/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 588 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (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: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) 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(e) -> e encArg(l) -> l encArg(m) -> m encArg(d) -> d encArg(A) -> A encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(k) -> k encArg(z(x_1, x_2)) -> z(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encArg(g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__a) -> a__a encArg(cons_a__b) -> a__b encArg(cons_a__c) -> a__c encArg(cons_a__k) -> a__k encArg(cons_a__d) -> a__d encArg(cons_a__A) -> a__A encArg(cons_a__h(x_1, x_2)) -> a__h(encArg(x_1), encArg(x_2)) encArg(cons_a__g(x_1, x_2, x_3)) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__z(x_1, x_2)) -> a__z(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__a -> a__a encode_a__c -> a__c encode_a__b -> a__b encode_e -> e encode_a__k -> a__k encode_l -> l encode_a__d -> a__d encode_m -> m encode_a__A -> a__A encode_a__h(x_1, x_2) -> a__h(encArg(x_1), encArg(x_2)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_a__g(x_1, x_2, x_3) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_a__z(x_1, x_2) -> a__z(encArg(x_1), encArg(x_2)) encode_A -> A encode_a -> a encode_b -> b encode_c -> c encode_k -> k encode_z(x_1, x_2) -> z(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_g(x_1, x_2, x_3) -> g(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: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) The (relative) TRS S consists of the following rules: encArg(e) -> e encArg(l) -> l encArg(m) -> m encArg(d) -> d encArg(A) -> A encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(k) -> k encArg(z(x_1, x_2)) -> z(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encArg(g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__a) -> a__a encArg(cons_a__b) -> a__b encArg(cons_a__c) -> a__c encArg(cons_a__k) -> a__k encArg(cons_a__d) -> a__d encArg(cons_a__A) -> a__A encArg(cons_a__h(x_1, x_2)) -> a__h(encArg(x_1), encArg(x_2)) encArg(cons_a__g(x_1, x_2, x_3)) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__z(x_1, x_2)) -> a__z(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__a -> a__a encode_a__c -> a__c encode_a__b -> a__b encode_e -> e encode_a__k -> a__k encode_l -> l encode_a__d -> a__d encode_m -> m encode_a__A -> a__A encode_a__h(x_1, x_2) -> a__h(encArg(x_1), encArg(x_2)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_a__g(x_1, x_2, x_3) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_a__z(x_1, x_2) -> a__z(encArg(x_1), encArg(x_2)) encode_A -> A encode_a -> a encode_b -> b encode_c -> c encode_k -> k encode_z(x_1, x_2) -> z(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_g(x_1, x_2, x_3) -> g(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: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) The (relative) TRS S consists of the following rules: encArg(e) -> e encArg(l) -> l encArg(m) -> m encArg(d) -> d encArg(A) -> A encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(k) -> k encArg(z(x_1, x_2)) -> z(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encArg(g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__a) -> a__a encArg(cons_a__b) -> a__b encArg(cons_a__c) -> a__c encArg(cons_a__k) -> a__k encArg(cons_a__d) -> a__d encArg(cons_a__A) -> a__A encArg(cons_a__h(x_1, x_2)) -> a__h(encArg(x_1), encArg(x_2)) encArg(cons_a__g(x_1, x_2, x_3)) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__z(x_1, x_2)) -> a__z(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__a -> a__a encode_a__c -> a__c encode_a__b -> a__b encode_e -> e encode_a__k -> a__k encode_l -> l encode_a__d -> a__d encode_m -> m encode_a__A -> a__A encode_a__h(x_1, x_2) -> a__h(encArg(x_1), encArg(x_2)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_a__g(x_1, x_2, x_3) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_a__z(x_1, x_2) -> a__z(encArg(x_1), encArg(x_2)) encode_A -> A encode_a -> a encode_b -> b encode_c -> c encode_k -> k encode_z(x_1, x_2) -> z(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: 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: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) The (relative) TRS S consists of the following rules: encArg(e) -> e encArg(l) -> l encArg(m) -> m encArg(d) -> d encArg(A) -> A encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(k) -> k encArg(z(x_1, x_2)) -> z(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encArg(g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__a) -> a__a