7.90/3.58 YES 9.56/4.05 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.56/4.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.56/4.05 9.56/4.05 9.56/4.05 H-Termination with start terms of the given HASKELL could be proven: 9.56/4.05 9.56/4.05 (0) HASKELL 9.56/4.05 (1) BR [EQUIVALENT, 0 ms] 9.56/4.05 (2) HASKELL 9.56/4.05 (3) COR [EQUIVALENT, 0 ms] 9.56/4.05 (4) HASKELL 9.56/4.05 (5) Narrow [SOUND, 0 ms] 9.56/4.05 (6) QDP 9.56/4.05 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 9.56/4.05 (8) AND 9.56/4.05 (9) QDP 9.56/4.05 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.56/4.05 (11) YES 9.56/4.05 (12) QDP 9.56/4.05 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.56/4.05 (14) YES 9.56/4.05 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (0) 9.56/4.05 Obligation: 9.56/4.05 mainModule Main 9.56/4.05 module Main where { 9.56/4.05 import qualified Prelude; 9.56/4.05 data List a = Cons a (List a) | Nil ; 9.56/4.05 9.56/4.05 data Main.Maybe a = Nothing | Just a ; 9.56/4.05 9.56/4.05 data MyBool = MyTrue | MyFalse ; 9.56/4.05 9.56/4.05 data Tup2 b a = Tup2 b a ; 9.56/4.05 9.56/4.05 esEsMyBool :: MyBool -> MyBool -> MyBool; 9.56/4.05 esEsMyBool MyFalse MyFalse = MyTrue; 9.56/4.05 esEsMyBool MyFalse MyTrue = MyFalse; 9.56/4.05 esEsMyBool MyTrue MyFalse = MyFalse; 9.56/4.05 esEsMyBool MyTrue MyTrue = MyTrue; 9.56/4.05 9.56/4.05 lookup k Nil = lookup3 k Nil; 9.56/4.05 lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys); 9.56/4.05 9.56/4.05 lookup0 k x y xys MyTrue = lookup k xys; 9.56/4.05 9.56/4.05 lookup1 k x y xys MyTrue = Main.Just y; 9.56/4.05 lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise; 9.56/4.05 9.56/4.05 lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsMyBool k x); 9.56/4.05 9.56/4.05 lookup3 k Nil = Main.Nothing; 9.56/4.05 lookup3 vy vz = lookup2 vy vz; 9.56/4.05 9.56/4.05 otherwise :: MyBool; 9.56/4.05 otherwise = MyTrue; 9.56/4.05 9.56/4.05 } 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (1) BR (EQUIVALENT) 9.56/4.05 Replaced joker patterns by fresh variables and removed binding patterns. 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (2) 9.56/4.05 Obligation: 9.56/4.05 mainModule Main 9.56/4.05 module Main where { 9.56/4.05 import qualified Prelude; 9.56/4.05 data List a = Cons a (List a) | Nil ; 9.56/4.05 9.56/4.05 data Main.Maybe a = Nothing | Just a ; 9.56/4.05 9.56/4.05 data MyBool = MyTrue | MyFalse ; 9.56/4.05 9.56/4.05 data Tup2 b a = Tup2 b a ; 9.56/4.05 9.56/4.05 esEsMyBool :: MyBool -> MyBool -> MyBool; 9.56/4.05 esEsMyBool MyFalse MyFalse = MyTrue; 9.56/4.05 esEsMyBool MyFalse MyTrue = MyFalse; 9.56/4.05 esEsMyBool MyTrue MyFalse = MyFalse; 9.56/4.05 esEsMyBool MyTrue MyTrue = MyTrue; 9.56/4.05 9.56/4.05 lookup k Nil = lookup3 k Nil; 9.56/4.05 lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys); 9.56/4.05 9.56/4.05 lookup0 k x y xys MyTrue = lookup k xys; 9.56/4.05 9.56/4.05 lookup1 k x y xys MyTrue = Main.Just y; 9.56/4.05 lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise; 9.56/4.05 9.56/4.05 lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsMyBool k x); 9.56/4.05 9.56/4.05 lookup3 k Nil = Main.Nothing; 9.56/4.05 lookup3 vy vz = lookup2 vy vz; 9.56/4.05 9.56/4.05 otherwise :: MyBool; 9.56/4.05 otherwise = MyTrue; 9.56/4.05 9.56/4.05 } 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (3) COR (EQUIVALENT) 9.56/4.05 Cond Reductions: 9.56/4.05 The following Function with conditions 9.56/4.05 "undefined |Falseundefined; 9.56/4.05 " 9.56/4.05 is transformed to 9.56/4.05 "undefined = undefined1; 9.56/4.05 " 9.56/4.05 "undefined0 True = undefined; 9.56/4.05 " 9.56/4.05 "undefined1 = undefined0 False; 9.56/4.05 " 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (4) 9.56/4.05 Obligation: 9.56/4.05 mainModule Main 9.56/4.05 module Main where { 9.56/4.05 import qualified Prelude; 9.56/4.05 data List a = Cons a (List a) | Nil ; 9.56/4.05 9.56/4.05 data Main.Maybe a = Nothing | Just a ; 9.56/4.05 9.56/4.05 data MyBool = MyTrue | MyFalse ; 9.56/4.05 9.56/4.05 data Tup2 b a = Tup2 b a ; 9.56/4.05 9.56/4.05 esEsMyBool :: MyBool -> MyBool -> MyBool; 9.56/4.05 