20.96/6.29 WORST_CASE(Omega(n^1), O(n^1)) 20.96/6.30 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 20.96/6.30 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 20.96/6.30 20.96/6.30 20.96/6.30 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.96/6.30 20.96/6.30 (0) CpxTRS 20.96/6.30 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 20.96/6.30 (2) CpxTRS 20.96/6.30 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 20.96/6.30 (4) CpxWeightedTrs 20.96/6.30 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 20.96/6.30 (6) CpxTypedWeightedTrs 20.96/6.30 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 20.96/6.30 (8) CpxTypedWeightedCompleteTrs 20.96/6.30 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 20.96/6.30 (10) CpxRNTS 20.96/6.30 (11) CompleteCoflocoProof [FINISHED, 176 ms] 20.96/6.30 (12) BOUNDS(1, n^1) 20.96/6.30 (13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 20.96/6.30 (14) CpxTRS 20.96/6.30 (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 20.96/6.30 (16) typed CpxTrs 20.96/6.30 (17) OrderProof [LOWER BOUND(ID), 0 ms] 20.96/6.30 (18) typed CpxTrs 20.96/6.30 (19) RewriteLemmaProof [LOWER BOUND(ID), 3275 ms] 20.96/6.30 (20) proven lower bound 20.96/6.30 (21) LowerBoundPropagationProof [FINISHED, 0 ms] 20.96/6.30 (22) BOUNDS(n^1, INF) 20.96/6.30 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (0) 20.96/6.30 Obligation: 20.96/6.30 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.96/6.30 20.96/6.30 20.96/6.30 The TRS R consists of the following rules: 20.96/6.30 20.96/6.30 a__f(X, X) -> a__f(a, b) 20.96/6.30 a__b -> a 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) 20.96/6.30 mark(b) -> a__b 20.96/6.30 mark(a) -> a 20.96/6.30 a__f(X1, X2) -> f(X1, X2) 20.96/6.30 a__b -> b 20.96/6.30 20.96/6.30 S is empty. 20.96/6.30 Rewrite Strategy: FULL 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 20.96/6.30 Converted rc-obligation to irc-obligation. 20.96/6.30 20.96/6.30 As the TRS is a non-duplicating overlay system, we have rc = irc. 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (2) 20.96/6.30 Obligation: 20.96/6.30 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 20.96/6.30 20.96/6.30 20.96/6.30 The TRS R consists of the following rules: 20.96/6.30 20.96/6.30 a__f(X, X) -> a__f(a, b) 20.96/6.30 a__b -> a 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) 20.96/6.30 mark(b) -> a__b 20.96/6.30 mark(a) -> a 20.96/6.30 a__f(X1, X2) -> f(X1, X2) 20.96/6.30 a__b -> b 20.96/6.30 20.96/6.30 S is empty. 20.96/6.30 Rewrite Strategy: INNERMOST 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 20.96/6.30 Transformed relative TRS to weighted TRS 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (4) 20.96/6.30 Obligation: 20.96/6.30 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 20.96/6.30 20.96/6.30 20.96/6.30 The TRS R consists of the following rules: 20.96/6.30 20.96/6.30 a__f(X, X) -> a__f(a, b) [1] 20.96/6.30 a__b -> a [1] 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) [1] 20.96/6.30 mark(b) -> a__b [1] 20.96/6.30 mark(a) -> a [1] 20.96/6.30 a__f(X1, X2) -> f(X1, X2) [1] 20.96/6.30 a__b -> b [1] 20.96/6.30 20.96/6.30 Rewrite Strategy: INNERMOST 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 20.96/6.30 Infered types. 