21.48/6.55 WORST_CASE(Omega(n^1), O(n^1)) 21.48/6.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 21.48/6.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 21.48/6.56 21.48/6.56 21.48/6.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.48/6.56 21.48/6.56 (0) CpxTRS 21.48/6.56 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 21.48/6.56 (2) CpxWeightedTrs 21.48/6.56 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 21.48/6.56 (4) CpxTypedWeightedTrs 21.48/6.56 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 21.48/6.56 (6) CpxTypedWeightedCompleteTrs 21.48/6.56 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 21.48/6.56 (8) CpxRNTS 21.48/6.56 (9) CompleteCoflocoProof [FINISHED, 291 ms] 21.48/6.56 (10) BOUNDS(1, n^1) 21.48/6.56 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 21.48/6.56 (12) CpxTRS 21.48/6.56 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 21.48/6.56 (14) typed CpxTrs 21.48/6.56 (15) OrderProof [LOWER BOUND(ID), 0 ms] 21.48/6.56 (16) typed CpxTrs 21.48/6.56 (17) RewriteLemmaProof [LOWER BOUND(ID), 1186 ms] 21.48/6.56 (18) BEST 21.48/6.56 (19) proven lower bound 21.48/6.56 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 21.48/6.56 (21) BOUNDS(n^1, INF) 21.48/6.56 (22) typed CpxTrs 21.48/6.56 (23) RewriteLemmaProof [LOWER BOUND(ID), 0 ms] 21.48/6.56 (24) BOUNDS(1, INF) 21.48/6.56 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (0) 21.48/6.56 Obligation: 21.48/6.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.48/6.56 21.48/6.56 21.48/6.56 The TRS R consists of the following rules: 21.48/6.56 21.48/6.56 minus(X, 0) -> X 21.48/6.56 minus(s(X), s(Y)) -> p(minus(X, Y)) 21.48/6.56 p(s(X)) -> X 21.48/6.56 div(0, s(Y)) -> 0 21.48/6.56 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) 21.48/6.56 21.48/6.56 S is empty. 21.48/6.56 Rewrite Strategy: INNERMOST 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 21.48/6.56 Transformed relative TRS to weighted TRS 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (2) 21.48/6.56 Obligation: 21.48/6.56 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 21.48/6.56 21.48/6.56 21.48/6.56 The TRS R consists of the following rules: 21.48/6.56 21.48/6.56 minus(X, 0) -> X [1] 21.48/6.56 minus(s(X), s(Y)) -> p(minus(X, Y)) [1] 21.48/6.56 p(s(X)) -> X [1] 21.48/6.56 div(0, s(Y)) -> 0 [1] 21.48/6.56 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) [1] 21.48/6.56 21.48/6.56 Rewrite Strategy: INNERMOST 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 21.48/6.56 Infered types. 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (4) 21.48/6.56 Obligation: 21.48/6.56 Runtime Complexity Weighted TRS with Types. 