60.40/16.49 WORST_CASE(Omega(n^1), O(n^1)) 60.40/16.50 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 60.40/16.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 60.40/16.50 60.40/16.50 60.40/16.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 60.40/16.50 60.40/16.50 (0) CpxTRS 60.40/16.50 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 60.40/16.50 (2) CpxWeightedTrs 60.40/16.50 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 60.40/16.50 (4) CpxTypedWeightedTrs 60.40/16.50 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 60.40/16.50 (6) CpxTypedWeightedCompleteTrs 60.40/16.50 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms] 60.40/16.50 (8) CpxRNTS 60.40/16.50 (9) CompleteCoflocoProof [FINISHED, 574 ms] 60.40/16.50 (10) BOUNDS(1, n^1) 60.40/16.50 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 60.40/16.50 (12) CpxTRS 60.40/16.50 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 60.40/16.50 (14) typed CpxTrs 60.40/16.50 (15) OrderProof [LOWER BOUND(ID), 0 ms] 60.40/16.50 (16) typed CpxTrs 60.40/16.50 (17) RewriteLemmaProof [LOWER BOUND(ID), 297 ms] 60.40/16.50 (18) BEST 60.40/16.50 (19) proven lower bound 60.40/16.50 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 60.40/16.50 (21) BOUNDS(n^1, INF) 60.40/16.50 (22) typed CpxTrs 60.40/16.50 (23) RewriteLemmaProof [LOWER BOUND(ID), 72 ms] 60.40/16.50 (24) typed CpxTrs 60.40/16.50 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (0) 60.40/16.50 Obligation: 60.40/16.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 60.40/16.50 60.40/16.50 60.40/16.50 The TRS R consists of the following rules: 60.40/16.50 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0), x, y) 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) 60.40/16.50 gr(0, x) -> false 60.40/16.50 gr(s(x), 0) -> true 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) 60.40/16.50 p(0) -> 0 60.40/16.50 p(s(x)) -> x 60.40/16.50 eq(0, 0) -> true 60.40/16.50 eq(s(x), 0) -> false 60.40/16.50 eq(0, s(x)) -> false 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) 60.40/16.50 and(true, true) -> true 60.40/16.50 and(false, x) -> false 60.40/16.50 and(x, false) -> false 60.40/16.50 60.40/16.50 S is empty. 60.40/16.50 Rewrite Strategy: INNERMOST 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 60.40/16.50 Transformed relative TRS to weighted TRS 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (2) 60.40/16.50 Obligation: 60.40/16.50 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 60.40/16.50 60.40/16.50 60.40/16.50 The TRS R consists of the following rules: 60.40/16.50 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1] 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) [1] 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) [1] 60.40/16.50 gr(0, x) -> false [1] 60.40/16.50 gr(s(x), 0) -> true [1] 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) [1] 60.40/16.50 p(0) -> 0 [1] 60.40/16.50 p(s(x)) -> x [1] 60.40/16.50 eq(0, 0) -> true [1] 60.40/16.50 eq(s(x), 0) -> false [1] 60.40/16.50 eq(0, s(x)) -> false [1] 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) [1] 60.40/16.50 and(true, true) -> true [1] 60.40/16.50 and(false, x) -> false [1] 60.40/16.50 and(x, false) -> false [1] 60.40/16.50 60.40/16.50 Rewrite Strategy: INNERMOST 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 60.40/16.50 Infered types. 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (4) 60.40/16.50 Obligation: 60.40/16.50 Runtime Complexity Weighted TRS with Types. 60.40/16.50 The TRS R consists of the following rules: 60.40/16.50 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1] 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) [1] 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) [1] 60.40/16.50 gr(0, x) -> false [1] 60.40/16.50 gr(s(x), 0) -> true [1] 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) [1] 60.40/16.50 p(0) -> 0 [1] 60.40/16.50 p(s(x)) -> x [1] 60.40/16.50 eq(0, 0) -> true [1] 60.40/16.50 eq(s(x), 0) -> false [1] 60.40/16.50 eq(0, s(x)) -> false [1] 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) [1] 60.40/16.50 and(true, true) -> true [1] 60.40/16.50 and(false, x) -> false [1] 60.40/16.50 and(x, false) -> false [1] 60.40/16.50 60.40/16.50 The TRS has the following type information: 60.40/16.50 cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2 60.40/16.50 true :: true:false 60.40/16.50 cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2 