9.79/4.41 YES 11.67/4.95 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 11.67/4.95 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.67/4.95 11.67/4.95 11.67/4.95 H-Termination with start terms of the given HASKELL could be proven: 11.67/4.95 11.67/4.95 (0) HASKELL 11.67/4.95 (1) CR [EQUIVALENT, 0 ms] 11.67/4.95 (2) HASKELL 11.67/4.95 (3) BR [EQUIVALENT, 0 ms] 11.67/4.95 (4) HASKELL 11.67/4.95 (5) COR [EQUIVALENT, 0 ms] 11.67/4.95 (6) HASKELL 11.67/4.95 (7) Narrow [SOUND, 0 ms] 11.67/4.95 (8) QDP 11.67/4.95 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.67/4.95 (10) YES 11.67/4.95 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (0) 11.67/4.95 Obligation: 11.67/4.95 mainModule Main 11.67/4.95 module FiniteMap where { 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 data FiniteMap b a = EmptyFM | Branch b a Int (FiniteMap b a) (FiniteMap b a) ; 11.67/4.95 11.67/4.95 minFM :: Ord b => FiniteMap b a -> Maybe b; 11.67/4.95 minFM EmptyFM = Nothing; 11.67/4.95 minFM (Branch key _ _ fm_l _) = case minFM fm_l of { 11.67/4.95 Nothing-> Just key; 11.67/4.95 Just key1-> Just key1; 11.67/4.95 } ; 11.67/4.95 11.67/4.95 } 11.67/4.95 module Maybe where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 module Main where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (1) CR (EQUIVALENT) 11.67/4.95 Case Reductions: 11.67/4.95 The following Case expression 11.67/4.95 "case minFM fm_l of { 11.67/4.95 Nothing -> Just key; 11.67/4.95 Just key1 -> Just key1} 11.67/4.95 " 11.67/4.95 is transformed to 11.67/4.95 "minFM0 key Nothing = Just key; 11.67/4.95 minFM0 key (Just key1) = Just key1; 11.67/4.95 " 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (2) 11.67/4.95 Obligation: 11.67/4.95 mainModule Main 11.67/4.95 module FiniteMap where { 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 data FiniteMap b a = EmptyFM | Branch b a Int (FiniteMap b a) (FiniteMap b a) ; 11.67/4.95 11.67/4.95 minFM :: Ord b => FiniteMap b a -> Maybe b; 11.67/4.95 minFM EmptyFM = Nothing; 11.67/4.95 minFM (Branch key _ _ fm_l _) = minFM0 key (minFM fm_l); 11.67/4.95 11.67/4.95 minFM0 key Nothing = Just key; 11.67/4.95 minFM0 key (Just key1) = Just key1; 11.67/4.95 11.67/4.95 } 11.67/4.95 module Maybe where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 module Main where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (3) BR (EQUIVALENT) 11.67/4.95 Replaced joker patterns by fresh variables and removed binding patterns. 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (4) 11.67/4.95 Obligation: 11.67/4.95 mainModule Main 11.67/4.95 module FiniteMap where { 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; 11.67/4.95 11.67/4.95 minFM :: Ord b => FiniteMap b a -> Maybe b; 11.67/4.95 minFM EmptyFM = Nothing; 11.67/4.95 minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l); 11.67/4.95 11.67/4.95 minFM0 key Nothing = Just key; 11.67/4.95 minFM0 key (Just key1) = Just key1; 11.67/4.95 11.67/4.95 } 11.67/4.95 module Maybe where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 module Main where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (5) COR (EQUIVALENT) 11.67/4.95 Cond Reductions: 11.67/4.95 The following Function with conditions 11.67/4.95 "undefined |Falseundefined; 11.67/4.95 " 11.67/4.95 is transformed to 11.67/4.95 "undefined = undefined1; 11.67/4.95 " 11.67/4.95 "undefined0 True = undefined; 11.67/4.95 " 11.67/4.95 "undefined1 = undefined0 False; 11.67/4.95 " 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (6) 11.67/4.95 Obligation: 11.67/4.95 mainModule Main 11.67/4.95 module FiniteMap where { 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; 11.67/4.95 