21.48/6.56 The TRS R consists of the following rules: 21.48/6.56 21.48/6.56 minus(X, 0) -> X [1] 21.48/6.56 minus(s(X), s(Y)) -> p(minus(X, Y)) [1] 21.48/6.56 p(s(X)) -> X [1] 21.48/6.56 div(0, s(Y)) -> 0 [1] 21.48/6.56 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) [1] 21.48/6.56 21.48/6.56 The TRS has the following type information: 21.48/6.56 minus :: 0:s -> 0:s -> 0:s 21.48/6.56 0 :: 0:s 21.48/6.56 s :: 0:s -> 0:s 21.48/6.56 p :: 0:s -> 0:s 21.48/6.56 div :: 0:s -> 0:s -> 0:s 21.48/6.56 21.48/6.56 Rewrite Strategy: INNERMOST 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (5) CompletionProof (UPPER BOUND(ID)) 21.48/6.56 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 21.48/6.56 21.48/6.56 minus(v0, v1) -> null_minus [0] 21.48/6.56 p(v0) -> null_p [0] 21.48/6.56 div(v0, v1) -> null_div [0] 21.48/6.56 21.48/6.56 And the following fresh constants: null_minus, null_p, null_div 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (6) 21.48/6.56 Obligation: 21.48/6.56 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 21.48/6.56 21.48/6.56 Runtime Complexity Weighted TRS with Types. 21.48/6.56 The TRS R consists of the following rules: 21.48/6.56 21.48/6.56 minus(X, 0) -> X [1] 21.48/6.56 minus(s(X), s(Y)) -> p(minus(X, Y)) [1] 21.48/6.56 p(s(X)) -> X [1] 21.48/6.56 div(0, s(Y)) -> 0 [1] 21.48/6.56 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) [1] 21.48/6.56 minus(v0, v1) -> null_minus [0] 21.48/6.56 p(v0) -> null_p [0] 21.48/6.56 div(v0, v1) -> null_div [0] 21.48/6.56 21.48/6.56 The TRS has the following type information: 21.48/6.56 minus :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div 21.48/6.56 0 :: 0:s:null_minus:null_p:null_div 21.48/6.56 s :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div 21.48/6.56 p :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div 21.48/6.56 div :: 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div -> 0:s:null_minus:null_p:null_div 21.48/6.56 null_minus :: 0:s:null_minus:null_p:null_div 21.48/6.56 null_p :: 0:s:null_minus:null_p:null_div 21.48/6.56 null_div :: 0:s:null_minus:null_p:null_div 21.48/6.56 21.48/6.56 Rewrite Strategy: INNERMOST 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 21.48/6.56 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 21.48/6.56 The constant constructors are abstracted as follows: 21.48/6.56 21.48/6.56 0 => 0 21.48/6.56 null_minus => 0 21.48/6.56 null_p => 0 21.48/6.56 null_div => 0 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (8) 21.48/6.56 Obligation: 21.48/6.56 Complexity RNTS consisting of the following rules: 21.48/6.56 21.48/6.56 div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 21.48/6.56 div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 21.48/6.56 div(z, z') -{ 1 }-> 1 + div(minus(X, Y), 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 21.48/6.56 minus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0 21.48/6.56 minus(z, z') -{ 1 }-> p(minus(X, Y)) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 21.48/6.56 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 21.48/6.56 p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 21.48/6.56 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 21.48/6.56 21.48/6.56 Only complete derivations are relevant for the runtime complexity. 