60.40/16.50 gr :: 0:s -> 0:s -> true:false 60.40/16.50 0 :: 0:s 60.40/16.50 p :: 0:s -> 0:s 60.40/16.50 false :: true:false 60.40/16.50 and :: true:false -> true:false -> true:false 60.40/16.50 eq :: 0:s -> 0:s -> true:false 60.40/16.50 s :: 0:s -> 0:s 60.40/16.50 60.40/16.50 Rewrite Strategy: INNERMOST 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (5) CompletionProof (UPPER BOUND(ID)) 60.40/16.50 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 60.40/16.50 60.40/16.50 cond1(v0, v1, v2) -> null_cond1 [0] 60.40/16.50 60.40/16.50 And the following fresh constants: null_cond1 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (6) 60.40/16.50 Obligation: 60.40/16.50 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 60.40/16.50 60.40/16.50 Runtime Complexity Weighted TRS with Types. 60.40/16.50 The TRS R consists of the following rules: 60.40/16.50 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1] 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) [1] 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) [1] 60.40/16.50 gr(0, x) -> false [1] 60.40/16.50 gr(s(x), 0) -> true [1] 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) [1] 60.40/16.50 p(0) -> 0 [1] 60.40/16.50 p(s(x)) -> x [1] 60.40/16.50 eq(0, 0) -> true [1] 60.40/16.50 eq(s(x), 0) -> false [1] 60.40/16.50 eq(0, s(x)) -> false [1] 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) [1] 60.40/16.50 and(true, true) -> true [1] 60.40/16.50 and(false, x) -> false [1] 60.40/16.50 and(x, false) -> false [1] 60.40/16.50 cond1(v0, v1, v2) -> null_cond1 [0] 60.40/16.50 60.40/16.50 The TRS has the following type information: 60.40/16.50 cond1 :: true:false -> 0:s -> 0:s -> null_cond1 60.40/16.50 true :: true:false 60.40/16.50 cond2 :: true:false -> 0:s -> 0:s -> null_cond1 60.40/16.50 gr :: 0:s -> 0:s -> true:false 60.40/16.50 0 :: 0:s 60.40/16.50 p :: 0:s -> 0:s 60.40/16.50 false :: true:false 60.40/16.50 and :: true:false -> true:false -> true:false 60.40/16.50 eq :: 0:s -> 0:s -> true:false 60.40/16.50 s :: 0:s -> 0:s 60.40/16.50 null_cond1 :: null_cond1 60.40/16.50 60.40/16.50 Rewrite Strategy: INNERMOST 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 60.40/16.50 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 60.40/16.50 The constant constructors are abstracted as follows: 60.40/16.50 60.40/16.50 true => 1 60.40/16.50 0 => 0 60.40/16.50 false => 0 60.40/16.50 null_cond1 => 0 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (8) 60.40/16.50 Obligation: 60.40/16.50 Complexity RNTS consisting of the following rules: 60.40/16.50 60.40/16.50 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 60.40/16.50 and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 60.40/16.50 and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 60.40/16.50 cond1(z, z', z'') -{ 1 }-> cond2(gr(y, 0), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 60.40/16.50 cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 60.40/16.50 cond2(z, z', z'') -{ 1 }-> cond2(gr(y, 0), p(x), p(y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 60.40/16.50 cond2(z, z', z'') -{ 1 }-> cond1(and(eq(x, y), gr(x, 0)), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 60.40/16.50 eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 60.40/16.50 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 60.40/16.50 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 60.40/16.50 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 60.40/16.50 gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 60.40/16.50 gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 60.40/16.50 gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 60.40/16.50 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 60.40/16.50 p(z) -{ 1 }-> 0 :|: z = 0 60.40/16.50 60.40/16.50 Only complete derivations are relevant for the runtime complexity. 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (9) CompleteCoflocoProof (FINISHED) 60.40/16.50 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 60.40/16.50 60.40/16.50 eq(start(V1, V, V2),0,[cond1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 60.40/16.50 eq(start(V1, V, V2),0,[cond2(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 60.40/16.50 eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]). 60.40/16.50 eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]). 60.40/16.50 eq(start(V1, V, V2),0,[eq(V1, V, Out)],[V1 >= 0,V >= 0]). 