11.67/4.95 minFM :: Ord b => FiniteMap b a -> Maybe b; 11.67/4.95 minFM EmptyFM = Nothing; 11.67/4.95 minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l); 11.67/4.95 11.67/4.95 minFM0 key Nothing = Just key; 11.67/4.95 minFM0 key (Just key1) = Just key1; 11.67/4.95 11.67/4.95 } 11.67/4.95 module Maybe where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Main; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 module Main where { 11.67/4.95 import qualified FiniteMap; 11.67/4.95 import qualified Maybe; 11.67/4.95 import qualified Prelude; 11.67/4.95 } 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (7) Narrow (SOUND) 11.67/4.95 Haskell To QDPs 11.67/4.95 11.67/4.95 digraph dp_graph { 11.67/4.95 node [outthreshold=100, inthreshold=100];1[label="FiniteMap.minFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 11.67/4.95 3[label="FiniteMap.minFM wv3",fontsize=16,color="burlywood",shape="triangle"];15[label="wv3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];3 -> 15[label="",style="solid", color="burlywood", weight=9]; 11.67/4.95 15 -> 4[label="",style="solid", color="burlywood", weight=3]; 11.67/4.95 16[label="wv3/FiniteMap.Branch wv30 wv31 wv32 wv33 wv34",fontsize=10,color="white",style="solid",shape="box"];3 -> 16[label="",style="solid", color="burlywood", weight=9]; 11.67/4.95 16 -> 5[label="",style="solid", color="burlywood", weight=3]; 11.67/4.95 4[label="FiniteMap.minFM FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3]; 11.67/4.95 5[label="FiniteMap.minFM (FiniteMap.Branch wv30 wv31 wv32 wv33 wv34)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 11.67/4.95 6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0]; 11.67/4.95 7[label="FiniteMap.minFM0 wv30 (FiniteMap.minFM wv33)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3]; 11.67/4.95 9 -> 3[label="",style="dashed", color="red", weight=0]; 11.67/4.95 9[label="FiniteMap.minFM wv33",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3]; 11.67/4.95 8[label="FiniteMap.minFM0 wv30 wv4",fontsize=16,color="burlywood",shape="triangle"];17[label="wv4/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 17[label="",style="solid", color="burlywood", weight=9]; 11.67/4.95 17 -> 11[label="",style="solid", color="burlywood", weight=3]; 11.67/4.95 18[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 18[label="",style="solid", color="burlywood", weight=9]; 11.67/4.95 18 -> 12[label="",style="solid", color="burlywood", weight=3]; 11.67/4.95 10[label="wv33",fontsize=16,color="green",shape="box"];11[label="FiniteMap.minFM0 wv30 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 11.67/4.95 12[label="FiniteMap.minFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3]; 11.67/4.95 13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];} 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (8) 11.67/4.95 Obligation: 11.67/4.95 Q DP problem: 11.67/4.95 The TRS P consists of the following rules: 11.67/4.95 11.67/4.95 new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_minFM(wv33, h) 11.67/4.95 11.67/4.95 R is empty. 11.67/4.95 Q is empty. 11.67/4.95 We have to consider all minimal (P,Q,R)-chains. 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (9) QDPSizeChangeProof (EQUIVALENT) 11.67/4.95 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. 11.67/4.95 11.67/4.95 From the DPs we obtained the following set of size-change graphs: 11.67/4.95 *new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_minFM(wv33, h) 11.67/4.95 The graph contains the following edges 1 > 1, 2 >= 2 11.67/4.95 11.67/4.95 11.67/4.95 ---------------------------------------- 11.67/4.95 11.67/4.95 (10) 11.67/4.95 YES 11.79/4.99 EOF