WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 105 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 329 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. The certificate found is represented by the following graph. 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(2914,2720,[2_1|4]), (2914,2738,[2_1|4]), (2914,2759,[2_1|4]), (2914,2783,[2_1|4]), (2914,2810,[2_1|4]), (2915,2916,[2_1|4]), (2916,2917,[1_1|4]), (2917,2918,[1_1|4]), (2918,2919,[0_1|4]), (2919,2920,[1_1|4]), (2920,2921,[2_1|4]), (2921,2922,[0_1|4]), (2922,2923,[1_1|4]), (2923,2924,[2_1|4]), (2924,2925,[0_1|4]), (2925,2926,[1_1|4]), (2926,2927,[2_1|4]), (2927,2928,[0_1|4]), (2928,2929,[1_1|4]), (2929,2930,[2_1|4]), (2930,2931,[0_1|4]), (2931,2932,[1_1|4]), (2932,2933,[2_1|4]), (2933,2934,[0_1|4]), (2934,2935,[1_1|4]), (2935,2936,[2_1|4]), (2936,2937,[0_1|4]), (2937,2938,[1_1|4]), (2938,2684,[2_1|4]), (2938,2693,[2_1|4]), (2938,2705,[2_1|4]), (2938,2720,[2_1|4]), (2938,2738,[2_1|4]), (2938,2759,[2_1|4]), (2938,2783,[2_1|4]), (2938,2810,[2_1|4]), (2939,2940,[2_1|4]), (2940,2941,[1_1|4]), (2941,2942,[1_1|4]), (2942,2943,[0_1|4]), (2943,2944,[1_1|4]), (2944,2945,[2_1|4]), (2945,2946,[0_1|4]), (2946,2947,[1_1|4]), (2947,2948,[2_1|4]), (2948,2949,[0_1|4]), (2949,2950,[1_1|4]), (2950,2951,[2_1|4]), (2951,2952,[0_1|4]), (2952,2953,[1_1|4]), (2953,2954,[2_1|4]), (2954,2955,[0_1|4]), (2955,2956,[1_1|4]), (2956,2957,[2_1|4]), (2957,2958,[0_1|4]), (2958,2959,[1_1|4]), (2959,2960,[2_1|4]), (2960,2961,[0_1|4]), (2961,2962,[1_1|4]), (2962,2963,[2_1|4]), (2963,2964,[0_1|4]), (2964,2965,[1_1|4]), (2965,2684,[2_1|4]), (2965,2693,[2_1|4]), (2965,2705,[2_1|4]), (2965,2720,[2_1|4]), (2965,2738,[2_1|4]), (2965,2759,[2_1|4]), (2965,2783,[2_1|4]), (2965,2810,[2_1|4]), (2966,2967,[2_1|4]), (2967,2968,[1_1|4]), (2968,2969,[1_1|4]), (2969,2970,[0_1|4]), (2970,2971,[1_1|4]), (2971,2972,[2_1|4]), (2972,2973,[0_1|4]), (2973,2974,[1_1|4]), (2974,2975,[2_1|4]), (2975,2976,[0_1|4]), (2976,2977,[1_1|4]), (2977,2978,[2_1|4]), (2978,2979,[0_1|4]), (2979,2980,[1_1|4]), (2980,2981,[2_1|4]), (2981,2982,[0_1|4]), (2982,2983,[1_1|4]), (2983,2984,[2_1|4]), (2984,2985,[0_1|4]), (2985,2986,[1_1|4]), (2986,2987,[2_1|4]), (2987,2988,[0_1|4]), (2988,2989,[1_1|4]), (2989,2990,[2_1|4]), (2990,2991,[0_1|4]), (2991,2992,[1_1|4]), (2992,2993,[2_1|4]), (2993,2994,[0_1|4]), (2994,2995,[1_1|4]), (2995,2684,[2_1|4]), (2995,2693,[2_1