60.40/16.50 eq(start(V1, V, V2),0,[and(V1, V, Out)],[V1 >= 0,V >= 0]). 60.40/16.50 eq(cond1(V1, V, V2, Out),1,[gr(V3, 0, Ret0),cond2(Ret0, V4, V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]). 60.40/16.50 eq(cond2(V1, V, V2, Out),1,[gr(V6, 0, Ret01),p(V5, Ret1),p(V6, Ret2),cond2(Ret01, Ret1, Ret2, Ret3)],[Out = Ret3,V = V5,V2 = V6,V1 = 1,V5 >= 0,V6 >= 0]). 60.40/16.50 eq(cond2(V1, V, V2, Out),1,[eq(V8, V7, Ret00),gr(V8, 0, Ret011),and(Ret00, Ret011, Ret02),cond1(Ret02, V8, V7, Ret4)],[Out = Ret4,V = V8,V2 = V7,V8 >= 0,V7 >= 0,V1 = 0]). 60.40/16.50 eq(gr(V1, V, Out),1,[],[Out = 0,V = V9,V9 >= 0,V1 = 0]). 60.40/16.50 eq(gr(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = 1 + V10,V = 0]). 60.40/16.50 eq(gr(V1, V, Out),1,[gr(V12, V11, Ret5)],[Out = Ret5,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]). 60.40/16.50 eq(p(V1, Out),1,[],[Out = 0,V1 = 0]). 60.40/16.50 eq(p(V1, Out),1,[],[Out = V13,V13 >= 0,V1 = 1 + V13]). 60.40/16.50 eq(eq(V1, V, Out),1,[],[Out = 1,V1 = 0,V = 0]). 60.40/16.50 eq(eq(V1, V, Out),1,[],[Out = 0,V14 >= 0,V1 = 1 + V14,V = 0]). 60.40/16.50 eq(eq(V1, V, Out),1,[],[Out = 0,V = 1 + V15,V15 >= 0,V1 = 0]). 60.40/16.50 eq(eq(V1, V, Out),1,[eq(V17, V16, Ret6)],[Out = Ret6,V = 1 + V16,V17 >= 0,V16 >= 0,V1 = 1 + V17]). 60.40/16.50 eq(and(V1, V, Out),1,[],[Out = 1,V1 = 1,V = 1]). 60.40/16.50 eq(and(V1, V, Out),1,[],[Out = 0,V = V18,V18 >= 0,V1 = 0]). 60.40/16.50 eq(and(V1, V, Out),1,[],[Out = 0,V19 >= 0,V1 = V19,V = 0]). 60.40/16.50 eq(cond1(V1, V, V2, Out),0,[],[Out = 0,V21 >= 0,V2 = V22,V20 >= 0,V1 = V21,V = V20,V22 >= 0]). 60.40/16.50 input_output_vars(cond1(V1,V,V2,Out),[V1,V,V2],[Out]). 60.40/16.50 input_output_vars(cond2(V1,V,V2,Out),[V1,V,V2],[Out]). 60.40/16.50 input_output_vars(gr(V1,V,Out),[V1,V],[Out]). 60.40/16.50 input_output_vars(p(V1,Out),[V1],[Out]). 60.40/16.50 input_output_vars(eq(V1,V,Out),[V1,V],[Out]). 60.40/16.50 input_output_vars(and(V1,V,Out),[V1,V],[Out]). 60.40/16.50 60.40/16.50 60.40/16.50 CoFloCo proof output: 60.40/16.50 Preprocessing Cost Relations 60.40/16.50 ===================================== 60.40/16.50 60.40/16.50 #### Computed strongly connected components 60.40/16.50 0. non_recursive : [and/3] 60.40/16.50 1. recursive : [eq/3] 60.40/16.50 2. recursive : [gr/3] 60.40/16.50 3. non_recursive : [p/2] 60.40/16.50 4. recursive : [cond1/4,cond2/4] 60.40/16.50 5. non_recursive : [start/3] 60.40/16.50 60.40/16.50 #### Obtained direct recursion through partial evaluation 60.40/16.50 0. SCC is partially evaluated into and/3 60.40/16.50 1. SCC is partially evaluated into eq/3 60.40/16.50 2. SCC is partially evaluated into gr/3 60.40/16.50 3. SCC is partially evaluated into p/2 60.40/16.50 4. SCC is partially evaluated into cond2/4 60.40/16.50 5. SCC is partially evaluated into start/3 60.40/16.50 60.40/16.50 Control-Flow Refinement of Cost Relations 60.40/16.50 ===================================== 60.40/16.50 60.40/16.50 ### Specialization of cost equations and/3 60.40/16.50 * CE 22 is refined into CE [23] 60.40/16.50 * CE 20 is refined into CE [24] 60.40/16.50 * CE 21 is refined into CE [25] 60.40/16.50 60.40/16.50 60.40/16.50 ### Cost equations --> "Loop" of and/3 60.40/16.50 * CEs [23] --> Loop 17 60.40/16.50 * CEs [24] --> Loop 18 60.40/16.50 * CEs [25] --> Loop 19 60.40/16.50 60.40/16.50 ### Ranking functions of CR and(V1,V,Out) 60.40/16.50 60.40/16.50 #### Partial ranking functions of CR and(V1,V,Out) 60.40/16.50 60.40/16.50 60.40/16.50 ### Specialization of cost equations eq/3 60.40/16.50 * CE 19 is refined into CE [26] 60.40/16.50 * CE 17 is refined into CE [27] 60.40/16.50 * CE 18 is refined into CE [28] 60.40/16.50 * CE 16 is refined into CE [29] 60.40/16.50 60.40/16.50 60.40/16.50 ### Cost equations --> "Loop" of eq/3 60.40/16.50 * CEs [27] --> Loop 20 60.40/16.50 * CEs [28] --> Loop 21 60.40/16.50 * CEs [29] --> Loop 22 60.40/16.50 * CEs [26] --> Loop 23 60.40/16.50 60.40/16.50 ### Ranking functions of CR eq(V1,V,Out) 60.40/16.50 * RF of phase [23]: [V,V1] 60.40/16.50 60.40/16.50 #### Partial ranking functions of CR eq(V1,V,Out) 60.40/16.50 * Partial RF of phase [23]: 60.40/16.50 - RF of loop [23:1]: 60.40/16.50 V 60.40/16.50 V1 60.40/16.50 60.40/16.50 60.40/16.50 ### Specialization of cost equations gr/3 60.40/16.50 * CE 10 is refined into CE [30] 60.40/16.50 * CE 9 is refined into CE [31] 60.40/16.50 * CE 8 is refined into CE [32] 60.40/16.50 60.40/16.50 60.40/16.50 ### Cost equations --> "Loop" of gr/3 60.40/16.50 * CEs [31] --> Loop 24 60.40/16.50 * CEs [32] --> Loop 25 60.40/16.50 * CEs [30] --> Loop 26 60.40/16.50 60.40/16.50 ### Ranking functions of CR gr(V1,V,Out) 60.40/16.50 * RF of phase [26]: [V,V1] 60.40/16.50 60.40/16.50 #### Partial ranking functions of CR gr(V1,V,Out) 60.40/16.50 * Partial RF of phase [26]: 60.40/16.50 - RF of loop [26:1]: 60.40/16.50 V 60.40/16.50 V1 60.40/16.50 60.40/16.50 60.40/16.50 ### Specialization of cost equations p/2 60.40/16.50 * CE 15 is refined into CE [33] 60.40/16.50 * CE 14 is refined into CE [34] 60.40/16.50 60.40/16.50 60.40/16.50 ### Cost equations --> "Loop" of p/2 60.40/16.50 * CEs [33] --> Loop 27 60.40/16.50 * CEs [34] --> Loop 28 60.40/16.50 60.40/16.50 ### Ranking functions of CR p(V1,Out) 60.40/16.50 60.40/16.50 #### Partial ranking functions of CR p(V1,Out) 60.40/16.50 60.40/16.50 60.40/16.50 ### Specialization of cost equations cond2/4 60.40/16.50 * CE 13 is refined into CE [35,36,37,38] 60.40/16.50 * CE 12 is refined into CE [39] 60.40/16.50 * CE 11 is refined into CE [40,41,42,43,44,45,46] 60.40/16.50 60.40/16.50 60.40/16.50 ### Cost equations --> "Loop" of cond2/4 60.40/16.50 * CEs [45] --> Loop 29 60.40/16.50 * CEs [44] --> Loop 30 60.40/16.50 * CEs [46] --> Loop 31 60.40/16.50 * CEs [43] --> Loop 32 60.40/16.50 * CEs [41,42] --> Loop 33 60.40/16.50 * CEs [40] --> Loop 34 60.40/16.50 * CEs [38] --> Loop 35 60.40/16.50 * CEs [36] --> Loop 36 60.40/16.50 * CEs [37] --> Loop 37 60.40/16.50 * CEs [35] --> Loop 38 60.40/16.50 * CEs [39] --> Loop 39 60.40/16.50 60.40/16.50 ### Ranking functions of CR cond2(V1,V,V2,Out) 60.40/16.50 * RF of phase [35]: [V,V2] 60.40/16.50 * RF of phase [37]: [V2] 60.40/16.50 60.40/16.50 #### Partial ranking functions of CR cond2(V1,V,V2,Out) 60.40/16.50 * Partial RF of phase [35]: 60.40/16.50 - RF of loop [35:1]: 60.40/16.50 V 60.40/16.50 V2 60.40/16.50 * Partial RF of phase [37]: 60.40/16.50 - RF of loop [37:1]: 60.40/16.50 V2 60.40/16.50 60.40/16.50 60.40/16.50 ### Specialization of cost equations start/3 60.40/16.50 * CE 1 is refined into CE [47] 60.40/16.50 * CE 2 is refined into CE [48,49,50,51,52,53,54] 60.40/16.50 * CE 3 is refined into CE [55,56,57,58,59,60,61,62,63,64,65,66,67,68] 60.40/16.50 * CE 4 is refined into CE [69,70,71,72] 60.40/16.50 * CE 5 is refined into CE [73,74] 60.40/16.50 * CE 6 is refined into CE [75,76,77,78,79,80] 60.40/16.50 * CE 7 is refined into CE [81,82,83] 60.40/16.50 60.40/16.50 60.40/16.50 ### Cost equations --> "Loop" of start/3 60.40/16.50 * CEs [54,68] --> Loop 40 60.40/16.50 * CEs [53,67] --> Loop 41 60.40/16.50 * CEs [52,66] --> Loop 42 60.40/16.50 * CEs [51,65] --> Loop 43 60.40/16.50 * CEs [49,64] --> Loop 44 60.40/16.50 * CEs [63,71,72,74,78,79,80,82] --> Loop 45 60.40/16.50 * CEs [47,50,62] --> Loop 46 60.40/16.50 * CEs [48,61,70,77,83] --> Loop 47 60.40/16.50 * CEs [55,56,57,58,59,60,69,73,75,76,81] --> Loop 48 60.40/16.50 60.40/16.50 ### Ranking functions of CR start(V1,V,V2) 60.40/16.50 60.40/16.50 #### Partial ranking functions of CR start(V1,V,V2) 60.40/16.50 60.40/16.50 60.40/16.50 Computing Bounds 60.40/16.50 ===================================== 60.40/16.50 60.40/16.50 #### Cost of chains of and(V1,V,Out): 60.40/16.50 * Chain [19]: 1 60.40/16.50 with precondition: [V1=0,Out=0,V>=0] 60.40/16.50 60.40/16.50 * Chain [18]: 1 60.40/16.50 with precondition: [V1=1,V=1,Out=1] 60.40/16.50 60.40/16.50 * Chain [17]: 1 60.40/16.50 with precondition: [V=0,Out=0,V1>=0] 60.40/16.50 60.40/16.50 60.40/16.50 #### Cost of chains of eq(V1,V,Out): 60.40/16.50 * Chain [[23],22]: 1*it(23)+1 60.40/16.50 Such that:it(23) =< V1 60.40/16.50 60.40/16.50 with precondition: [Out=1,V1=V,V1>=1] 60.40/16.50 60.40/16.50 * Chain [[23],21]: 1*it(23)+1 60.40/16.50 Such that:it(23) =< V1 60.40/16.50 60.40/16.50 with precondition: [Out=0,V1>=1,V>=V1+1] 60.40/16.50 60.40/16.50 * Chain [[23],20]: 1*it(23)+1 60.40/16.50 Such that:it(23) =< V 60.40/16.50 60.40/16.50 with precondition: [Out=0,V>=1,V1>=V+1] 60.40/16.50 60.40/16.50 * Chain [22]: 1 60.40/16.50 with precondition: [V1=0,V=0,Out=1] 60.40/16.50 60.40/16.50 * Chain [21]: 1 60.40/16.50 with precondition: [V1=0,Out=0,V>=1] 60.40/16.50 60.40/16.50 * Chain [20]: 1 60.40/16.50 with precondition: [V=0,Out=0,V1>=1] 60.40/16.50 60.40/16.50 60.40/16.50 #### Cost of chains of gr(V1,V,Out): 60.40/16.50 * Chain [[26],25]: 1*it(26)+1 60.40/16.50 Such that:it(26) =< V1 