882.99/291.63 WORST_CASE(Omega(n^1), O(n^2)) 882.99/291.64 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 882.99/291.64 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 882.99/291.64 882.99/291.64 882.99/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.99/291.64 882.99/291.64 (0) CpxTRS 882.99/291.64 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 882.99/291.64 (2) CpxTRS 882.99/291.64 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 1 ms] 882.99/291.64 (4) CpxWeightedTrs 882.99/291.64 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 882.99/291.64 (6) CpxTypedWeightedTrs 882.99/291.64 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 882.99/291.64 (8) CpxTypedWeightedCompleteTrs 882.99/291.64 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 882.99/291.64 (10) CpxRNTS 882.99/291.64 (11) CompleteCoflocoProof [FINISHED, 218 ms] 882.99/291.64 (12) BOUNDS(1, n^2) 882.99/291.64 (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 882.99/291.64 (14) TRS for Loop Detection 882.99/291.64 (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 882.99/291.64 (16) BEST 882.99/291.64 (17) proven lower bound 882.99/291.64 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 882.99/291.64 (19) BOUNDS(n^1, INF) 882.99/291.64 (20) TRS for Loop Detection 882.99/291.64 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (0) 882.99/291.64 Obligation: 882.99/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.99/291.64 882.99/291.64 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) 882.99/291.64 cond(false, x, y) -> 0 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) 882.99/291.64 ge(u, 0) -> true 882.99/291.64 ge(0, s(v)) -> false 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) 882.99/291.64 882.99/291.64 S is empty. 882.99/291.64 Rewrite Strategy: FULL 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 882.99/291.64 Converted rc-obligation to irc-obligation. 882.99/291.64 882.99/291.64 The duplicating contexts are: 882.99/291.64 minus([], y) 882.99/291.64 minus(x, []) 882.99/291.64 882.99/291.64 882.99/291.64 The defined contexts are: 882.99/291.64 cond([], x1, x2) 882.99/291.64 882.99/291.64 882.99/291.64 [] just represents basic- or constructor-terms in the following defined contexts: 882.99/291.64 cond([], x1, x2) 882.99/291.64 882.99/291.64 882.99/291.64 As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (2) 882.99/291.64 Obligation: 882.99/291.64 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). 882.99/291.64 882.99/291.64 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) 882.99/291.64 cond(false, x, y) -> 0 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) 882.99/291.64 ge(u, 0) -> true 882.99/291.64 ge(0, s(v)) -> false 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) 882.99/291.64 882.99/291.64 S is empty. 882.99/291.64 Rewrite Strategy: INNERMOST 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 882.99/291.64 Transformed relative TRS to weighted TRS 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (4) 882.99/291.64 Obligation: 882.99/291.64 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 882.99/291.64 882.99/291.64 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) [1] 882.99/291.64 cond(false, x, y) -> 0 [1] 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) [1] 882.99/291.64 ge(u, 0) -> true [1] 882.99/291.64 ge(0, s(v)) -> false [1] 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) [1] 882.99/291.64 882.99/291.64 Rewrite Strategy: INNERMOST 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 882.99/291.64 Infered types. 