|4]), (2995,2705,[2_1|4]), (2995,2720,[2_1|4]), (2995,2738,[2_1|4]), (2995,2759,[2_1|4]), (2995,2783,[2_1|4]), (2995,2810,[2_1|4]), (2996,2997,[2_1|4]), (2997,2998,[1_1|4]), (2998,2999,[1_1|4]), (2999,3000,[0_1|4]), (3000,3001,[1_1|4]), (3001,3002,[2_1|4]), (3002,3003,[0_1|4]), (3003,3004,[1_1|4]), (3004,2840,[2_1|4]), (3004,2849,[2_1|4]), (3004,2861,[2_1|4]), (3004,2876,[2_1|4]), (3004,2894,[2_1|4]), (3004,2915,[2_1|4]), (3004,2939,[2_1|4]), (3004,2966,[2_1|4]), (3005,3006,[2_1|4]), (3006,3007,[1_1|4]), (3007,3008,[1_1|4]), (3008,3009,[0_1|4]), (3009,3010,[1_1|4]), (3010,3011,[2_1|4]), (3011,3012,[0_1|4]), (3012,3013,[1_1|4]), (3013,3014,[2_1|4]), (3014,3015,[0_1|4]), (3015,3016,[1_1|4]), (3016,2840,[2_1|4]), (3016,2849,[2_1|4]), (3016,2861,[2_1|4]), (3016,2876,[2_1|4]), (3016,2894,[2_1|4]), (3016,2915,[2_1|4]), (3016,2939,[2_1|4]), (3016,2966,[2_1|4]), (3017,3018,[2_1|4]), (3018,3019,[1_1|4]), (3019,3020,[1_1|4]), (3020,3021,[0_1|4]), (3021,3022,[1_1|4]), (3022,3023,[2_1|4]), (3023,3024,[0_1|4]), (3024,3025,[1_1|4]), (3025,3026,[2_1|4]), (3026,3027,[0_1|4]), (3027,3028,[1_1|4]), (3028,3029,[2_1|4]), (3029,3030,[0_1|4]), (3030,3031,[1_1|4]), (3031,2840,[2_1|4]), (3031,2849,[2_1|4]), (3031,2861,[2_1|4]), (3031,2876,[2_1|4]), (3031,2894,[2_1|4]), (3031,2915,[2_1|4]), (3031,2939,[2_1|4]), (3031,2966,[2_1|4]), (3032,3033,[2_1|4]), (3033,3034,[1_1|4]), (3034,3035,[1_1|4]), (3035,3036,[0_1|4]), (3036,3037,[1_1|4]), (3037,3038,[2_1|4]), (3038,3039,[0_1|4]), (3039,3040,[1_1|4]), (3040,3041,[2_1|4]), (3041,3042,[0_1|4]), (3042,3043,[1_1|4]), (3043,3044,[2_1|4]), (3044,3045,[0_1|4]), (3045,3046,[1_1|4]), (3046,3047,[2_1|4]), (3047,3048,[0_1|4]), (3048,3049,[1_1|4]), (3049,2840,[2_1|4]), (3049,2849,[2_1|4]), (3049,2861,[2_1|4]), (3049,2876,[2_1|4]), (3049,2894,[2_1|4]), (3049,2915,[2_1|4]), (3049,2939,[2_1|4]), (3049,2966,[2_1|4]), (3050,3051,[2_1|4]), (3051,3052,[1_1|4]), (3052,3053,[1_1|4]), (3053,3054,[0_1|4]), (3054,3055,[1_1|4]), (3055,3056,[2_1|4]), (3056,3057,[0_1|4]), (3057,3058,[1_1|4]), (3058,3059,[2_1|4]), (3059,3060,[0_1|4]), (3060,3061,[1_1|4]), (3061,3062,[2_1|4]), (3062,3063,[0_1|4]), (3063,3064,[1_1|4]), (3064,3065,[2_1|4]), (3065,3066,[0_1|4]), (3066,3067,[1_1|4]), (3067,3068,[2_1|4]), (3068,3069,[0_1|4]), (3069,3070,[1_1|4]), (3070,2840,[2_1|4]), (3070,2849,[2_1|4]), (3070,2861,[2_1|4]), (3070,2876,[2_1|4]), (3070,2894,[2_1|4]), (3070,2915,[2_1|4]), (