60.40/16.50 60.40/16.50 with precondition: [Out=0,V1>=1,V>=V1] 60.40/16.50 60.40/16.50 * Chain [[26],24]: 1*it(26)+1 60.40/16.50 Such that:it(26) =< V 60.40/16.50 60.40/16.50 with precondition: [Out=1,V>=1,V1>=V+1] 60.40/16.50 60.40/16.50 * Chain [25]: 1 60.40/16.50 with precondition: [V1=0,Out=0,V>=0] 60.40/16.50 60.40/16.50 * Chain [24]: 1 60.40/16.50 with precondition: [V=0,Out=1,V1>=1] 60.40/16.50 60.40/16.50 60.40/16.50 #### Cost of chains of p(V1,Out): 60.40/16.50 * Chain [28]: 1 60.40/16.50 with precondition: [V1=0,Out=0] 60.40/16.50 60.40/16.50 * Chain [27]: 1 60.40/16.50 with precondition: [V1=Out+1,V1>=1] 60.40/16.50 60.40/16.50 60.40/16.50 #### Cost of chains of cond2(V1,V,V2,Out): 60.40/16.50 * Chain [[37],38,34]: 4*it(37)+8 60.40/16.50 Such that:it(37) =< V2 60.40/16.50 60.40/16.50 with precondition: [V1=1,V=0,Out=0,V2>=1] 60.40/16.50 60.40/16.50 * Chain [[35],[37],38,34]: 4*it(35)+4*it(37)+8 60.40/16.50 Such that:it(37) =< -V+V2 60.40/16.50 it(35) =< V 60.40/16.50 60.40/16.50 with precondition: [V1=1,Out=0,V>=1,V2>=V+1] 60.40/16.50 60.40/16.50 * Chain [[35],38,34]: 4*it(35)+8 60.40/16.50 Such that:it(35) =< V 60.40/16.50 60.40/16.50 with precondition: [V1=1,Out=0,V=V2,V>=1] 60.40/16.50 60.40/16.50 * Chain [[35],36,34]: 4*it(35)+8 60.40/16.50 Such that:it(35) =< V 60.40/16.50 60.40/16.50 with precondition: [V1=1,Out=0,V=V2+1,V>=2] 60.40/16.50 60.40/16.50 * Chain [[35],36,32]: 4*it(35)+8 60.40/16.50 Such that:it(35) =< V2 60.40/16.50 60.40/16.50 with precondition: [V1=1,Out=0,V2>=1,V>=V2+2] 60.40/16.50 60.40/16.50 * Chain [39,[35],38,34]: 5*it(35)+14 60.40/16.50 Such that:aux(1) =< V2 60.40/16.50 it(35) =< aux(1) 60.40/16.50 60.40/16.50 with precondition: [V1=0,Out=0,V=V2,V>=1] 60.40/16.50 60.40/16.50 * Chain [38,34]: 8 60.40/16.50 with precondition: [V1=1,V=0,V2=0,Out=0] 60.40/16.50 60.40/16.50 * Chain [36,34]: 8 60.40/16.50 with precondition: [V1=1,V=1,V2=0,Out=0] 60.40/16.50 60.40/16.50 * Chain [36,32]: 8 60.40/16.50 with precondition: [V1=1,V2=0,Out=0,V>=2] 60.40/16.50 60.40/16.50 * Chain [34]: 4 60.40/16.50 with precondition: [V1=0,V=0,V2=0,Out=0] 60.40/16.50 60.40/16.50 * Chain [33]: 4 60.40/16.50 with precondition: [V1=0,V=0,Out=0,V2>=1] 60.40/16.50 60.40/16.50 * Chain [32]: 4 60.40/16.50 with precondition: [V1=0,V2=0,Out=0,V>=1] 60.40/16.50 60.40/16.50 * Chain [31]: 1*s(2)+4 60.40/16.50 Such that:s(2) =< V2 60.40/16.50 60.40/16.50 with precondition: [V1=0,Out=0,V=V2,V>=1] 60.40/16.50 60.40/16.50 * Chain [30]: 1*s(3)+4 60.40/16.50 Such that:s(3) =< V 60.40/16.50 60.40/16.50 with precondition: [V1=0,Out=0,V>=1,V2>=V+1] 60.40/16.50 60.40/16.50 * Chain [29]: 1*s(4)+4 60.40/16.50 Such that:s(4) =< V2 60.40/16.50 60.40/16.50 with precondition: [V1=0,Out=0,V2>=1,V>=V2+1] 60.40/16.50 60.40/16.50 60.40/16.50 #### Cost of chains of start(V1,V,V2): 60.40/16.50 * Chain [48]: 7*s(9)+1*s(10)+14 60.40/16.50 Such that:s(10) =< V 60.40/16.50 aux(3) =< V2 60.40/16.50 s(9) =< aux(3) 60.40/16.50 60.40/16.50 with precondition: [V1=0] 60.40/16.50 60.40/16.50 * Chain [47]: 8 60.40/16.50 with precondition: [V=0,V1>=0] 60.40/16.50 60.40/16.50 * Chain [46]: 8*s(12)+10 60.40/16.50 Such that:aux(4) =< V2 60.40/16.50 s(12) =< aux(4) 60.40/16.50 60.40/16.50 with precondition: [V1>=0,V>=0,V2>=0] 60.40/16.50 60.40/16.50 * Chain [45]: 2*s(14)+3*s(15)+8 60.40/16.50 Such that:aux(5) =< V1 60.40/16.50 aux(6) =< V 60.40/16.50 s(14) =< aux(5) 60.40/16.50 s(15) =< aux(6) 60.40/16.50 60.40/16.50 with precondition: [V1>=1] 60.40/16.50 60.40/16.50 * Chain [44]: 8 60.40/16.50 with precondition: [V1=1,V2=0,V>=1] 60.40/16.50 60.40/16.50 * Chain [43]: 8*s(19)+10 60.40/16.50 Such that:aux(7) =< V2 60.40/16.50 s(19) =< aux(7) 60.40/16.50 60.40/16.50 with precondition: [V1=1,V=V2,V>=1] 60.40/16.50 60.40/16.50 * Chain [42]: 8*s(21)+10 60.40/16.50 Such that:aux(8) =< V2+1 60.40/16.50 s(21) =< aux(8) 60.40/16.50 60.40/16.50 with precondition: [V1=1,V=V2+1,V>=2] 60.40/16.50 60.40/16.50 * Chain [41]: 8*s(23)+8*s(24)+10 60.40/16.50 Such that:aux(9) =< -V+V2 60.40/16.50 aux(10) =< V 60.40/16.50 s(23) =< aux(9) 60.40/16.50 s(24) =< aux(10) 60.40/16.50 60.40/16.50 with precondition: [V1=1,V>=1,V2>=V+1] 60.40/16.50 60.40/16.50 * Chain [40]: 8*s(27)+10 60.40/16.50 Such that:aux(11) =< V2 60.40/16.50 s(27) =< aux(11) 60.40/16.50 60.40/16.50 with precondition: [V1=1,V2>=1,V>=V2+2] 60.40/16.50 60.40/16.50 60.40/16.50 Closed-form bounds of start(V1,V,V2): 60.40/16.50 ------------------------------------- 