901.97/291.52 WORST_CASE(Omega(n^1), O(n^2)) 901.97/291.54 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 901.97/291.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 901.97/291.54 901.97/291.54 901.97/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 901.97/291.54 901.97/291.54 (0) CpxTRS 901.97/291.54 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 901.97/291.54 (2) CpxWeightedTrs 901.97/291.54 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 901.97/291.54 (4) CpxTypedWeightedTrs 901.97/291.54 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 901.97/291.54 (6) CpxTypedWeightedCompleteTrs 901.97/291.54 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 901.97/291.54 (8) CpxRNTS 901.97/291.54 (9) CompleteCoflocoProof [FINISHED, 239 ms] 901.97/291.54 (10) BOUNDS(1, n^2) 901.97/291.54 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 901.97/291.54 (12) TRS for Loop Detection 901.97/291.54 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 901.97/291.54 (14) BEST 901.97/291.54 (15) proven lower bound 901.97/291.54 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 901.97/291.54 (17) BOUNDS(n^1, INF) 901.97/291.54 (18) TRS for Loop Detection 901.97/291.54 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (0) 901.97/291.54 Obligation: 901.97/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 901.97/291.54 901.97/291.54 901.97/291.54 The TRS R consists of the following rules: 901.97/291.54 901.97/291.54 le(0, y) -> true 901.97/291.54 le(s(x), 0) -> false 901.97/291.54 le(s(x), s(y)) -> le(x, y) 901.97/291.54 int(x, y) -> if(le(x, y), x, y) 901.97/291.54 if(true, x, y) -> cons(x, int(s(x), y)) 901.97/291.54 if(false, x, y) -> nil 901.97/291.54 901.97/291.54 S is empty. 901.97/291.54 Rewrite Strategy: INNERMOST 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 901.97/291.54 Transformed relative TRS to weighted TRS 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (2) 901.97/291.54 Obligation: 901.97/291.54 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 901.97/291.54 901.97/291.54 901.97/291.54 The TRS R consists of the following rules: 901.97/291.54 901.97/291.54 le(0, y) -> true [1] 901.97/291.54 le(s(x), 0) -> false [1] 901.97/291.54 le(s(x), s(y)) -> le(x, y) [1] 901.97/291.54 int(x, y) -> if(le(x, y), x, y) [1] 901.97/291.54 if(true, x, y) -> cons(x, int(s(x), y)) [1] 901.97/291.54 if(false, x, y) -> nil [1] 901.97/291.54 901.97/291.54 Rewrite Strategy: INNERMOST 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 901.97/291.54 Infered types. 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (4) 901.97/291.54 Obligation: 901.97/291.54 Runtime Complexity Weighted TRS with Types. 901.97/291.54 The TRS R consists of the following rules: 901.97/291.54 901.97/291.54 le(0, y) -> true [1] 901.97/291.54 le(s(x), 0) -> false [1] 901.97/291.54 le(s(x), s(y)) -> le(x, y) [1] 901.97/291.54 int(x, y) -> if(le(x, y), x, y) [1] 901.97/291.54 if(true, x, y) -> cons(x, int(s(x), y)) [1] 901.97/291.54 if(false, x, y) -> nil [1] 901.97/291.54 901.97/291.54 The TRS has the following type information: 901.97/291.54 le :: 0:s -> 0:s -> true:false 901.97/291.54 0 :: 0:s 901.97/291.54 true :: true:false 901.97/291.54 s :: 0:s -> 0:s 901.97/291.54 false :: true:false 901.97/291.54 int :: 0:s -> 0:s -> cons:nil 901.97/291.54 if :: true:false -> 0:s -> 0:s -> cons:nil 901.97/291.54 cons :: 0:s -> cons:nil -> cons:nil 901.97/291.54 nil :: cons:nil 901.97/291.54 901.97/291.54 Rewrite Strategy: INNERMOST 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (5) CompletionProof (UPPER BOUND(ID)) 901.97/291.54 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 901.97/291.54 none 901.97/291.54 901.97/291.54 And the following fresh constants: none 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (6) 901.97/291.54 Obligation: 901.97/291.54 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 901.97/291.54 901.97/291.54 Runtime Complexity Weighted TRS with Types. 901.97/291.54 The TRS R consists of the following rules: 901.97/291.54 901.97/291.54 le(0, y) -> true [1] 901.97/291.54 le(s(x), 0) -> false [1] 901.97/291.54 le(s(x), s(y)) -> le(x, y) [1] 901.97/291.54 int(x, y) -> if(le(x, y), x, y) [1] 901.97/291.54 if(true, x, y) -> cons(x, int(s(x), y)) [1] 901.97/291.54 if(false, x, y) -> nil [1] 901.97/291.54 901.97/291.54 The TRS has the following type information: 901.97/291.54 le :: 0:s -> 0:s -> true:false 901.97/291.54 0 :: 0:s 901.97/291.54 true :: true:false 901.97/291.54 s :: 0:s -> 0:s 901.97/291.54 false :: true:false 901.97/291.54 int :: 0:s -> 0:s -> cons:nil 901.97/291.54 if :: true:false -> 0:s -> 0:s -> cons:nil 901.97/291.54 cons :: 0:s -> cons:nil -> cons:nil 901.97/291.54 nil :: cons:nil 901.97/291.54 901.97/291.54 Rewrite Strategy: INNERMOST 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 901.97/291.54 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 901.97/291.54 The constant constructors are abstracted as follows: 901.97/291.54 901.97/291.54 0 => 0 901.97/291.54 true => 1 901.97/291.54 false => 0 901.97/291.54 nil => 0 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (8) 901.97/291.54 Obligation: 901.97/291.54 Complexity RNTS consisting of the following rules: 901.97/291.54 901.97/291.54 if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 901.97/291.54 if(z, z', z'') -{ 1 }-> 1 + x + int(1 + x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 901.97/291.54 int(z, z') -{ 1 }-> if(le(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y 901.97/291.54 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 901.97/291.54 le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y 901.97/291.54 le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 901.97/291.54 901.97/291.54 Only complete derivations are relevant for the runtime complexity. 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (9) CompleteCoflocoProof (FINISHED) 901.97/291.54 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 901.97/291.54 901.97/291.54 eq(start(V1, V, V10),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]). 901.97/291.54 eq(start(V1, V, V10),0,[int(V1, V, Out)],[V1 >= 0,V >= 0]). 901.97/291.54 eq(start(V1, V, V10),0,[if(V1, V, V10, Out)],[V1 >= 0,V >= 0,V10 >= 0]). 901.97/291.54 eq(le(V1, V, Out),1,[],[Out = 1,V2 >= 0,V1 = 0,V = V2]). 901.97/291.54 eq(le(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = 1 + V3,V = 0]). 901.97/291.54 eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). 901.97/291.54 eq(int(V1, V, Out),1,[le(V7, V6, Ret0),if(Ret0, V7, V6, Ret1)],[Out = Ret1,V7 >= 0,V6 >= 0,V1 = V7,V = V6]). 901.97/291.54 eq(if(V1, V, V10, Out),1,[int(1 + V8, V9, Ret11)],[Out = 1 + Ret11 + V8,V = V8,V10 = V9,V1 = 1,V8 >= 0,V9 >= 0]). 901.97/291.54 eq(if(V1, V, V10, Out),1,[],[Out = 0,V = V12,V10 = V11,V12 >= 0,V11 >= 0,V1 = 0]). 901.97/291.54 input_output_vars(le(V1,V,Out),[V1,V],[Out]). 