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (9) CompleteCoflocoProof (FINISHED) 21.48/6.56 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 21.48/6.56 21.48/6.56 eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 21.48/6.56 eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]). 21.48/6.56 eq(start(V1, V),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]). 21.48/6.56 eq(minus(V1, V, Out),1,[],[Out = X1,X1 >= 0,V1 = X1,V = 0]). 21.48/6.56 eq(minus(V1, V, Out),1,[minus(X2, Y1, Ret0),p(Ret0, Ret)],[Out = Ret,V1 = 1 + X2,Y1 >= 0,V = 1 + Y1,X2 >= 0]). 21.48/6.56 eq(p(V1, Out),1,[],[Out = X3,V1 = 1 + X3,X3 >= 0]). 21.48/6.56 eq(div(V1, V, Out),1,[],[Out = 0,Y2 >= 0,V = 1 + Y2,V1 = 0]). 21.48/6.56 eq(div(V1, V, Out),1,[minus(X4, Y3, Ret10),div(Ret10, 1 + Y3, Ret1)],[Out = 1 + Ret1,V1 = 1 + X4,Y3 >= 0,V = 1 + Y3,X4 >= 0]). 21.48/6.56 eq(minus(V1, V, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V1 = V3,V = V2]). 21.48/6.56 eq(p(V1, Out),0,[],[Out = 0,V4 >= 0,V1 = V4]). 21.48/6.56 eq(div(V1, V, Out),0,[],[Out = 0,V6 >= 0,V5 >= 0,V1 = V6,V = V5]). 21.48/6.56 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 21.48/6.56 input_output_vars(p(V1,Out),[V1],[Out]). 21.48/6.56 input_output_vars(div(V1,V,Out),[V1,V],[Out]). 21.48/6.56 21.48/6.56 21.48/6.56 CoFloCo proof output: 21.48/6.56 Preprocessing Cost Relations 21.48/6.56 ===================================== 21.48/6.56 21.48/6.56 #### Computed strongly connected components 21.48/6.56 0. non_recursive : [p/2] 21.48/6.56 1. recursive [non_tail] : [minus/3] 21.48/6.56 2. recursive : [(div)/3] 21.48/6.56 3. non_recursive : [start/2] 21.48/6.56 21.48/6.56 #### Obtained direct recursion through partial evaluation 21.48/6.56 0. SCC is partially evaluated into p/2 21.48/6.56 1. SCC is partially evaluated into minus/3 21.48/6.56 2. SCC is partially evaluated into (div)/3 21.48/6.56 3. SCC is partially evaluated into start/2 21.48/6.56 21.48/6.56 Control-Flow Refinement of Cost Relations 21.48/6.56 ===================================== 21.48/6.56 21.48/6.56 ### Specialization of cost equations p/2 21.48/6.56 * CE 7 is refined into CE [12] 21.48/6.56 * CE 8 is refined into CE [13] 21.48/6.56 21.48/6.56 21.48/6.56 ### Cost equations --> "Loop" of p/2 21.48/6.56 * CEs [12] --> Loop 9 21.48/6.56 * CEs [13] --> Loop 10 21.48/6.56 21.48/6.56 ### Ranking functions of CR p(V1,Out) 21.48/6.56 21.48/6.56 #### Partial ranking functions of CR p(V1,Out) 21.48/6.56 21.48/6.56 21.48/6.56 ### Specialization of cost equations minus/3 21.48/6.56 * CE 6 is refined into CE [14] 21.48/6.56 * CE 4 is refined into CE [15] 21.48/6.56 * CE 5 is refined into CE [16,17] 21.48/6.56 21.48/6.56 21.48/6.56 ### Cost equations --> "Loop" of minus/3 21.48/6.56 * CEs [17] --> Loop 11 21.48/6.56 * CEs [16] --> Loop 12 21.48/6.56 * CEs [14] --> Loop 13 21.48/6.56 * CEs [15] --> Loop 14 21.48/6.56 