esEsMyBool MyFalse MyFalse = MyTrue; 9.56/4.05 esEsMyBool MyFalse MyTrue = MyFalse; 9.56/4.05 esEsMyBool MyTrue MyFalse = MyFalse; 9.56/4.05 esEsMyBool MyTrue MyTrue = MyTrue; 9.56/4.05 9.56/4.05 lookup k Nil = lookup3 k Nil; 9.56/4.05 lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys); 9.56/4.05 9.56/4.05 lookup0 k x y xys MyTrue = lookup k xys; 9.56/4.05 9.56/4.05 lookup1 k x y xys MyTrue = Main.Just y; 9.56/4.05 lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise; 9.56/4.05 9.56/4.05 lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsMyBool k x); 9.56/4.05 9.56/4.05 lookup3 k Nil = Main.Nothing; 9.56/4.05 lookup3 vy vz = lookup2 vy vz; 9.56/4.05 9.56/4.05 otherwise :: MyBool; 9.56/4.05 otherwise = MyTrue; 9.56/4.05 9.56/4.05 } 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (5) Narrow (SOUND) 9.56/4.05 Haskell To QDPs 9.56/4.05 9.56/4.05 digraph dp_graph { 9.56/4.05 node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.56/4.05 3[label="lookup vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.56/4.05 4[label="lookup vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];34[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 34 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 35[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 35 -> 6[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 5[label="lookup vx3 (Cons vx40 vx41)",fontsize=16,color="burlywood",shape="box"];36[label="vx40/Tup2 vx400 vx401",fontsize=10,color="white",style="solid",shape="box"];5 -> 36[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 36 -> 7[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 6[label="lookup vx3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 9.56/4.05 7[label="lookup vx3 (Cons (Tup2 vx400 vx401) vx41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 9.56/4.05 8[label="lookup3 vx3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 9.56/4.05 9[label="lookup2 vx3 (Cons (Tup2 vx400 vx401) vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 9.56/4.05 10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 vx3 vx400 vx401 vx41 (esEsMyBool vx3 vx400)",fontsize=16,color="burlywood",shape="box"];37[label="vx3/MyTrue",fontsize=10,color="white",style="solid",shape="box"];11 -> 37[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 37 -> 12[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 38[label="vx3/MyFalse",fontsize=10,color="white",style="solid",shape="box"];11 -> 38[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 38 -> 13[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 12[label="lookup1 MyTrue vx400 vx401 vx41 (esEsMyBool MyTrue vx400)",fontsize=16,color="burlywood",shape="box"];39[label="vx400/MyTrue",fontsize=10,color="white",style="solid",shape="box"];12 -> 39[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 39 -> 14[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 40[label="vx400/MyFalse",fontsize=10,color="white",style="solid",shape="box"];12 -> 40[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 40 -> 15[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 13[label="lookup1 MyFalse vx400 vx401 vx41 (esEsMyBool MyFalse vx400)",fontsize=16,color="burlywood",shape="box"];41[label="vx400/MyTrue",fontsize=10,color="white",style="solid",shape="box"];13 -> 41[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 41 -> 16[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 42[label="vx400/MyFalse",fontsize=10,color="white",style="solid",shape="box"];13 -> 42[label="",style="solid", color="burlywood", weight=9]; 9.56/4.05 42 -> 17[label="",style="solid", color="burlywood", weight=3]; 9.56/4.05 14[label="lookup1 MyTrue MyTrue vx401 vx41 (esEsMyBool MyTrue MyTrue)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 9.56/4.05 15[label="lookup1 MyTrue MyFalse vx401 vx41 (esEsMyBool MyTrue MyFalse)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 9.56/4.05 16[label="lookup1 MyFalse MyTrue vx401 vx41 (esEsMyBool MyFalse MyTrue)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 9.56/4.05 17[label="lookup1 MyFalse MyFalse vx401 vx41 (esEsMyBool MyFalse