901.97/291.54 input_output_vars(int(V1,V,Out),[V1,V],[Out]). 901.97/291.54 input_output_vars(if(V1,V,V10,Out),[V1,V,V10],[Out]). 901.97/291.54 901.97/291.54 901.97/291.54 CoFloCo proof output: 901.97/291.54 Preprocessing Cost Relations 901.97/291.54 ===================================== 901.97/291.54 901.97/291.54 #### Computed strongly connected components 901.97/291.54 0. recursive : [le/3] 901.97/291.54 1. recursive : [if/4,int/3] 901.97/291.54 2. non_recursive : [start/3] 901.97/291.54 901.97/291.54 #### Obtained direct recursion through partial evaluation 901.97/291.54 0. SCC is partially evaluated into le/3 901.97/291.54 1. SCC is partially evaluated into int/3 901.97/291.54 2. SCC is partially evaluated into start/3 901.97/291.54 901.97/291.54 Control-Flow Refinement of Cost Relations 901.97/291.54 ===================================== 901.97/291.54 901.97/291.54 ### Specialization of cost equations le/3 901.97/291.54 * CE 9 is refined into CE [10] 901.97/291.54 * CE 8 is refined into CE [11] 901.97/291.54 * CE 7 is refined into CE [12] 901.97/291.54 901.97/291.54 901.97/291.54 ### Cost equations --> "Loop" of le/3 901.97/291.54 * CEs [11] --> Loop 8 901.97/291.54 * CEs [12] --> Loop 9 901.97/291.54 * CEs [10] --> Loop 10 901.97/291.54 901.97/291.54 ### Ranking functions of CR le(V1,V,Out) 901.97/291.54 * RF of phase [10]: [V,V1] 901.97/291.54 901.97/291.54 #### Partial ranking functions of CR le(V1,V,Out) 901.97/291.54 * Partial RF of phase [10]: 901.97/291.54 - RF of loop [10:1]: 901.97/291.54 V 901.97/291.54 V1 901.97/291.54 901.97/291.54 901.97/291.54 ### Specialization of cost equations int/3 901.97/291.54 * CE 6 is refined into CE [13,14] 901.97/291.54 * CE 5 is refined into CE [15,16] 901.97/291.54 901.97/291.54 901.97/291.54 ### Cost equations --> "Loop" of int/3 901.97/291.54 * CEs [16] --> Loop 11 901.97/291.54 * CEs [15] --> Loop 12 901.97/291.54 * CEs [14] --> Loop 13 901.97/291.54 * CEs [13] --> Loop 14 901.97/291.54 901.97/291.54 ### Ranking functions of CR int(V1,V,Out) 901.97/291.54 * RF of phase [13]: [-V1+V+1] 901.97/291.54 901.97/291.54 #### Partial ranking functions of CR int(V1,V,Out) 901.97/291.54 * Partial RF of phase [13]: 901.97/291.54 - RF of loop [13:1]: 901.97/291.54 -V1+V+1 901.97/291.54 901.97/291.54 901.97/291.54 ### Specialization of cost equations start/3 901.97/291.54 * CE 2 is refined into CE [17,18,19] 901.97/291.54 * CE 1 is refined into CE [20] 901.97/291.54 * CE 3 is refined into CE [21,22,23,24] 901.97/291.54 * CE 4 is refined into CE [25,26,27,28,29] 901.97/291.54 901.97/291.54 901.97/291.54 ### Cost equations --> "Loop" of start/3 901.97/291.54 * CEs [23,28] --> Loop 15 901.97/291.54 * CEs [22,27] --> Loop 16 901.97/291.54 * CEs [18,24,29] --> Loop 17 901.97/291.54 * CEs [19] --> Loop 18 901.97/291.54 * CEs [17] --> Loop 19 901.97/291.54 * CEs [20,21,25,26] --> Loop 20 901.97/291.54 901.97/291.54 ### Ranking functions of CR start(V1,V,V10) 901.97/291.54 901.97/291.54 #### Partial ranking functions of CR start(V1,V,V10) 901.97/291.54 901.97/291.54 901.97/291.54 Computing Bounds 901.97/291.54 ===================================== 901.97/291.54 901.97/291.54 #### Cost of chains of le(V1,V,Out): 901.97/291.54 * Chain [[10],9]: 1*it(10)+1 901.97/291.54 Such that:it(10) =< V1 901.97/291.54 901.97/291.54 with precondition: [Out=1,V1>=1,V>=V1] 901.97/291.54 901.97/291.54 * Chain [[10],8]: 1*it(10)+1 901.97/291.54 