3070,2939,[2_1|4]), (3070,2966,[2_1|4]), (3071,3072,[2_1|4]), (3072,3073,[1_1|4]), (3073,3074,[1_1|4]), (3074,3075,[0_1|4]), (3075,3076,[1_1|4]), (3076,3077,[2_1|4]), (3077,3078,[0_1|4]), (3078,3079,[1_1|4]), (3079,3080,[2_1|4]), (3080,3081,[0_1|4]), (3081,3082,[1_1|4]), (3082,3083,[2_1|4]), (3083,3084,[0_1|4]), (3084,3085,[1_1|4]), (3085,3086,[2_1|4]), (3086,3087,[0_1|4]), (3087,3088,[1_1|4]), (3088,3089,[2_1|4]), (3089,3090,[0_1|4]), (3090,3091,[1_1|4]), (3091,3092,[2_1|4]), (3092,3093,[0_1|4]), (3093,3094,[1_1|4]), (3094,2840,[2_1|4]), (3094,2849,[2_1|4]), (3094,2861,[2_1|4]), (3094,2876,[2_1|4]), (3094,2894,[2_1|4]), (3094,2915,[2_1|4]), (3094,2939,[2_1|4]), (3094,2966,[2_1|4]), (3095,3096,[2_1|4]), (3096,3097,[1_1|4]), (3097,3098,[1_1|4]), (3098,3099,[0_1|4]), (3099,3100,[1_1|4]), (3100,3101,[2_1|4]), (3101,3102,[0_1|4]), (3102,3103,[1_1|4]), (3103,3104,[2_1|4]), (3104,3105,[0_1|4]), (3105,3106,[1_1|4]), (3106,3107,[2_1|4]), (3107,3108,[0_1|4]), (3108,3109,[1_1|4]), (3109,3110,[2_1|4]), (3110,3111,[0_1|4]), (3111,3112,[1_1|4]), (3112,3113,[2_1|4]), (3113,3114,[0_1|4]), (3114,3115,[1_1|4]), (3115,3116,[2_1|4]), (3116,3117,[0_1|4]), (3117,3118,[1_1|4]), (3118,3119,[2_1|4]), (3119,3120,[0_1|4]), (3120,3121,[1_1|4]), (3121,2840,[2_1|4]), (3121,2849,[2_1|4]), (3121,2861,[2_1|4]), (3121,2876,[2_1|4]), (3121,2894,[2_1|4]), (3121,2915,[2_1|4]), (3121,2939,[2_1|4]), (3121,2966,[2_1|4]), (3122,3123,[2_1|4]), (3123,3124,[1_1|4]), (3124,3125,[1_1|4]), (3125,3126,[0_1|4]), (3126,3127,[1_1|4]), (3127,3128,[2_1|4]), (3128,3129,[0_1|4]), (3129,3130,[1_1|4]), (3130,3131,[2_1|4]), (3131,3132,[0_1|4]), (3132,3133,[1_1|4]), (3133,3134,[2_1|4]), (3134,3135,[0_1|4]), (3135,3136,[1_1|4]), (3136,3137,[2_1|4]), (3137,3138,[0_1|4]), (3138,3139,[1_1|4]), (3139,3140,[2_1|4]), (3140,3141,[0_1|4]), (3141,3142,[1_1|4]), (3142,3143,[2_1|4]), (3143,3144,[0_1|4]), (3144,3145,[1_1|4]), (3145,3146,[2_1|4]), (3146,3147,[0_1|4]), (3147,3148,[1_1|4]), (3148,3149,[2_1|4]), (3149,3150,[0_1|4]), (3150,3151,[1_1|4]), (3151,2840,[2_1|4]), (3151,2849,[2_1|4]), (3151,2861,[2_1|4]), (3151,2876,[2_1|4]), (3151,2894,[2_1|4]), (3151,2915,[2_1|4]), (3151,2939,[2_1|4]), (3151,2966,[2_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(2(1(x1)))) ->^+ 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]. The pumping substitution is [x1 / 1(x1)]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: FULL