60.40/16.50 * Chain [48] with precondition: [V1=0] 60.40/16.50 - Upper bound: nat(V)+14+nat(V2)*7 60.40/16.50 - Complexity: n 60.40/16.50 * Chain [47] with precondition: [V=0,V1>=0] 60.40/16.50 - Upper bound: 8 60.40/16.50 - Complexity: constant 60.40/16.50 * Chain [46] with precondition: [V1>=0,V>=0,V2>=0] 60.40/16.50 - Upper bound: 8*V2+10 60.40/16.50 - Complexity: n 60.40/16.50 * Chain [45] with precondition: [V1>=1] 60.40/16.50 - Upper bound: 2*V1+8+nat(V)*3 60.40/16.50 - Complexity: n 60.40/16.50 * Chain [44] with precondition: [V1=1,V2=0,V>=1] 60.40/16.50 - Upper bound: 8 60.40/16.50 - Complexity: constant 60.40/16.50 * Chain [43] with precondition: [V1=1,V=V2,V>=1] 60.40/16.50 - Upper bound: 8*V2+10 60.40/16.50 - Complexity: n 60.40/16.50 * Chain [42] with precondition: [V1=1,V=V2+1,V>=2] 60.40/16.50 - Upper bound: 8*V2+18 60.40/16.50 - Complexity: n 60.40/16.50 * Chain [41] with precondition: [V1=1,V>=1,V2>=V+1] 60.40/16.50 - Upper bound: 8*V2+10 60.40/16.50 - Complexity: n 60.40/16.50 * Chain [40] with precondition: [V1=1,V2>=1,V>=V2+2] 60.40/16.50 - Upper bound: 8*V2+10 60.40/16.50 - Complexity: n 60.40/16.50 60.40/16.50 ### Maximum cost of start(V1,V,V2): max([max([nat(V2)*8+2,nat(V2+1)*8+2]),nat(V)+max([nat(V)*2+max([2*V1,nat(V)*5+2+nat(-V+V2)*8]),nat(V2)*7+6])])+8 60.40/16.50 Asymptotic class: n 60.40/16.50 * Total analysis performed in 487 ms. 60.40/16.50 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (10) 60.40/16.50 BOUNDS(1, n^1) 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 60.40/16.50 Renamed function symbols to avoid clashes with predefined symbol. 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (12) 60.40/16.50 Obligation: 60.40/16.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 60.40/16.50 60.40/16.50 60.40/16.50 The TRS R consists of the following rules: 60.40/16.50 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0'), x, y) 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y)) 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y) 60.40/16.50 gr(0', x) -> false 60.40/16.50 gr(s(x), 0') -> true 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) 60.40/16.50 p(0') -> 0' 60.40/16.50 p(s(x)) -> x 60.40/16.50 eq(0', 0') -> true 60.40/16.50 eq(s(x), 0') -> false 60.40/16.50 eq(0', s(x)) -> false 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) 60.40/16.50 and(true, true) -> true 60.40/16.50 and(false, x) -> false 60.40/16.50 and(x, false) -> false 60.40/16.50 60.40/16.50 S is empty. 60.40/16.50 Rewrite Strategy: INNERMOST 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 60.40/16.50 Infered types. 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (14) 60.40/16.50 Obligation: 60.40/16.50 Innermost TRS: 60.40/16.50 Rules: 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0'), x, y) 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y)) 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y) 60.40/16.50 gr(0', x) -> false 60.40/16.50 gr(s(x), 0') -> true 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) 60.40/16.50 p(0') -> 0' 60.40/16.50 p(s(x)) -> x 60.40/16.50 eq(0', 0') -> true 60.40/16.50 eq(s(x), 0') -> false 60.40/16.50 eq(0', s(x)) -> false 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) 60.40/16.50 and(true, true) -> true 60.40/16.50 and(false, x) -> false 60.40/16.50 and(x, false) -> false 60.40/16.50 60.40/16.50 Types: 60.40/16.50 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 true :: true:false 60.40/16.50 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 gr :: 0':s -> 0':s -> true:false 60.40/16.50 0' :: 0':s 60.40/16.50 p :: 0':s -> 0':s 60.40/16.50 false :: true:false 60.40/16.50 and :: true:false -> true:false -> true:false 60.40/16.50 eq :: 0':s -> 0':s -> true:false 60.40/16.50 s :: 0':s -> 0':s 60.40/16.50 hole_cond1:cond21_0 :: cond1:cond2 60.40/16.50 hole_true:false2_0 :: true:false 60.40/16.50 hole_0':s3_0 :: 0':s 60.40/16.50 gen_0':s4_0 :: Nat -> 0':s 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (15) OrderProof (LOWER BOUND(ID)) 60.40/16.50 Heuristically decided to analyse the following defined symbols: 60.40/16.50 cond1, cond2, gr, eq 60.40/16.50 60.40/16.50 They will be analysed ascendingly in the following order: 60.40/16.50 cond1 = cond2 60.40/16.50 gr < cond1 60.40/16.50 gr < cond2 60.40/16.50 eq < cond2 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (16) 60.40/16.50 Obligation: 60.40/16.50 Innermost