Such that:it(10) =< V 901.97/291.54 901.97/291.54 with precondition: [Out=0,V>=1,V1>=V+1] 901.97/291.54 901.97/291.54 * Chain [9]: 1 901.97/291.54 with precondition: [V1=0,Out=1,V>=0] 901.97/291.54 901.97/291.54 * Chain [8]: 1 901.97/291.54 with precondition: [V=0,Out=0,V1>=1] 901.97/291.54 901.97/291.54 901.97/291.54 #### Cost of chains of int(V1,V,Out): 901.97/291.54 * Chain [[13],11]: 3*it(13)+1*s(1)+1*s(4)+3 901.97/291.54 Such that:it(13) =< -V1+V+1 901.97/291.54 s(1) =< V 901.97/291.54 aux(1) =< V+1 901.97/291.54 s(4) =< it(13)*aux(1) 901.97/291.54 901.97/291.54 with precondition: [V1>=1,V>=V1,Out+2*V1>=3*V+1] 901.97/291.54 901.97/291.54 * Chain [14,[13],11]: 4*it(13)+1*s(4)+6 901.97/291.54 Such that:aux(1) =< V+1 901.97/291.54 aux(2) =< V 901.97/291.54 it(13) =< aux(2) 901.97/291.54 s(4) =< it(13)*aux(1) 901.97/291.54 901.97/291.54 with precondition: [V1=0,V>=1,Out>=3*V] 901.97/291.54 901.97/291.54 * Chain [14,12]: 6 901.97/291.54 with precondition: [V1=0,V=0,Out=1] 901.97/291.54 901.97/291.54 * Chain [12]: 3 901.97/291.54 with precondition: [V=0,Out=0,V1>=1] 901.97/291.54 901.97/291.54 * Chain [11]: 1*s(1)+3 901.97/291.54 Such that:s(1) =< V 901.97/291.54 901.97/291.54 with precondition: [Out=0,V>=1,V1>=V+1] 901.97/291.54 901.97/291.54 901.97/291.54 #### Cost of chains of start(V1,V,V10): 901.97/291.54 * Chain [20]: 4*s(7)+1*s(8)+6 901.97/291.54 Such that:s(6) =< V 901.97/291.54 s(5) =< V+1 901.97/291.54 s(7) =< s(6) 901.97/291.54 s(8) =< s(7)*s(5) 901.97/291.54 901.97/291.54 with precondition: [V1=0,V>=0] 901.97/291.54 901.97/291.54 * Chain [19]: 4 901.97/291.54 with precondition: [V1=1,V10=0,V>=0] 901.97/291.54 901.97/291.54 * Chain [18]: 3*s(9)+1*s(10)+1*s(12)+4 901.97/291.54 Such that:s(9) =< -V+V10 901.97/291.54 s(10) =< V10 901.97/291.54 s(11) =< V10+1 901.97/291.54 s(12) =< s(9)*s(11) 901.97/291.54 901.97/291.54 with precondition: [V1=1,V>=0,V10>=V+1] 901.97/291.54 901.97/291.54 * Chain [17]: 1*s(13)+1*s(14)+3*s(15)+1*s(16)+1*s(18)+4 901.97/291.54 Such that:s(15) =< -V1+V+1 901.97/291.54 s(14) =< V1 901.97/291.54 s(16) =< V 901.97/291.54 s(17) =< V+1 901.97/291.54 s(13) =< V10 901.97/291.54 s(18) =< s(15)*s(17) 901.97/291.54 901.97/291.54 with precondition: [V1>=1,V>=V1] 901.97/291.54 901.97/291.54 * Chain [16]: 3 901.97/291.54 with precondition: [V=0,V1>=1] 901.97/291.54 901.97/291.54 * Chain [15]: 2*s(19)+3 901.97/291.54 Such that:aux(3) =< V 901.97/291.54 s(19) =< aux(3) 901.97/291.54 901.97/291.54 with precondition: [V>=1,V1>=V+1] 901.97/291.54 901.97/291.54 901.97/291.54 Closed-form bounds of start(V1,V,V10): 901.97/291.54 ------------------------------------- 901.97/291.54 * Chain [20] with precondition: [V1=0,V>=0] 901.97/291.54 - Upper bound: 4*V+6+(V+1)*V 901.97/291.54 - Complexity: n^2 901.97/291.54 * Chain [19] with precondition: [V1=1,V10=0,V>=0] 901.97/291.54 - Upper bound: 4 901.97/291.54 - Complexity: constant 901.97/291.54 * Chain [18] with precondition: [V1=1,V>=0,V10>=V+1] 901.97/291.54 - Upper bound: V10+4+(-V+V10)*(V10+1)+(-3*V+3*V10) 901.97/291.54 - Complexity: n^2 901.97/291.54 * Chain [17] with precondition: [V1>=1,V>=V1] 901.97/291.54 - Upper bound: V1+V+4+nat(V10)+(-V1+V+1)*(V+1)+(-3*V1+3*V+3) 901.97/291.54 - Complexity: n^2 901.97/291.54 * Chain [16] with precondition: [V=0,V1>=1] 901.97/291.54 - Upper bound: 3 901.97/291.54 - Complexity: constant 901.97/291.54 * Chain [15] with precondition: [V>=1,V1>=V+1] 901.97/291.54 - Upper bound: 