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (6) 20.96/6.30 Obligation: 20.96/6.30 Runtime Complexity Weighted TRS with Types. 20.96/6.30 The TRS R consists of the following rules: 20.96/6.30 20.96/6.30 a__f(X, X) -> a__f(a, b) [1] 20.96/6.30 a__b -> a [1] 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) [1] 20.96/6.30 mark(b) -> a__b [1] 20.96/6.30 mark(a) -> a [1] 20.96/6.30 a__f(X1, X2) -> f(X1, X2) [1] 20.96/6.30 a__b -> b [1] 20.96/6.30 20.96/6.30 The TRS has the following type information: 20.96/6.30 a__f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 a :: a:b:f 20.96/6.30 b :: a:b:f 20.96/6.30 a__b :: a:b:f 20.96/6.30 mark :: a:b:f -> a:b:f 20.96/6.30 f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 20.96/6.30 Rewrite Strategy: INNERMOST 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (7) CompletionProof (UPPER BOUND(ID)) 20.96/6.30 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 20.96/6.30 none 20.96/6.30 20.96/6.30 And the following fresh constants: none 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (8) 20.96/6.30 Obligation: 20.96/6.30 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 20.96/6.30 20.96/6.30 Runtime Complexity Weighted TRS with Types. 20.96/6.30 The TRS R consists of the following rules: 20.96/6.30 20.96/6.30 a__f(X, X) -> a__f(a, b) [1] 20.96/6.30 a__b -> a [1] 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) [1] 20.96/6.30 mark(b) -> a__b [1] 20.96/6.30 mark(a) -> a [1] 20.96/6.30 a__f(X1, X2) -> f(X1, X2) [1] 20.96/6.30 a__b -> b [1] 20.96/6.30 20.96/6.30 The TRS has the following type information: 20.96/6.30 a__f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 a :: a:b:f 20.96/6.30 b :: a:b:f 20.96/6.30 a__b :: a:b:f 20.96/6.30 mark :: a:b:f -> a:b:f 20.96/6.30 f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 20.96/6.30 Rewrite Strategy: INNERMOST 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 20.96/6.30 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 20.96/6.30 The constant constructors are abstracted as follows: 20.96/6.30 20.96/6.30 a => 0 20.96/6.30 b => 1 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (10) 20.96/6.30 Obligation: 20.96/6.30 Complexity RNTS consisting of the following rules: 20.96/6.30 20.96/6.30 a__b -{ 1 }-> 1 :|: 20.96/6.30 a__b -{ 1 }-> 0 :|: 20.96/6.30 a__f(z, z') -{ 1 }-> a__f(0, 1) :|: z' = X, X >= 0, z = X 20.96/6.30 a__f(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 20.96/6.30 mark(z) -{ 1 }-> a__f(mark(X1), X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 20.96/6.30 mark(z) -{ 1 }-> a__b :|: z = 1 20.96/6.30 mark(z) -{ 1 }-> 0 :|: z = 0 20.96/6.30 20.96/6.30 Only complete derivations are relevant for the runtime complexity. 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (11) CompleteCoflocoProof (FINISHED) 20.96/6.30 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 20.96/6.30 20.96/6.30 eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]). 20.96/6.30 eq(start(V1, V),0,[fun1(Out)],[]). 20.96/6.30 eq(start(V1, V),0,[mark(V1, Out)],[V1 >= 0]). 20.96/6.30 eq(fun(V1, V, Out),1,[fun(0, 1, Ret)],[Out = Ret,V = X3,X3 >= 0,V1 = X3]). 20.96/6.30 eq(fun1(Out),1,[],[Out = 0]). 20.96/6.30 eq(mark(V1, Out),1,[mark(X11, Ret0),fun(Ret0, X21, Ret1)],[Out = Ret1,X11 >= 0,X21 >= 0,V1 = 1 + X11 + X21]). 20.96/6.30 eq(mark(V1, Out),1,[fun1(Ret2)],[Out = Ret2,V1 = 1]). 20.96/6.30 eq(mark(V1, Out),1,[],[Out = 0,V1 = 0]). 20.96/6.30 eq(fun(V1, V, Out),1,[],[Out = 1 + X12 + X22,X12 >= 0,X22 >= 0,V1 = X12,V = X22]). 20.96/6.30 eq(fun1(Out),1,[],[Out = 1]). 20.96/6.30 input_output_vars(fun(V1,V,Out),[V1,V],[Out]). 20.96/6.30 input_output_vars(fun1(Out),[],[Out]). 20.96/6.30 input_output_vars(mark(V1,Out),[V1],[Out]). 20.96/6.30 20.96/6.30 20.96/6.30 CoFloCo proof output: 20.96/6.30 Preprocessing Cost Relations 20.96/6.30 ===================================== 20.96/6.30 20.96/6.30 #### Computed strongly connected components 20.96/6.30 0. recursive : [fun/3] 20.96/6.30 1. non_recursive : [fun1/1] 20.96/6.30 2. recursive [non_tail] : [mark/2] 20.96/6.30 3. non_recursive : [start/2] 20.96/6.30 20.96/6.30 #### Obtained direct recursion through partial evaluation 20.96/6.30 0. SCC is partially evaluated into fun/3 20.96/6.30 1. SCC is partially evaluated into fun1/1 20.96/6.30 2. SCC is partially evaluated into mark/2 20.96/6.30 3. SCC is partially evaluated into start/2 20.96/6.30 20.96/6.30 Control-Flow Refinement of Cost Relations 20.96/6.30 ===================================== 20.96/6.30 20.96/6.30 ### Specialization of cost equations fun/3 20.96/6.30 * CE 5 is refined into CE [11] 20.96/6.30 * CE 4 is refined into CE [12] 20.96/6.30 20.96/6.30 20.96/6.30 ### Cost equations --> "Loop" of fun/3 20.96/6.30 * CEs [12] --> Loop 9 20.96/6.30 * CEs [11] --> Loop 10 20.96/6.30 20.96/6.30 ### Ranking functions of CR fun(V1,V,Out) 20.96/6.30 20.96/6.30 #### Partial ranking functions of CR fun(V1,V,Out) 20.96/6.30 20.96/6.30 20.96/6.30 ### Specialization of cost equations fun1/1 20.96/6.30 * CE 7 is refined into CE [13] 20.96/6.30 * CE 6 is refined into CE [14] 20.96/6.30 20.96/6.30 20.96/6.30 ### Cost equations --> "Loop" of fun1/1 20.96/6.30 * CEs [13] --> Loop 11 20.96/6.30 * CEs [14] --> Loop 12 20.96/6.30 20.96/6.30 ### Ranking functions of CR fun1(Out) 20.96/6.30 20.96/6.30 #### Partial ranking functions of CR fun1(Out) 20.96/6.30 20.96/6.30 20.96/6.30 ### Specialization of cost equations mark/2 20.96/6.30 * CE 9 is refined into CE [15,16] 20.96/6.30 * CE 10 is refined into CE [17] 20.96/6.30 * CE 8 is refined into CE [18,19] 20.96/6.30 20.96/6.30 20.96/6.30 ### Cost equations --> "Loop" of mark/2 20.96/6.30 * CEs [19] --> Loop 13 20.96/6.30 * CEs [18] --> Loop 14 20.96/6.30 * CEs [16] --> Loop 15 20.96/6.30 * CEs [15] --> Loop 16 20.96/6.30 * CEs [17] --> Loop 17 20.96/6.30 20.96/6.30 ### Ranking functions of CR mark(V1,Out) 20.96/6.30 * RF of phase [13,14]: [V1] 20.96/6.30 20.96/6.30 #### Partial ranking functions of CR mark(V1,Out) 20.96/6.30 * Partial RF of phase [13,14]: 20.96/6.30 - RF of loop [13:1,14:1]: 20.96/6.30 V1 20.96/6.30 20.96/6.30 20.96/6.30 ### Specialization of cost equations start/2 20.96/6.30 * CE 1 is refined into CE [20,21] 20.96/6.30 * CE 2 is refined into CE [22,23] 20.96/6.30 * CE 3 is refined into CE [24,25,26] 20.96/6.30 20.96/6.30 20.96/6.30 ### Cost equations --> "Loop" of start/2 20.96/6.30 * CEs [20,21,22,23,24,25,26] --> Loop 18 20.96/6.30 20.96/6.30 ### Ranking functions of CR start(V1,V) 20.96/6.30 20.96/6.30 #### Partial ranking functions of CR start(V1,V) 