TRS: 60.40/16.50 Rules: 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0'), x, y) 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y)) 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y) 60.40/16.50 gr(0', x) -> false 60.40/16.50 gr(s(x), 0') -> true 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) 60.40/16.50 p(0') -> 0' 60.40/16.50 p(s(x)) -> x 60.40/16.50 eq(0', 0') -> true 60.40/16.50 eq(s(x), 0') -> false 60.40/16.50 eq(0', s(x)) -> false 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) 60.40/16.50 and(true, true) -> true 60.40/16.50 and(false, x) -> false 60.40/16.50 and(x, false) -> false 60.40/16.50 60.40/16.50 Types: 60.40/16.50 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 true :: true:false 60.40/16.50 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 gr :: 0':s -> 0':s -> true:false 60.40/16.50 0' :: 0':s 60.40/16.50 p :: 0':s -> 0':s 60.40/16.50 false :: true:false 60.40/16.50 and :: true:false -> true:false -> true:false 60.40/16.50 eq :: 0':s -> 0':s -> true:false 60.40/16.50 s :: 0':s -> 0':s 60.40/16.50 hole_cond1:cond21_0 :: cond1:cond2 60.40/16.50 hole_true:false2_0 :: true:false 60.40/16.50 hole_0':s3_0 :: 0':s 60.40/16.50 gen_0':s4_0 :: Nat -> 0':s 60.40/16.50 60.40/16.50 60.40/16.50 Generator Equations: 60.40/16.50 gen_0':s4_0(0) <=> 0' 60.40/16.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 60.40/16.50 60.40/16.50 60.40/16.50 The following defined symbols remain to be analysed: 60.40/16.50 gr, cond1, cond2, eq 60.40/16.50 60.40/16.50 They will be analysed ascendingly in the following order: 60.40/16.50 cond1 = cond2 60.40/16.50 gr < cond1 60.40/16.50 gr < cond2 60.40/16.50 eq < cond2 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (17) RewriteLemmaProof (LOWER BOUND(ID)) 60.40/16.50 Proved the following rewrite lemma: 60.40/16.50 gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 60.40/16.50 60.40/16.50 Induction Base: 60.40/16.50 gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 60.40/16.50 false 60.40/16.50 60.40/16.50 Induction Step: 60.40/16.50 gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1) 60.40/16.50 gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH 60.40/16.50 false 60.40/16.50 60.40/16.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (18) 60.40/16.50 Complex Obligation (BEST) 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (19) 60.40/16.50 Obligation: 60.40/16.50 Proved the lower bound n^1 for the following obligation: 60.40/16.50 60.40/16.50 Innermost TRS: 60.40/16.50 Rules: 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0'), x, y) 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y)) 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y) 60.40/16.50 gr(0', x) -> false 60.40/16.50 gr(s(x), 0') -> true 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) 60.40/16.50 p(0') -> 0' 60.40/16.50 p(s(x)) -> x 60.40/16.50 eq(0', 0') -> true 60.40/16.50 eq(s(x), 0') -> false 60.40/16.50 eq(0', s(x)) -> false 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) 60.40/16.50 and(true, true) -> true 60.40/16.50 and(false, x) -> false 60.40/16.50 and(x, false) -> false 60.40/16.50 60.40/16.50 Types: 60.40/16.50 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 true :: true:false 60.40/16.50 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 gr :: 0':s -> 0':s -> true:false 60.40/16.50 0' :: 0':s 60.40/16.50 p :: 0':s -> 0':s 60.40/16.50 false :: true:false 60.40/16.50 and :: true:false -> true:false -> true:false 60.40/16.50 eq :: 0':s -> 0':s -> true:false 60.40/16.50 s :: 0':s -> 0':s 60.40/16.50 hole_cond1:cond21_0 :: cond1:cond2 60.40/16.50 hole_true:false2_0 :: true:false 60.40/16.50 hole_0':s3_0 :: 0':s 60.40/16.50 gen_0':s4_0 :: Nat -> 0':s 60.40/16.50 60.40/16.50 60.40/16.50 Generator Equations: 60.40/16.50 gen_0':s4_0(0) <=> 0' 60.40/16.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 60.40/16.50 60.40/16.50 60.40/16.50 The following defined symbols remain to be analysed: 60.40/16.50 gr, cond1, cond2, eq 60.40/16.50 60.40/16.50 They will be analysed ascendingly in the following order: 60.40/16.50 cond1 = cond2 60.40/16.50 gr < cond1 60.40/16.50 gr < cond2 60.40/16.50 eq < cond2 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (20) LowerBoundPropagationProof (FINISHED) 60.40/16.50 Propagated lower bound. 