21.48/6.56 ### Ranking functions of CR minus(V1,V,Out) 21.48/6.56 * RF of phase [11]: [V,V1] 21.48/6.56 * RF of phase [12]: [V,V1] 21.48/6.56 21.48/6.56 #### Partial ranking functions of CR minus(V1,V,Out) 21.48/6.56 * Partial RF of phase [11]: 21.48/6.56 - RF of loop [11:1]: 21.48/6.56 V 21.48/6.56 V1 21.48/6.56 * Partial RF of phase [12]: 21.48/6.56 - RF of loop [12:1]: 21.48/6.56 V 21.48/6.56 V1 21.48/6.56 21.48/6.56 21.48/6.56 ### Specialization of cost equations (div)/3 21.48/6.56 * CE 9 is refined into CE [18] 21.48/6.56 * CE 11 is refined into CE [19] 21.48/6.56 * CE 10 is refined into CE [20,21,22] 21.48/6.56 21.48/6.56 21.48/6.56 ### Cost equations --> "Loop" of (div)/3 21.48/6.56 * CEs [22] --> Loop 15 21.48/6.56 * CEs [21] --> Loop 16 21.48/6.56 * CEs [20] --> Loop 17 21.48/6.56 * CEs [18,19] --> Loop 18 21.48/6.56 21.48/6.56 ### Ranking functions of CR div(V1,V,Out) 21.48/6.56 * RF of phase [15]: [V1/3-2/3,V1/3-2/3*V+2/3] 21.48/6.56 * RF of phase [17]: [V1] 21.48/6.56 21.48/6.56 #### Partial ranking functions of CR div(V1,V,Out) 21.48/6.56 * Partial RF of phase [15]: 21.48/6.56 - RF of loop [15:1]: 21.48/6.56 V1/3-2/3 21.48/6.56 V1/3-2/3*V+2/3 21.48/6.56 * Partial RF of phase [17]: 21.48/6.56 - RF of loop [17:1]: 21.48/6.56 V1 21.48/6.56 21.48/6.56 21.48/6.56 ### Specialization of cost equations start/2 21.48/6.56 * CE 1 is refined into CE [23,24,25] 21.48/6.56 * CE 2 is refined into CE [26,27] 21.48/6.56 * CE 3 is refined into CE [28,29,30,31,32] 21.48/6.56 21.48/6.56 21.48/6.56 ### Cost equations --> "Loop" of start/2 21.48/6.56 * CEs [28] --> Loop 19 21.48/6.56 * CEs [23,24,25,26,27,29,30,31,32] --> Loop 20 21.48/6.56 21.48/6.56 ### Ranking functions of CR start(V1,V) 21.48/6.56 21.48/6.56 #### Partial ranking functions of CR start(V1,V) 21.48/6.56 21.48/6.56 21.48/6.56 Computing Bounds 21.48/6.56 ===================================== 21.48/6.56 21.48/6.56 #### Cost of chains of p(V1,Out): 21.48/6.56 * Chain [10]: 0 21.48/6.56 with precondition: [Out=0,V1>=0] 21.48/6.56 21.48/6.56 * Chain [9]: 1 21.48/6.56 with precondition: [V1=Out+1,V1>=1] 21.48/6.56 21.48/6.56 21.48/6.56 #### Cost of chains of minus(V1,V,Out): 21.48/6.56 * Chain [[12],[11],14]: 3*it(11)+1 21.48/6.56 Such that:aux(1) =< V 21.48/6.56 it(11) =< aux(1) 21.48/6.56 21.48/6.56 with precondition: [Out=0,V>=2,V1>=V+1] 21.48/6.56 21.48/6.56 * Chain [[12],14]: 1*it(12)+1 21.48/6.56 Such that:it(12) =< V 21.48/6.56 21.48/6.56 with precondition: [Out=0,V>=1,V1>=V] 21.48/6.56 21.48/6.56 * Chain [[12],13]: 1*it(12)+0 21.48/6.56 Such that:it(12) =< V 21.48/6.56 21.48/6.56 with precondition: [Out=0,V1>=1,V>=1] 21.48/6.56 21.48/6.56 * Chain [[11],14]: 2*it(11)+1 21.48/6.56 Such that:it(11) =< V 21.48/6.56 21.48/6.56 with precondition: [V1=2*V+Out,V>=1,V1>=2*V] 21.48/6.56 21.48/6.56 * Chain [14]: 1 21.48/6.56 with precondition: [V=0,V1=Out,V1>=0] 21.48/6.56 21.48/6.56 * Chain [13]: 0 21.48/6.56 with precondition: [Out=0,V1>=0,V>=0] 21.48/6.56 