2*V+3 901.97/291.54 - Complexity: n 901.97/291.54 901.97/291.54 ### Maximum cost of start(V1,V,V10): max([max([1,nat(V10)+1+nat(-V+V10)*nat(V10+1)+nat(-V+V10)*3]),max([2*V+3+(V+1)*V+V,V1+1+nat(V10)+(V+1)*nat(-V1+V+1)+nat(-V1+V+1)*3])+V])+3 901.97/291.54 Asymptotic class: n^2 901.97/291.54 * Total analysis performed in 170 ms. 901.97/291.54 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (10) 901.97/291.54 BOUNDS(1, n^2) 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 901.97/291.54 Transformed a relative TRS into a decreasing-loop problem. 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (12) 901.97/291.54 Obligation: 901.97/291.54 Analyzing the following TRS for decreasing loops: 901.97/291.54 901.97/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 901.97/291.54 901.97/291.54 901.97/291.54 The TRS R consists of the following rules: 901.97/291.54 901.97/291.54 le(0, y) -> true 901.97/291.54 le(s(x), 0) -> false 901.97/291.54 le(s(x), s(y)) -> le(x, y) 901.97/291.54 int(x, y) -> if(le(x, y), x, y) 901.97/291.54 if(true, x, y) -> cons(x, int(s(x), y)) 901.97/291.54 if(false, x, y) -> nil 901.97/291.54 901.97/291.54 S is empty. 901.97/291.54 Rewrite Strategy: INNERMOST 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (13) DecreasingLoopProof (LOWER BOUND(ID)) 901.97/291.54 The following loop(s) give(s) rise to the lower bound Omega(n^1): 901.97/291.54 901.97/291.54 The rewrite sequence 901.97/291.54 901.97/291.54 le(s(x), s(y)) ->^+ le(x, y) 901.97/291.54 901.97/291.54 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 901.97/291.54 901.97/291.54 The pumping substitution is [x / s(x), y / s(y)]. 901.97/291.54 901.97/291.54 The result substitution is [ ]. 901.97/291.54 901.97/291.54 901.97/291.54 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (14) 901.97/291.54 Complex Obligation (BEST) 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (15) 901.97/291.54 Obligation: 901.97/291.54 Proved the lower bound n^1 for the following obligation: 901.97/291.54 901.97/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 901.97/291.54 901.97/291.54 901.97/291.54 The TRS R consists of the following rules: 901.97/291.54 901.97/291.54 le(0, y) -> true 901.97/291.54 le(s(x), 0) -> false 901.97/291.54 le(s(x), s(y)) -> le(x, y) 901.97/291.54 int(x, y) -> if(le(x, y), x, y) 901.97/291.54 if(true, x, y) -> cons(x, int(s(x), y)) 901.97/291.54 if(false, x, y) -> nil 901.97/291.54 901.97/291.54 S is empty. 901.97/291.54 Rewrite Strategy: INNERMOST 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (16) LowerBoundPropagationProof (FINISHED) 901.97/291.54 Propagated lower bound. 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (17) 901.97/291.54 BOUNDS(n^1, INF) 901.97/291.54 901.97/291.54 ---------------------------------------- 901.97/291.54 901.97/291.54 (18) 901.97/291.54 Obligation: 901.97/291.54 Analyzing the following TRS for decreasing loops: 901.97/291.54 901.97/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 901.97/291.54 901.97/291.54 901.97/291.54 The TRS R consists of the following rules: 901.97/291.54 901.97/291.54 le(0, y) -> true 901.97/291.54 le(s(x), 0) -> false 901.97/291.54 le(s(x), s(y)) -> le(x, y) 901.97/291.54 int(x, y) -> if(le(x, y), x, y) 901.97/291.54 if(true, x, y) -> cons(x, int(s(x), y)) 901.97/291.54 if(false, x, y) -> nil 901.97/291.54 901.97/291.54 S is empty. 901.97/291.54 Rewrite Strategy: INNERMOST 902.25/291.58 EOF