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (6) 882.99/291.64 Obligation: 882.99/291.64 Runtime Complexity Weighted TRS with Types. 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) [1] 882.99/291.64 cond(false, x, y) -> 0 [1] 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) [1] 882.99/291.64 ge(u, 0) -> true [1] 882.99/291.64 ge(0, s(v)) -> false [1] 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) [1] 882.99/291.64 882.99/291.64 The TRS has the following type information: 882.99/291.64 minus :: s:0 -> s:0 -> s:0 882.99/291.64 cond :: false:true -> s:0 -> s:0 -> s:0 882.99/291.64 ge :: s:0 -> s:0 -> false:true 882.99/291.64 s :: s:0 -> s:0 882.99/291.64 false :: false:true 882.99/291.64 0 :: s:0 882.99/291.64 true :: false:true 882.99/291.64 882.99/291.64 Rewrite Strategy: INNERMOST 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (7) CompletionProof (UPPER BOUND(ID)) 882.99/291.64 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 882.99/291.64 none 882.99/291.64 882.99/291.64 And the following fresh constants: none 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (8) 882.99/291.64 Obligation: 882.99/291.64 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 882.99/291.64 882.99/291.64 Runtime Complexity Weighted TRS with Types. 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) [1] 882.99/291.64 cond(false, x, y) -> 0 [1] 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) [1] 882.99/291.64 ge(u, 0) -> true [1] 882.99/291.64 ge(0, s(v)) -> false [1] 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) [1] 882.99/291.64 882.99/291.64 The TRS has the following type information: 882.99/291.64 minus :: s:0 -> s:0 -> s:0 882.99/291.64 cond :: false:true -> s:0 -> s:0 -> s:0 882.99/291.64 ge :: s:0 -> s:0 -> false:true 882.99/291.64 s :: s:0 -> s:0 882.99/291.64 false :: false:true 882.99/291.64 0 :: s:0 882.99/291.64 true :: false:true 882.99/291.64 882.99/291.64 Rewrite Strategy: INNERMOST 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 882.99/291.64 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 882.99/291.64 The constant constructors are abstracted as follows: 882.99/291.64 882.99/291.64 false => 0 882.99/291.64 0 => 0 882.99/291.64 true => 1 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (10) 882.99/291.64 Obligation: 882.99/291.64 Complexity RNTS consisting of the following rules: 882.99/291.64 882.99/291.64 cond(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 882.99/291.64 cond(z, z', z'') -{ 1 }-> 1 + minus(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 882.99/291.64 ge(z, z') -{ 1 }-> ge(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 882.99/291.64 ge(z, z') -{ 1 }-> 1 :|: z = u, z' = 0, u >= 0 882.99/291.64 ge(z, z') -{ 1 }-> 0 :|: v >= 0, z' = 1 + v, z = 0 882.99/291.64 minus(z, z') -{ 1 }-> cond(ge(x, 1 + y), x, y) :|: x >= 0, y >= 0, z = x, z' = y 882.99/291.64 882.99/291.64 Only complete derivations are relevant for the runtime complexity. 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (11) CompleteCoflocoProof (FINISHED) 882.99/291.64 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 882.99/291.64 882.99/291.64 eq(start(V1, V, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 882.99/291.64 eq(start(V1, V, V5),0,[cond(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]). 