21.48/6.56 21.48/6.56 #### Cost of chains of div(V1,V,Out): 21.48/6.56 * Chain [[17],18]: 2*it(17)+1 21.48/6.56 Such that:it(17) =< Out 21.48/6.56 21.48/6.56 with precondition: [V=1,Out>=1,V1>=Out] 21.48/6.56 21.48/6.56 * Chain [[17],16,18]: 2*it(17)+5*s(6)+3 21.48/6.56 Such that:s(5) =< 1 21.48/6.56 it(17) =< Out 21.48/6.56 s(6) =< s(5) 21.48/6.56 21.48/6.56 with precondition: [V=1,Out>=2,V1>=Out] 21.48/6.56 21.48/6.56 * Chain [[15],18]: 2*it(15)+2*s(9)+1 21.48/6.56 Such that:it(15) =< V1/3-2/3*V+2/3 21.48/6.56 s(9) =< 2/3*V1-V/3+2/3 21.48/6.56 21.48/6.56 with precondition: [V>=2,Out>=1,V1+4>=3*Out+2*V] 21.48/6.56 21.48/6.56 * Chain [[15],16,18]: 2*it(15)+5*s(6)+2*s(9)+3 21.48/6.56 Such that:it(15) =< V1/3-2/3*V+2/3 21.48/6.56 s(9) =< 2/3*V1 21.48/6.56 s(5) =< V 21.48/6.56 s(6) =< s(5) 21.48/6.56 21.48/6.56 with precondition: [V>=2,Out>=2,V1+6>=3*Out+2*V] 21.48/6.56 21.48/6.56 * Chain [18]: 1 21.48/6.56 with precondition: [Out=0,V1>=0,V>=0] 21.48/6.56 21.48/6.56 * Chain [16,18]: 5*s(6)+3 21.48/6.56 Such that:s(5) =< V 21.48/6.56 s(6) =< s(5) 21.48/6.56 21.48/6.56 with precondition: [Out=1,V1>=1,V>=1] 21.48/6.56 21.48/6.56 21.48/6.56 #### Cost of chains of start(V1,V): 21.48/6.56 * Chain [20]: 17*s(15)+4*s(19)+2*s(20)+2*s(22)+3 21.48/6.56 Such that:s(22) =< 2/3*V1 21.48/6.56 s(20) =< 2/3*V1-V/3+2/3 21.48/6.56 aux(4) =< V1/3-2/3*V+2/3 21.48/6.56 aux(5) =< V 21.48/6.56 s(19) =< aux(4) 21.48/6.56 s(15) =< aux(5) 21.48/6.56 21.48/6.56 with precondition: [V1>=0] 21.48/6.56 21.48/6.56 * Chain [19]: 4*s(27)+5*s(28)+3 21.48/6.56 Such that:s(25) =< 1 21.48/6.56 s(26) =< V1 21.48/6.56 s(27) =< s(26) 21.48/6.56 s(28) =< s(25) 21.48/6.56 21.48/6.56 with precondition: [V=1,V1>=1] 21.48/6.56 21.48/6.56 21.48/6.56 Closed-form bounds of start(V1,V): 21.48/6.56 ------------------------------------- 21.48/6.56 * Chain [20] with precondition: [V1>=0] 21.48/6.56 - Upper bound: nat(V)*17+3+4/3*V1+nat(2/3*V1-V/3+2/3)*2+nat(V1/3-2/3*V+2/3)*4 21.48/6.56 - Complexity: n 21.48/6.56 * Chain [19] with precondition: [V=1,V1>=1] 21.48/6.56 - Upper bound: 4*V1+8 21.48/6.56 - Complexity: n 21.48/6.56 21.48/6.56 ### Maximum cost of start(V1,V): max([4*V1+5,4/3*V1+nat(V)*17+nat(2/3*V1-V/3+2/3)*2+nat(V1/3-2/3*V+2/3)*4])+3 21.48/6.56 Asymptotic class: n 21.48/6.56 * Total analysis performed in 216 ms. 21.48/6.56 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (10) 21.48/6.56 BOUNDS(1, n^1) 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 21.48/6.56 Renamed function symbols to avoid clashes with predefined symbol. 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (12) 21.48/6.56 Obligation: 21.48/6.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 21.48/6.56 21.48/6.56 21.48/6.56 The TRS R consists of the following rules: 21.48/6.56 21.48/6.56 minus(X, 0') -> X 21.48/6.56 minus(s(X), s(Y)) -> p(minus(X, Y)) 21.48/6.56 p(s(X)) -> X 21.48/6.56 div(0', s(Y)) -> 0' 21.48/6.56 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) 21.48/6.56 21.48/6.56 S is empty. 