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (21) 60.40/16.50 BOUNDS(n^1, INF) 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (22) 60.40/16.50 Obligation: 60.40/16.50 Innermost TRS: 60.40/16.50 Rules: 60.40/16.50 cond1(true, x, y) -> cond2(gr(y, 0'), x, y) 60.40/16.50 cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y)) 60.40/16.50 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y) 60.40/16.50 gr(0', x) -> false 60.40/16.50 gr(s(x), 0') -> true 60.40/16.50 gr(s(x), s(y)) -> gr(x, y) 60.40/16.50 p(0') -> 0' 60.40/16.50 p(s(x)) -> x 60.40/16.50 eq(0', 0') -> true 60.40/16.50 eq(s(x), 0') -> false 60.40/16.50 eq(0', s(x)) -> false 60.40/16.50 eq(s(x), s(y)) -> eq(x, y) 60.40/16.50 and(true, true) -> true 60.40/16.50 and(false, x) -> false 60.40/16.50 and(x, false) -> false 60.40/16.50 60.40/16.50 Types: 60.40/16.50 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 true :: true:false 60.40/16.50 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.50 gr :: 0':s -> 0':s -> true:false 60.40/16.50 0' :: 0':s 60.40/16.50 p :: 0':s -> 0':s 60.40/16.50 false :: true:false 60.40/16.50 and :: true:false -> true:false -> true:false 60.40/16.50 eq :: 0':s -> 0':s -> true:false 60.40/16.50 s :: 0':s -> 0':s 60.40/16.50 hole_cond1:cond21_0 :: cond1:cond2 60.40/16.50 hole_true:false2_0 :: true:false 60.40/16.50 hole_0':s3_0 :: 0':s 60.40/16.50 gen_0':s4_0 :: Nat -> 0':s 60.40/16.50 60.40/16.50 60.40/16.50 Lemmas: 60.40/16.50 gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 60.40/16.50 60.40/16.50 60.40/16.50 Generator Equations: 60.40/16.50 gen_0':s4_0(0) <=> 0' 60.40/16.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 60.40/16.50 60.40/16.50 60.40/16.50 The following defined symbols remain to be analysed: 60.40/16.50 eq, cond1, cond2 60.40/16.50 60.40/16.50 They will be analysed ascendingly in the following order: 60.40/16.50 cond1 = cond2 60.40/16.50 eq < cond2 60.40/16.50 60.40/16.50 ---------------------------------------- 60.40/16.50 60.40/16.50 (23) RewriteLemmaProof (LOWER BOUND(ID)) 60.40/16.50 Proved the following rewrite lemma: 60.40/16.50 eq(gen_0':s4_0(n295_0), gen_0':s4_0(n295_0)) -> true, rt in Omega(1 + n295_0) 60.40/16.50 60.40/16.50 Induction Base: 60.40/16.50 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 60.40/16.50 true 60.40/16.50 60.40/16.50 Induction Step: 60.40/16.50 eq(gen_0':s4_0(+(n295_0, 1)), gen_0':s4_0(+(n295_0, 1))) ->_R^Omega(1) 60.40/16.50 eq(gen_0':s4_0(n295_0), gen_0':s4_0(n295_0)) ->_IH 60.40/16.50 true 60.40/16.50 60.40/16.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 60.40/16.51 ---------------------------------------- 60.40/16.51 60.40/16.51 (24) 60.40/16.51 Obligation: 60.40/16.51 Innermost TRS: 60.40/16.51 Rules: 60.40/16.51 cond1(true, x, y) -> cond2(gr(y, 0'), x, y) 60.40/16.51 cond2(true, x, y) -> cond2(gr(y, 0'), p(x), p(y)) 60.40/16.51 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0')), x, y) 60.40/16.51 gr(0', x) -> false 60.40/16.51 gr(s(x), 0') -> true 60.40/16.51 gr(s(x), s(y)) -> gr(x, y) 60.40/16.51 p(0') -> 0' 60.40/16.51 p(s(x)) -> x 60.40/16.51 eq(0', 0') -> true 60.40/16.51 eq(s(x), 0') -> false 60.40/16.51 eq(0', s(x)) -> false 60.40/16.51 eq(s(x), s(y)) -> eq(x, y) 60.40/16.51 and(true, true) -> true 60.40/16.51 and(false, x) -> false 60.40/16.51 and(x, false) -> false 60.40/16.51 60.40/16.51 Types: 60.40/16.51 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.51 true :: true:false 60.40/16.51 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2 60.40/16.51 gr :: 0':s -> 0':s -> true:false 60.40/16.51 0' :: 0':s 60.40/16.51 p :: 0':s -> 0':s 60.40/16.51 false :: true:false 60.40/16.51 and :: true:false -> true:false -> true:false 60.40/16.51 eq :: 0':s -> 0':s -> true:false 60.40/16.51 s :: 0':s -> 0':s 60.40/16.51 hole_cond1:cond21_0 :: cond1:cond2 60.40/16.51 hole_true:false2_0 :: true:false 60.40/16.51 hole_0':s3_0 :: 0':s 60.40/16.51 gen_0':s4_0 :: Nat -> 0':s 60.40/16.51 60.40/16.51 60.40/16.51 Lemmas: 60.40/16.51 gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 60.40/16.51 eq(gen_0':s4_0(n295_0), gen_0':s4_0(n295_0)) -> true, rt in Omega(1 + n295_0) 60.40/16.51 60.40/16.51 60.40/16.51 Generator Equations: 60.40/16.51 gen_0':s4_0(0) <=> 0' 60.40/16.51 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 60.40/16.51 60.40/16.51 60.40/16.51 The following defined symbols remain to be analysed: 60.40/16.51 cond2, cond1 60.40/16.51 60.40/16.51 They will be analysed ascendingly in the following order: 60.40/16.51 cond1 = cond2 60.50/16.55 EOF