882.99/291.64 eq(start(V1, V, V5),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]). 882.99/291.64 eq(minus(V1, V, Out),1,[ge(V3, 1 + V2, Ret0),cond(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]). 882.99/291.64 eq(cond(V1, V, V5, Out),1,[],[Out = 0,V = V4,V5 = V6,V4 >= 0,V6 >= 0,V1 = 0]). 882.99/291.64 eq(cond(V1, V, V5, Out),1,[minus(V8, 1 + V7, Ret1)],[Out = 1 + Ret1,V = V8,V5 = V7,V1 = 1,V8 >= 0,V7 >= 0]). 882.99/291.64 eq(ge(V1, V, Out),1,[],[Out = 1,V1 = V9,V = 0,V9 >= 0]). 882.99/291.64 eq(ge(V1, V, Out),1,[],[Out = 0,V10 >= 0,V = 1 + V10,V1 = 0]). 882.99/291.64 eq(ge(V1, V, Out),1,[ge(V12, V11, Ret2)],[Out = Ret2,V11 >= 0,V = 1 + V11,V1 = 1 + V12,V12 >= 0]). 882.99/291.64 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 882.99/291.64 input_output_vars(cond(V1,V,V5,Out),[V1,V,V5],[Out]). 882.99/291.64 input_output_vars(ge(V1,V,Out),[V1,V],[Out]). 882.99/291.64 882.99/291.64 882.99/291.64 CoFloCo proof output: 882.99/291.64 Preprocessing Cost Relations 882.99/291.64 ===================================== 882.99/291.64 882.99/291.64 #### Computed strongly connected components 882.99/291.64 0. recursive : [ge/3] 882.99/291.64 1. recursive : [cond/4,minus/3] 882.99/291.64 2. non_recursive : [start/3] 882.99/291.64 882.99/291.64 #### Obtained direct recursion through partial evaluation 882.99/291.64 0. SCC is partially evaluated into ge/3 882.99/291.64 1. SCC is partially evaluated into minus/3 882.99/291.64 2. SCC is partially evaluated into start/3 882.99/291.64 882.99/291.64 Control-Flow Refinement of Cost Relations 882.99/291.64 ===================================== 882.99/291.64 882.99/291.64 ### Specialization of cost equations ge/3 882.99/291.64 * CE 9 is refined into CE [10] 882.99/291.64 * CE 7 is refined into CE [11] 882.99/291.64 * CE 8 is refined into CE [12] 882.99/291.64 882.99/291.64 882.99/291.64 ### Cost equations --> "Loop" of ge/3 882.99/291.64 * CEs [11] --> Loop 8 882.99/291.64 * CEs [12] --> Loop 9 882.99/291.64 * CEs [10] --> Loop 10 882.99/291.64 882.99/291.64 ### Ranking functions of CR ge(V1,V,Out) 882.99/291.64 * RF of phase [10]: [V,V1] 882.99/291.64 882.99/291.64 #### Partial ranking functions of CR ge(V1,V,Out) 882.99/291.64 * Partial RF of phase [10]: 882.99/291.64 - RF of loop [10:1]: 882.99/291.64 V 882.99/291.64 V1 882.99/291.64 882.99/291.64 882.99/291.64 ### Specialization of cost equations minus/3 882.99/291.64 * CE 6 is refined into CE [13,14] 882.99/291.64 * CE 5 is refined into CE [15] 882.99/291.64 882.99/291.64 882.99/291.64 ### Cost equations --> "Loop" of minus/3 882.99/291.64 * CEs [15] --> Loop 11 882.99/291.64 * CEs [14] --> Loop 12 882.99/291.64 * CEs [13] --> Loop 13 882.99/291.64 882.99/291.64 ### Ranking functions of CR minus(V1,V,Out) 882.99/291.64 * RF of phase [11]: [V1-V] 882.99/291.64 882.99/291.64 #### Partial ranking functions of CR minus(V1,V,Out) 882.99/291.64 * Partial RF of phase [11]: 882.99/291.64 - RF of loop [11:1]: 882.99/291.64 V1-V 882.99/291.64 882.99/291.64 882.99/291.64 ### Specialization of cost equations start/3 882.99/291.64 * CE 1 is refined into CE [16,17,18] 882.99/291.64 * CE 2 is refined into CE [19] 882.99/291.64 * CE 3 is refined into CE [20,21,22] 882.99/291.64 * CE 4 is refined into CE [23,24,25,26] 882.99/291.64 882.99/291.64 882.99/291.64 ### Cost equations --> "Loop" of start/3 882.99/291.64 * CEs [26] --> Loop 14 882.99/291.64 * CEs [22] --> Loop 15 882.99/291.64 * CEs [18] --> Loop 16 882.99/291.64 * CEs [17,21,25] --> Loop 17 882.99/291.64 * CEs [16,24] --> Loop 18 882.99/291.64 * CEs [19,20,23] --> Loop 19 882.99/291.64 882.99/291.64 ### Ranking functions of CR start(V1,V,V5) 882.99/291.64 882.99/291.64 #### Partial ranking functions of CR start(V1,V,V5) 882.99/291.64 882.99/291.64 882.99/291.64 Computing Bounds 882.99/291.64 ===================================== 882.99/291.64 882.99/291.64 #### Cost of chains of ge(V1,V,Out): 882.99/291.64 * Chain [[10],9]: 1*it(10)+1 882.99/291.64 Such that:it(10) =< V1 882.99/291.64 882.99/291.64 with precondition: [Out=0,V1>=1,V>=V1+1] 882.99/291.64 882.99/291.64 * Chain [[10],8]: 1*it(10)+1 882.99/291.64 Such that:it(10) =< V 882.99/291.64 882.99/291.64 with precondition: [Out=1,V>=1,V1>=V] 882.99/291.64 882.99/291.64 * Chain [9]: 1 882.99/291.64 with precondition: [V1=0,Out=0,V>=1] 882.99/291.64 882.99/291.64 * Chain [8]: 1 882.99/291.64 with precondition: [V=0,Out=1,V1>=0] 882.99/291.64 882.99/291.64 882.99/291.64 #### Cost of chains of minus(V1,V,Out): 882.99/291.64 * Chain [[11],12]: 3*it(11)+1*s(1)+1*s(4)+3 882.99/291.64 Such that:it(11) =< Out 882.99/291.64 aux(2) =< V+Out 882.99/291.64 s(1) =< aux(2) 882.99/291.64 s(4) =< it(11)*aux(2) 882.99/291.64 882.99/291.64 with precondition: [V1=Out+V,V>=0,V1>=V+1] 882.99/291.64 882.99/291.64 * Chain [13]: 3 882.99/291.64 with precondition: [V1=0,Out=0,V>=0] 882.99/291.64 882.99/291.64 * Chain [12]: 1*s(1)+3 882.99/291.64 Such that:s(1) =< V1 882.99/291.64 882.99/291.64 with precondition: [Out=0,V1>=1,V>=V1] 882.99/291.64 882.99/291.64 882.99/291.64 #### Cost of chains of start(V1,V,V5): 882.99/291.64 * Chain [19]: 3 882.99/291.64 with precondition: [V1=0,V>=0] 882.99/291.64 882.99/291.64 * Chain [18]: 4 882.99/291.64 with precondition: [V=0,V1>=0] 882.99/291.64 882.99/291.64 * Chain [17]: 1*s(5)+2*s(6)+4 882.99/291.64 Such that:s(5) =< V 882.99/291.64 aux(3) =< V1 882.99/291.64 s(6) =< aux(3) 882.99/291.64 882.99/291.64 with precondition: [V1>=1,V>=V1] 882.99/291.64 882.99/291.64 * Chain [16]: 3*s(8)+1*s(10)+1*s(11)+4 882.99/291.64 Such that:s(9) =< V 882.99/291.64 s(8) =< V-V5 882.99/291.64 s(10) =< s(9) 882.99/291.64 s(11) =< s(8)*s(9) 882.99/291.64 882.99/291.64 with precondition: [V1=1,V5>=0,V>=V5+2] 882.99/291.64 882.99/291.64 * Chain [15]: 3*s(12)+1*s(14)+1*s(15)+3 882.99/291.64 Such that:s(13) =< V1 882.99/291.64 s(12) =< V1-V 882.99/291.64 s(14) =< s(13) 882.99/291.64 s(15) =< s(12)*s(13) 882.99/291.64 882.99/291.64 with precondition: [V>=0,V1>=V+1] 882.99/291.64 882.99/291.64 * Chain [14]: 1*s(16)+1 882.99/291.64 Such that:s(16) =< V 882.99/291.64 882.99/291.64 with precondition: [V>=1,V1>=V] 882.99/291.64 882.99/291.64 882.99/291.64 Closed-form bounds of start(V1,V,V5): 882.99/291.64 ------------------------------------- 882.99/291.64 * Chain [19] with precondition: [V1=0,V>=0] 882.99/291.64 - Upper bound: 3 882.99/291.64 - Complexity: constant 882.99/291.64 * Chain [18] with precondition: [V=0,V1>=0] 882.99/291.64 - Upper bound: 4 882.99/291.64 - Complexity: constant 882.99/291.64 * Chain [17] with precondition: [V1>=1,V>=V1] 882.99/291.64 - Upper bound: 2*V1+V+4 882.99/291.64 - Complexity: n 882.99/291.64 * Chain [16] with precondition: [V1=1,V5>=0,V>=V5+2] 882.99/291.64 - Upper bound: 3*V-3*V5+(V+4+(V-V5)*V) 882.99/291.64 - Complexity: n^2 882.99/291.64 * Chain [15] with precondition: [V>=0,V1>=V+1] 