21.48/6.56 Rewrite Strategy: INNERMOST 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 21.48/6.56 Infered types. 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (14) 21.48/6.56 Obligation: 21.48/6.56 Innermost TRS: 21.48/6.56 Rules: 21.48/6.56 minus(X, 0') -> X 21.48/6.56 minus(s(X), s(Y)) -> p(minus(X, Y)) 21.48/6.56 p(s(X)) -> X 21.48/6.56 div(0', s(Y)) -> 0' 21.48/6.56 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) 21.48/6.56 21.48/6.56 Types: 21.48/6.56 minus :: 0':s -> 0':s -> 0':s 21.48/6.56 0' :: 0':s 21.48/6.56 s :: 0':s -> 0':s 21.48/6.56 p :: 0':s -> 0':s 21.48/6.56 div :: 0':s -> 0':s -> 0':s 21.48/6.56 hole_0':s1_0 :: 0':s 21.48/6.56 gen_0':s2_0 :: Nat -> 0':s 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (15) OrderProof (LOWER BOUND(ID)) 21.48/6.56 Heuristically decided to analyse the following defined symbols: 21.48/6.56 minus, div 21.48/6.56 21.48/6.56 They will be analysed ascendingly in the following order: 21.48/6.56 minus < div 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (16) 21.48/6.56 Obligation: 21.48/6.56 Innermost TRS: 21.48/6.56 Rules: 21.48/6.56 minus(X, 0') -> X 21.48/6.56 minus(s(X), s(Y)) -> p(minus(X, Y)) 21.48/6.56 p(s(X)) -> X 21.48/6.56 div(0', s(Y)) -> 0' 21.48/6.56 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) 21.48/6.56 21.48/6.56 Types: 21.48/6.56 minus :: 0':s -> 0':s -> 0':s 21.48/6.56 0' :: 0':s 21.48/6.56 s :: 0':s -> 0':s 21.48/6.56 p :: 0':s -> 0':s 21.48/6.56 div :: 0':s -> 0':s -> 0':s 21.48/6.56 hole_0':s1_0 :: 0':s 21.48/6.56 gen_0':s2_0 :: Nat -> 0':s 21.48/6.56 21.48/6.56 21.48/6.56 Generator Equations: 21.48/6.56 gen_0':s2_0(0) <=> 0' 21.48/6.56 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 21.48/6.56 21.48/6.56 21.48/6.56 The following defined symbols remain to be analysed: 21.48/6.56 minus, div 21.48/6.56 21.48/6.56 They will be analysed ascendingly in the following order: 21.48/6.56 minus < div 21.48/6.56 21.48/6.56 ---------------------------------------- 21.48/6.56 21.48/6.56 (17) RewriteLemmaProof (LOWER BOUND(ID)) 21.48/6.56 Proved the following rewrite lemma: 21.48/6.56 minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) 21.48/6.56 21.48/6.56 Induction Base: 21.48/6.56 minus(gen_0':s2_0(+(1, 0)), gen_0':s2_0(+(1, 0))) 21.48/6.56 21.48/6.56 Induction Step: 21.48/6.56 minus(gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) 21.48/6.56 p(minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0)))) ->_IH 21.48/6.56 p(*3_0) 21.48/6.56 21.48/6.