encArg(cons_a__b) -> a__b encArg(cons_a__c) -> a__c encArg(cons_a__k) -> a__k encArg(cons_a__d) -> a__d encArg(cons_a__A) -> a__A encArg(cons_a__h(x_1, x_2)) -> a__h(encArg(x_1), encArg(x_2)) encArg(cons_a__g(x_1, x_2, x_3)) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__z(x_1, x_2)) -> a__z(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__a -> a__a encode_a__c -> a__c encode_a__b -> a__b encode_e -> e encode_a__k -> a__k encode_l -> l encode_a__d -> a__d encode_m -> m encode_a__A -> a__A encode_a__h(x_1, x_2) -> a__h(encArg(x_1), encArg(x_2)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_a__g(x_1, x_2, x_3) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_a__z(x_1, x_2) -> a__z(encArg(x_1), encArg(x_2)) encode_A -> A encode_a -> a encode_b -> b encode_c -> c encode_k -> k encode_z(x_1, x_2) -> z(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(z(X1, X2)) ->^+ a__z(mark(X1), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / z(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) 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, INF). The TRS R consists of the following rules: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) The (relative) TRS S consists of the following rules: encArg(e) -> e encArg(l) -> l encArg(m) -> m encArg(d) -> d encArg(A) -> A encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(k) -> k encArg(z(x_1, x_2)) -> z(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encArg(g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__a) -> a__a encArg(cons_a__b) -> a__b encArg(cons_a__c) -> a__c encArg(cons_a__k) -> a__k encArg(cons_a__d) -> a__d encArg(cons_a__A) -> a__A encArg(cons_a__h(x_1, x_2)) -> a__h(encArg(x_1), encArg(x_2)) encArg(cons_a__g(x_1, x_2, x_3)) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__z(x_1, x_2)) -> a__z(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__a -> a__a encode_a__c -> a__c encode_a__b -> a__b encode_e -> e encode_a__k -> a__k encode_l -> l encode_a__d -> a__d encode_m -> m encode_a__A -> a__A encode_a__h(x_1, x_2) -> a__h(encArg(x_1), encArg(x_2)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_a__g(x_1, x_2, x_3) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_a__z(x_1, x_2) -> a__z(encArg(x_1), encArg(x_2)) encode_A -> A encode_a -> a encode_b -> b encode_c -> c encode_k -> k encode_z(x_1, x_2) -> z(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: 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: a__a -> a__c a__b -> a__c a__c -> e a__k -> l a__d -> m a__a -> a__d a__b -> a__d a__c -> l a__k -> m a__A -> a__h(a__f(a__a), a__f(a__b)) a__h(X, X) -> a__g(mark(X), mark(X), a__f(a__k)) a__g(d, X, X) -> a__A a__f(X) -> a__z(mark(X), X) a__z(e, X) -> mark(X) mark(A) -> a__A mark(a) -> a__a mark(b) -> a__b mark(c) -> a__c mark(d) -> a__d mark(k) -> a__k mark(z(X1, X2)) -> a__z(mark(X1), X2) mark(f(X)) -> a__f(mark(X)) mark(h(X1, X2)) -> a__h(mark(X1), mark(X2)) mark(g(X1, X2, X3)) -> a__g(mark(X1), mark(X2), mark(X3)) mark(e) -> e mark(l) -> l mark(m) -> m a__A -> A a__a -> a a__b -> b a__c -> c a__d -> d a__k -> k a__z(X1, X2) -> z(X1, X2) a__f(X) -> f(X) a__h(X1, X2) -> h(X1, X2) a__g(X1, X2, X3) -> g(X1, X2, X3) The (relative) TRS S consists of the following rules: encArg(e) -> e encArg(l) -> l encArg(m) -> m encArg(d) -> d encArg(A) -> A encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(k) -> k encArg(z(x_1, x_2)) -> z(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encArg(g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__a) -> a__a encArg(cons_a__b) -> a__b encArg(cons_a__c) -> a__c encArg(cons_a__k) -> a__k encArg(cons_a__d) -> a__d encArg(cons_a__A) -> a__A encArg(cons_a__h(x_1, x_2)) -> a__h(encArg(x_1), encArg(x_2)) encArg(cons_a__g(x_1, x_2, x_3)) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__z(x_1, x_2)) -> a__z(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__a -> a__a encode_a__c -> a__c encode_a__b -> a__b encode_e -> e encode_a__k -> a__k encode_l -> l encode_a__d -> a__d encode_m -> m encode_a__A -> a__A encode_a__h(x_1, x_2) -> a__h(encArg(x_1), encArg(x_2)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_a__g(x_1, x_2, x_3) -> a__g(encArg(x_1), encArg(x_2), encArg(x_3)) encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_a__z(x_1, x_2) -> a__z(encArg(x_1), encArg(x_2)) encode_A -> A encode_a -> a encode_b -> b encode_c -> c encode_k -> k encode_z(x_1, x_2) -> z(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST