10.72/4.43 YES 12.67/4.99 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 12.67/4.99 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.67/4.99 12.67/4.99 12.67/4.99 H-Termination with start terms of the given HASKELL could be proven: 12.67/4.99 12.67/4.99 (0) HASKELL 12.67/4.99 (1) BR [EQUIVALENT, 0 ms] 12.67/4.99 (2) HASKELL 12.67/4.99 (3) COR [EQUIVALENT, 26 ms] 12.67/4.99 (4) HASKELL 12.67/4.99 (5) NumRed [SOUND, 0 ms] 12.67/4.99 (6) HASKELL 12.67/4.99 (7) Narrow [SOUND, 0 ms] 12.67/4.99 (8) QDP 12.67/4.99 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.67/4.99 (10) YES 12.67/4.99 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (0) 12.67/4.99 Obligation: 12.67/4.99 mainModule Main 12.67/4.99 module Maybe where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 module List where { 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 genericLength :: Num b => [a] -> b; 12.67/4.99 genericLength [] = 0; 12.67/4.99 genericLength (_ : l) = 1 + genericLength l; 12.67/4.99 12.67/4.99 } 12.67/4.99 module Main where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (1) BR (EQUIVALENT) 12.67/4.99 Replaced joker patterns by fresh variables and removed binding patterns. 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (2) 12.67/4.99 Obligation: 12.67/4.99 mainModule Main 12.67/4.99 module Maybe where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 module List where { 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 genericLength :: Num b => [a] -> b; 12.67/4.99 genericLength [] = 0; 12.67/4.99 genericLength (vy : l) = 1 + genericLength l; 12.67/4.99 12.67/4.99 } 12.67/4.99 module Main where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (3) COR (EQUIVALENT) 12.67/4.99 Cond Reductions: 12.67/4.99 The following Function with conditions 12.67/4.99 "undefined |Falseundefined; 12.67/4.99 " 12.67/4.99 is transformed to 12.67/4.99 "undefined = undefined1; 12.67/4.99 " 12.67/4.99 "undefined0 True = undefined; 12.67/4.99 " 12.67/4.99 "undefined1 = undefined0 False; 12.67/4.99 " 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (4) 12.67/4.99 Obligation: 12.67/4.99 mainModule Main 12.67/4.99 module Maybe where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 module List where { 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 genericLength :: Num b => [a] -> b; 12.67/4.99 genericLength [] = 0; 12.67/4.99 genericLength (vy : l) = 1 + genericLength l; 12.67/4.99 12.67/4.99 } 12.67/4.99 module Main where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (5) NumRed (SOUND) 12.67/4.99 Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (6) 12.67/4.99 Obligation: 12.67/4.99 mainModule Main 12.67/4.99 module Maybe where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 module List where { 12.67/4.99 import qualified Main; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 genericLength :: Num b => [a] -> b; 12.67/4.99 genericLength [] = fromInt (Pos Zero); 12.67/4.99 genericLength (vy : l) = fromInt (Pos (Succ Zero)) + genericLength l; 12.67/4.99 12.67/4.99 } 12.67/4.99 module Main where { 12.67/4.99 import qualified List; 12.67/4.99 import qualified Maybe; 12.67/4.99 import qualified Prelude; 12.67/4.99 } 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (7) Narrow (SOUND) 12.67/4.99 Haskell To QDPs 12.67/4.99 12.67/4.99 digraph dp_graph { 12.67/4.99 node [outthreshold=100, inthreshold=100];1[label="List.genericLength",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 12.67/4.99 3[label="List.genericLength vz3",fontsize=16,color="burlywood",shape="triangle"];36[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];3 -> 36[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 36 -> 4[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 37[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 37[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 37 -> 5[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 4[label="List.genericLength (vz30 : vz31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3]; 12.67/4.99 5[label="List.genericLength []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 12.67/4.99 6 -> 8[label="",style="dashed", color="red", weight=0]; 12.67/4.99 6[label="fromInt (Pos (Succ Zero)) + List.genericLength vz31",fontsize=16,color="magenta"];6 -> 9[label="",style="dashed", color="magenta", weight=3]; 12.67/4.99 7[label="fromInt (Pos Zero)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3]; 12.67/4.99 9 -> 3[label="",style="dashed", color="red", weight=0]; 12.67/4.99 9[label="List.genericLength vz31",fontsize=16,color="magenta"];9 -> 11[label="",style="dashed", color="magenta", weight=3]; 12.67/4.99 8[label="fromInt (Pos (Succ Zero)) + vz4",fontsize=16,color="black",shape="triangle"];8 -> 12[label="",style="solid", color="black", weight=3]; 12.67/4.99 10[label="Pos Zero",fontsize=16,color="green",shape="box"];11[label="vz31",fontsize=16,color="green",shape="box"];12[label="primPlusInt (fromInt (Pos (Succ Zero))) vz4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 12.67/4.99 13[label="primPlusInt (Pos (Succ Zero)) vz4",fontsize=16,color="burlywood",shape="box"];38[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];13 -> 38[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 38 -> 14[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 39[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];13 -> 39[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 39 -> 15[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 14[label="primPlusInt (Pos (Succ Zero)) (Pos vz40)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3]; 12.67/4.99 15[label="primPlusInt (Pos (Succ Zero)) (Neg vz40)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3]; 12.67/4.99 16[label="Pos (primPlusNat (Succ Zero) vz40)",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3]; 12.67/4.99 17[label="primMinusNat (Succ Zero) vz40",fontsize=16,color="burlywood",shape="box"];40[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];17 -> 40[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 40 -> 19[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 41[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 41[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 41 -> 20[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 18[label="primPlusNat (Succ Zero) vz40",fontsize=16,color="burlywood",shape="box"];42[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];18 -> 42[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 42 -> 21[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 43[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 43[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 43 -> 22[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 19[label="primMinusNat (Succ Zero) (Succ vz400)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 12.67/4.99 20[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 12.67/4.99 21[label="primPlusNat (Succ Zero) (Succ vz400)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 12.67/4.99 22[label="primPlusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 12.67/4.99 23[label="primMinusNat Zero vz400",fontsize=16,color="burlywood",shape="box"];44[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 44[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 44 -> 27[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 45[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 45[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 45 -> 28[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 24[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];25[label="Succ (Succ (primPlusNat Zero vz400))",fontsize=16,color="green",shape="box"];25 -> 29[label="",style="dashed", color="green", weight=3]; 12.67/4.99 26[label="Succ Zero",fontsize=16,color="green",shape="box"];27[label="primMinusNat Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3]; 12.67/4.99 28[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3]; 12.67/4.99 29[label="primPlusNat Zero vz400",fontsize=16,color="burlywood",shape="box"];46[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 46[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 46 -> 32[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 47[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 47[label="",style="solid", color="burlywood", weight=9]; 12.67/4.99 47 -> 33[label="",style="solid", color="burlywood", weight=3]; 12.67/4.99 30[label="Neg (Succ vz4000)",fontsize=16,color="green",shape="box"];31[label="Pos Zero",fontsize=16,color="green",shape="box"];32[label="primPlusNat Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3]; 12.67/4.99 33[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3]; 12.67/4.99 34[label="Succ vz4000",fontsize=16,color="green",shape="box"];35[label="Zero",fontsize=16,color="green",shape="box"];} 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (8) 12.67/4.99 Obligation: 12.67/4.99 Q DP problem: 12.67/4.99 The TRS P consists of the following rules: 12.67/4.99 12.67/4.99 new_genericLength(:(vz30, vz31), ba) -> new_genericLength(vz31, ba) 12.67/4.99 12.67/4.99 R is empty. 12.67/4.99 Q is empty. 12.67/4.99 We have to consider all minimal (P,Q,R)-chains. 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (9) QDPSizeChangeProof (EQUIVALENT) 12.67/4.99 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 12.67/4.99 12.67/4.99 From the DPs we obtained the following set of size-change graphs: 12.67/4.99 *new_genericLength(:(vz30, vz31), ba) -> new_genericLength(vz31, ba) 12.67/4.99 The graph contains the following edges 1 > 1, 2 >= 2 12.67/4.99 12.67/4.99 12.67/4.99 ---------------------------------------- 12.67/4.99 12.67/4.99 (10) 12.67/4.99 YES 12.78/5.04 EOF