882.99/291.64 - Upper bound: 3*V1-3*V+(V1+3+(V1-V)*V1) 882.99/291.64 - Complexity: n^2 882.99/291.64 * Chain [14] with precondition: [V>=1,V1>=V] 882.99/291.64 - Upper bound: V+1 882.99/291.64 - Complexity: n 882.99/291.64 882.99/291.64 ### Maximum cost of start(V1,V,V5): max([max([1,nat(V1-V)*V1+V1+nat(V1-V)*3])+2,max([2*V1+3,nat(V-V5)*V+3+nat(V-V5)*3])+V])+1 882.99/291.64 Asymptotic class: n^2 882.99/291.64 * Total analysis performed in 148 ms. 882.99/291.64 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (12) 882.99/291.64 BOUNDS(1, n^2) 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 882.99/291.64 Transformed a relative TRS into a decreasing-loop problem. 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (14) 882.99/291.64 Obligation: 882.99/291.64 Analyzing the following TRS for decreasing loops: 882.99/291.64 882.99/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.99/291.64 882.99/291.64 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) 882.99/291.64 cond(false, x, y) -> 0 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) 882.99/291.64 ge(u, 0) -> true 882.99/291.64 ge(0, s(v)) -> false 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) 882.99/291.64 882.99/291.64 S is empty. 882.99/291.64 Rewrite Strategy: FULL 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (15) DecreasingLoopProof (LOWER BOUND(ID)) 882.99/291.64 The following loop(s) give(s) rise to the lower bound Omega(n^1): 882.99/291.64 882.99/291.64 The rewrite sequence 882.99/291.64 882.99/291.64 ge(s(u), s(v)) ->^+ ge(u, v) 882.99/291.64 882.99/291.64 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 882.99/291.64 882.99/291.64 The pumping substitution is [u / s(u), v / s(v)]. 882.99/291.64 882.99/291.64 The result substitution is [ ]. 882.99/291.64 882.99/291.64 882.99/291.64 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (16) 882.99/291.64 Complex Obligation (BEST) 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (17) 882.99/291.64 Obligation: 882.99/291.64 Proved the lower bound n^1 for the following obligation: 882.99/291.64 882.99/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.99/291.64 882.99/291.64 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) 882.99/291.64 cond(false, x, y) -> 0 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) 882.99/291.64 ge(u, 0) -> true 882.99/291.64 ge(0, s(v)) -> false 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) 882.99/291.64 882.99/291.64 S is empty. 882.99/291.64 Rewrite Strategy: FULL 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (18) LowerBoundPropagationProof (FINISHED) 882.99/291.64 Propagated lower bound. 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (19) 882.99/291.64 BOUNDS(n^1, INF) 882.99/291.64 882.99/291.64 ---------------------------------------- 882.99/291.64 882.99/291.64 (20) 882.99/291.64 Obligation: 882.99/291.64 Analyzing the following TRS for decreasing loops: 882.99/291.64 882.99/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.99/291.64 882.99/291.64 882.99/291.64 The TRS R consists of the following rules: 882.99/291.64 882.99/291.64 minus(x, y) -> cond(ge(x, s(y)), x, y) 882.99/291.64 cond(false, x, y) -> 0 882.99/291.64 cond(true, x, y) -> s(minus(x, s(y))) 882.99/291.64 ge(u, 0) -> true 882.99/291.64 ge(0, s(v)) -> false 882.99/291.64 ge(s(u), s(v)) -> ge(u, v) 882.99/291.64 882.99/291.64 S is empty. 882.99/291.64 Rewrite Strategy: FULL 882.99/291.68 EOF