rt in Omega(1 + n4_0) Generator Equations: gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0) <=> leaf gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(x, 1)) <=> cons(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(x)) The following defined symbols remain to be analysed: if1, less_leaves, if2, encArg They will be analysed ascendingly in the following order: less_leaves = if1 less_leaves = if2 less_leaves < encArg if1 = if2 if1 < encArg if2 < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n1230_0)) -> false, rt in Omega(1 + n1230_0) Induction Base: less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, 0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0)) ->_R^Omega(1) if1(isLeaf(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, 0))), isLeaf(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, 0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0)) ->_R^Omega(1) if1(false, isLeaf(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(1), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0)) ->_R^Omega(1) if1(false, true, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(1), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0)) ->_R^Omega(1) false Induction Step: less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, +(n1230_0, 1))), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n1230_0, 1))) ->_R^Omega(1) if1(isLeaf(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, +(n1230_0, 1)))), isLeaf(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n1230_0, 1))), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, +(n1230_0, 1))), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n1230_0, 1))) ->_R^Omega(1) if1(false, isLeaf(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(2, n1230_0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))) ->_R^Omega(1) if1(false, false, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(2, n1230_0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))) ->_R^Omega(1) if2(false, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(2, n1230_0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))) ->_R^Omega(1) less_leaves(concat(left(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(2, n1230_0))), right(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(2, n1230_0)))), concat(left(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))), right(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))))) ->_R^Omega(1) less_leaves(concat(leaf, right(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(2, n1230_0)))), concat(left(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))), right(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))))) ->_R^Omega(1) less_leaves(concat(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))), concat(left(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))), right(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))))) ->_L^Omega(1) less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(0, +(1, n1230_0))), concat(left(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))), right(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))))) ->_R^Omega(1) less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0)), concat(leaf, right(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0))))) ->_R^Omega(1) less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0)), concat(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n1230_0))) ->_L^Omega(1) less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(0, n1230_0))) ->_IH false We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: isLeaf(leaf) -> true isLeaf(cons(u, v)) -> false left(cons(u, v)) -> u right(cons(u, v)) -> v concat(leaf, y) -> y concat(cons(u, v), y) -> cons(u, concat(v, y)) less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) if1(b, true, u, v) -> false if1(b, false, u, v) -> if2(b, u, v) if2(true, u, v) -> true if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) encArg(leaf) -> leaf encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(cons_isLeaf(x_1)) -> isLeaf(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_concat(x_1, x_2)) -> concat(encArg(x_1), encArg(x_2)) encArg(cons_less_leaves(x_1, x_2)) -> less_leaves(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isLeaf(x_1) -> isLeaf(encArg(x_1)) encode_leaf -> leaf encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_concat(x_1, x_2) -> concat(encArg(x_1), encArg(x_2)) encode_less_leaves(x_1, x_2) -> less_leaves(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Types: isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encArg :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 cons_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_isLeaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_leaf :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_true :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_cons :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_false :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_left :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_right :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_concat :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_less_leaves :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if1 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 encode_if2 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 hole_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if21_0 :: leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0 :: Nat -> leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if2 Lemmas: concat(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n4_0), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(b)) -> gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n4_0, b)), rt in Omega(1 + n4_0) less_leaves(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(1, n1230_0)), gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n1230_0)) -> false, rt in Omega(1 + n1230_0) Generator Equations: gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0) <=> leaf gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(x, 1)) <=> cons(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(x)) The following defined symbols remain to be analysed: if1, if2, encArg They will be analysed ascendingly in the following order: less_leaves = if1 less_leaves = if2 less_leaves < encArg if1 = if2 if1 < encArg if2 < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n6469_0)) -> gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n6469_0), rt in Omega(0) Induction Base: encArg(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(0)) ->_R^Omega(0) leaf Induction Step: encArg(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(+(n6469_0, 1))) ->_R^Omega(0) cons(encArg(leaf), encArg(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n6469_0))) ->_R^Omega(0) cons(leaf, encArg(gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(n6469_0))) ->_IH cons(leaf, gen_leaf:true:cons:false:cons_isLeaf:cons_left:cons_right:cons_concat:cons_less_leaves:cons_if1:cons_if22_0(c6470_0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)