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 21.48/6.57 ---------------------------------------- 21.48/6.57 21.48/6.57 (18) 21.48/6.57 Complex Obligation (BEST) 21.48/6.57 21.48/6.57 ---------------------------------------- 21.48/6.57 21.48/6.57 (19) 21.48/6.57 Obligation: 21.48/6.57 Proved the lower bound n^1 for the following obligation: 21.48/6.57 21.48/6.57 Innermost TRS: 21.48/6.57 Rules: 21.48/6.57 minus(X, 0') -> X 21.48/6.57 minus(s(X), s(Y)) -> p(minus(X, Y)) 21.48/6.57 p(s(X)) -> X 21.48/6.57 div(0', s(Y)) -> 0' 21.48/6.57 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) 21.48/6.57 21.48/6.57 Types: 21.48/6.57 minus :: 0':s -> 0':s -> 0':s 21.48/6.57 0' :: 0':s 21.48/6.57 s :: 0':s -> 0':s 21.48/6.57 p :: 0':s -> 0':s 21.48/6.57 div :: 0':s -> 0':s -> 0':s 21.48/6.57 hole_0':s1_0 :: 0':s 21.48/6.57 gen_0':s2_0 :: Nat -> 0':s 21.48/6.57 21.48/6.57 21.48/6.57 Generator Equations: 21.48/6.57 gen_0':s2_0(0) <=> 0' 21.48/6.57 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 21.48/6.57 21.48/6.57 21.48/6.57 The following defined symbols remain to be analysed: 21.48/6.57 minus, div 21.48/6.57 21.48/6.57 They will be analysed ascendingly in the following order: 21.48/6.57 minus < div 21.48/6.57 21.48/6.57 ---------------------------------------- 21.48/6.57 21.48/6.57 (20) LowerBoundPropagationProof (FINISHED) 21.48/6.57 Propagated lower bound. 21.48/6.57 ---------------------------------------- 21.48/6.57 21.48/6.57 (21) 21.48/6.57 BOUNDS(n^1, INF) 21.48/6.57 21.48/6.57 ---------------------------------------- 21.48/6.57 21.48/6.57 (22) 21.48/6.57 Obligation: 21.48/6.57 Innermost TRS: 21.48/6.57 Rules: 21.48/6.57 minus(X, 0') -> X 21.48/6.57 minus(s(X), s(Y)) -> p(minus(X, Y)) 21.48/6.57 p(s(X)) -> X 21.48/6.57 div(0', s(Y)) -> 0' 21.48/6.57 div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) 21.48/6.57 21.48/6.57 Types: 21.48/6.57 minus :: 0':s -> 0':s -> 0':s 21.48/6.57 0' :: 0':s 21.48/6.57 s :: 0':s -> 0':s 21.48/6.57 p :: 0':s -> 0':s 21.48/6.57 div :: 0':s -> 0':s -> 0':s 21.48/6.57 hole_0':s1_0 :: 0':s 21.48/6.57 gen_0':s2_0 :: Nat -> 0':s 21.48/6.57 21.48/6.57 21.48/6.57 Lemmas: 21.48/6.57 minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) 21.48/6.57 21.48/6.57 21.48/6.57 Generator Equations: 21.48/6.57 gen_0':s2_0(0) <=> 0' 21.48/6.57 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 21.48/6.57 21.48/6.57 21.48/6.57 The following defined symbols remain to be analysed: 21.48/6.57 div 21.48/6.57 ---------------------------------------- 21.48/6.57 21.48/6.57 (23) RewriteLemmaProof (LOWER BOUND(ID)) 21.48/6.57 Proved the following rewrite lemma: 21.48/6.57 div(gen_0':s2_0(n2194_0), gen_0':s2_0(1)) -> gen_0':s2_0(n2194_0), rt in Omega(1 + n2194_0) 21.48/6.57 21.48/6.57 Induction Base: 21.48/6.57 div(gen_0':s2_0(0), gen_0':s2_0(1)) ->_R^Omega(1) 21.48/6.57 0' 21.48/6.57 21.48/6.57 Induction Step: 21.48/6.57 div(gen_0':s2_0(+(n2194_0, 1)), gen_0':s2_0(1)) ->_R^Omega(1) 21.48/6.57 s(div(minus(gen_0':s2_0(n2194_0), gen_0':s2_0(0)), s(gen_0':s2_0(0)))) ->_R^Omega(1) 21.48/6.57 s(div(gen_0':s2_0(n2194_0), s(gen_0':s2_0(0)))) ->_IH 21.48/6.57 s(gen_0':s2_0(c2195_0)) 21.48/6.57 21.48/6.57 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 21.48/6.57 ---------------------------------------- 21.48/6.57 21.48/6.57 (24) 21.48/6.57 BOUNDS(1, INF) 21.93/6.62 EOF