20.96/6.30 20.96/6.30 20.96/6.30 Computing Bounds 20.96/6.30 ===================================== 20.96/6.30 20.96/6.30 #### Cost of chains of fun(V1,V,Out): 20.96/6.30 * Chain [10]: 1 20.96/6.30 with precondition: [V+V1+1=Out,V1>=0,V>=0] 20.96/6.30 20.96/6.30 * Chain [9,10]: 2 20.96/6.30 with precondition: [Out=2,V1=V,V1>=0] 20.96/6.30 20.96/6.30 20.96/6.30 #### Cost of chains of fun1(Out): 20.96/6.30 * Chain [12]: 1 20.96/6.30 with precondition: [Out=0] 20.96/6.30 20.96/6.30 * Chain [11]: 1 20.96/6.30 with precondition: [Out=1] 20.96/6.30 20.96/6.30 20.96/6.30 #### Cost of chains of mark(V1,Out): 20.96/6.30 * Chain [[13,14],17]: 5*it(13)+1 20.96/6.30 Such that:aux(3) =< V1 20.96/6.30 it(13) =< aux(3) 20.96/6.30 20.96/6.30 with precondition: [V1>=1,Out>=1,V1+1>=Out] 20.96/6.30 20.96/6.30 * Chain [[13,14],16]: 5*it(13)+2 20.96/6.30 Such that:aux(4) =< V1 20.96/6.30 it(13) =< aux(4) 20.96/6.30 20.96/6.30 with precondition: [V1>=2,Out>=1,V1>=Out] 20.96/6.30 20.96/6.30 * Chain [[13,14],15]: 5*it(13)+2 20.96/6.30 Such that:aux(5) =< V1 20.96/6.30 it(13) =< aux(5) 20.96/6.30 20.96/6.30 with precondition: [Out>=2,V1>=Out] 20.96/6.30 20.96/6.30 * Chain [17]: 1 20.96/6.30 with precondition: [V1=0,Out=0] 20.96/6.30 20.96/6.30 * Chain [16]: 2 20.96/6.30 with precondition: [V1=1,Out=0] 20.96/6.30 20.96/6.30 * Chain [15]: 2 20.96/6.30 with precondition: [V1=1,Out=1] 20.96/6.30 20.96/6.30 20.96/6.30 #### Cost of chains of start(V1,V): 20.96/6.30 * Chain [18]: 15*s(8)+2 20.96/6.30 Such that:s(7) =< V1 20.96/6.30 s(8) =< s(7) 20.96/6.30 20.96/6.30 with precondition: [] 20.96/6.30 20.96/6.30 20.96/6.30 Closed-form bounds of start(V1,V): 20.96/6.30 ------------------------------------- 20.96/6.30 * Chain [18] with precondition: [] 20.96/6.30 - Upper bound: nat(V1)*15+2 20.96/6.30 - Complexity: n 20.96/6.30 20.96/6.30 ### Maximum cost of start(V1,V): nat(V1)*15+2 20.96/6.30 Asymptotic class: n 20.96/6.30 * Total analysis performed in 99 ms. 20.96/6.30 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (12) 20.96/6.30 BOUNDS(1, n^1) 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (13) RenamingProof (BOTH BOUNDS(ID, ID)) 20.96/6.30 Renamed function symbols to avoid clashes with predefined symbol. 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (14) 20.96/6.30 Obligation: 20.96/6.30 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 20.96/6.30 20.96/6.30 20.96/6.30 The TRS R consists of the following rules: 20.96/6.30 20.96/6.30 a__f(X, X) -> a__f(a, b) 20.96/6.30 a__b -> a 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) 20.96/6.30 mark(b) -> a__b 20.96/6.30 mark(a) -> a 20.96/6.30 a__f(X1, X2) -> f(X1, X2) 20.96/6.30 a__b -> b 20.96/6.30 20.96/6.30 S is empty. 20.96/6.30 Rewrite Strategy: FULL 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (15) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 20.96/6.30 Infered types. 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (16) 20.96/6.30 Obligation: 20.96/6.30 TRS: 20.96/6.30 Rules: 20.96/6.30 a__f(X, X) -> a__f(a, b) 20.96/6.30 a__b -> a 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) 20.96/6.30 mark(b) -> a__b 20.96/6.30 mark(a) -> a 20.96/6.30 a__f(X1, X2) -> f(X1, X2) 20.96/6.30 a__b -> b 20.96/6.30 20.96/6.30 Types: 20.96/6.30 a__f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 a :: a:b:f 20.96/6.30 b :: a:b:f 20.96/6.30 a__b :: a:b:f 20.96/6.30 mark :: a:b:f -> a:b:f 20.96/6.30 f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 hole_a:b:f1_0 :: a:b:f 20.96/6.30 gen_a:b:f2_0 :: Nat -> a:b:f 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (17) OrderProof (LOWER BOUND(ID)) 20.96/6.30 Heuristically decided to analyse the following defined symbols: 20.96/6.30 a__f, mark 20.96/6.30 20.96/6.30 They will be analysed ascendingly in the following order: 20.96/6.30 a__f < mark 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (18) 20.96/6.30 Obligation: 20.96/6.30 TRS: 20.96/6.30 Rules: 20.96/6.30 a__f(X, X) -> a__f(a, b) 20.96/6.30 a__b -> a 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) 20.96/6.30 mark(b) -> a__b 20.96/6.30 mark(a) -> a 20.96/6.30 a__f(X1, X2) -> f(X1, X2) 20.96/6.30 a__b -> b 20.96/6.30 20.96/6.30 Types: 20.96/6.30 a__f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 a :: a:b:f 20.96/6.30 b :: a:b:f 20.96/6.30 a__b :: a:b:f 20.96/6.30 mark :: a:b:f -> a:b:f 20.96/6.30 f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 hole_a:b:f1_0 :: a:b:f 20.96/6.30 gen_a:b:f2_0 :: Nat -> a:b:f 20.96/6.30 20.96/6.30 20.96/6.30 Generator Equations: 20.96/6.30 gen_a:b:f2_0(0) <=> a 20.96/6.30 gen_a:b:f2_0(+(x, 1)) <=> f(gen_a:b:f2_0(x), a) 20.96/6.30 20.96/6.30 20.96/6.30 The following defined symbols remain to be analysed: 20.96/6.30 a__f, mark 20.96/6.30 20.96/6.30 They will be analysed ascendingly in the following order: 20.96/6.30 a__f < mark 20.96/6.30 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (19) RewriteLemmaProof (LOWER BOUND(ID)) 20.96/6.30 Proved the following rewrite lemma: 20.96/6.30 mark(gen_a:b:f2_0(+(1, n22_0))) -> *3_0, rt in Omega(n22_0) 20.96/6.30 20.96/6.30 Induction Base: 20.96/6.30 mark(gen_a:b:f2_0(+(1, 0))) 20.96/6.30 20.96/6.30 Induction Step: 20.96/6.30 mark(gen_a:b:f2_0(+(1, +(n22_0, 1)))) ->_R^Omega(1) 20.96/6.30 a__f(mark(gen_a:b:f2_0(+(1, n22_0))), a) ->_IH 20.96/6.30 a__f(*3_0, a) 20.96/6.30 20.96/6.30 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (20) 20.96/6.30 Obligation: 20.96/6.30 Proved the lower bound n^1 for the following obligation: 20.96/6.30 20.96/6.30 TRS: 20.96/6.30 Rules: 20.96/6.30 a__f(X, X) -> a__f(a, b) 20.96/6.30 a__b -> a 20.96/6.30 mark(f(X1, X2)) -> a__f(mark(X1), X2) 20.96/6.30 mark(b) -> a__b 20.96/6.30 mark(a) -> a 20.96/6.30 a__f(X1, X2) -> f(X1, X2) 20.96/6.30 a__b -> b 20.96/6.30 20.96/6.30 Types: 20.96/6.30 a__f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 a :: a:b:f 20.96/6.30 b :: a:b:f 20.96/6.30 a__b :: a:b:f 20.96/6.30 mark :: a:b:f -> a:b:f 20.96/6.30 f :: a:b:f -> a:b:f -> a:b:f 20.96/6.30 hole_a:b:f1_0 :: a:b:f 20.96/6.30 gen_a:b:f2_0 :: Nat -> a:b:f 20.96/6.30 20.96/6.30 20.96/6.30 Generator Equations: 20.96/6.30 gen_a:b:f2_0(0) <=> a 20.96/6.30 gen_a:b:f2_0(+(x, 1)) <=> f(gen_a:b:f2_0(x), a) 20.96/6.30 20.96/6.30 20.96/6.30 The following defined symbols remain to be analysed: 20.96/6.30 mark 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (21) LowerBoundPropagationProof (FINISHED) 20.96/6.30 Propagated lower bound. 20.96/6.30 ---------------------------------------- 20.96/6.30 20.96/6.30 (22) 20.96/6.30 BOUNDS(n^1, INF) 21.24/9.41 EOF