MyFalse)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 9.56/4.05 18[label="lookup1 MyTrue MyTrue vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 9.56/4.05 19[label="lookup1 MyTrue MyFalse vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 9.56/4.05 20[label="lookup1 MyFalse MyTrue vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 9.56/4.05 21[label="lookup1 MyFalse MyFalse vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 9.56/4.05 22[label="Just vx401",fontsize=16,color="green",shape="box"];23[label="lookup0 MyTrue MyFalse vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3]; 9.56/4.05 24[label="lookup0 MyFalse MyTrue vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3]; 9.56/4.05 25[label="Just vx401",fontsize=16,color="green",shape="box"];26[label="lookup0 MyTrue MyFalse vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3]; 9.56/4.05 27[label="lookup0 MyFalse MyTrue vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3]; 9.56/4.05 28 -> 4[label="",style="dashed", color="red", weight=0]; 9.56/4.05 28[label="lookup MyTrue vx41",fontsize=16,color="magenta"];28 -> 30[label="",style="dashed", color="magenta", weight=3]; 9.56/4.05 28 -> 31[label="",style="dashed", color="magenta", weight=3]; 9.56/4.05 29 -> 4[label="",style="dashed", color="red", weight=0]; 9.56/4.05 29[label="lookup MyFalse vx41",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3]; 9.56/4.05 29 -> 33[label="",style="dashed", color="magenta", weight=3]; 9.56/4.05 30[label="vx41",fontsize=16,color="green",shape="box"];31[label="MyTrue",fontsize=16,color="green",shape="box"];32[label="vx41",fontsize=16,color="green",shape="box"];33[label="MyFalse",fontsize=16,color="green",shape="box"];} 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (6) 9.56/4.05 Obligation: 9.56/4.05 Q DP problem: 9.56/4.05 The TRS P consists of the following rules: 9.56/4.05 9.56/4.05 new_lookup(MyFalse, Cons(Tup2(MyTrue, vx401), vx41), h) -> new_lookup(MyFalse, vx41, h) 9.56/4.05 new_lookup(MyTrue, Cons(Tup2(MyFalse, vx401), vx41), h) -> new_lookup(MyTrue, vx41, h) 9.56/4.05 9.56/4.05 R is empty. 9.56/4.05 Q is empty. 9.56/4.05 We have to consider all minimal (P,Q,R)-chains. 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (7) DependencyGraphProof (EQUIVALENT) 9.56/4.05 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (8) 9.56/4.05 Complex Obligation (AND) 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (9) 9.56/4.05 Obligation: 9.56/4.05 Q DP problem: 9.56/4.05 The TRS P consists of the following rules: 9.56/4.05 9.56/4.05 new_lookup(MyTrue, Cons(Tup2(MyFalse, vx401), vx41), h) -> new_lookup(MyTrue, vx41, h) 9.56/4.05 9.56/4.05 R is empty. 9.56/4.05 Q is empty. 9.56/4.05 We have to consider all minimal (P,Q,R)-chains. 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (10) QDPSizeChangeProof (EQUIVALENT) 9.56/4.05 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. 9.56/4.05 9.56/4.05 From the DPs we obtained the following set of size-change graphs: 9.56/4.05 *new_lookup(MyTrue, Cons(Tup2(MyFalse, vx401), vx41), h) -> new_lookup(MyTrue, vx41, h) 9.56/4.05 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 9.56/4.05 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (11) 9.56/4.05 YES 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (12) 9.56/4.05 Obligation: 9.56/4.05 Q DP problem: 9.56/4.05 The TRS P consists of the following rules: 9.56/4.05 9.56/4.05 new_lookup(MyFalse, Cons(Tup2(MyTrue, vx401), vx41), h) -> new_lookup(MyFalse, vx41, h) 9.56/4.05 9.56/4.05 R is empty. 9.56/4.05 Q is empty. 9.56/4.05 We have to consider all minimal (P,Q,R)-chains. 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (13) QDPSizeChangeProof (EQUIVALENT) 9.56/4.05 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. 9.56/4.05 9.56/4.05 From the DPs we obtained the following set of size-change graphs: 9.56/4.05 *new_lookup(MyFalse, Cons(Tup2(MyTrue, vx401), vx41), h) -> new_lookup(MyFalse, vx41, h) 9.56/4.05 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 9.56/4.05 9.56/4.05 9.56/4.05 ---------------------------------------- 9.56/4.05 9.56/4.05 (14) 9.56/4.05 YES 9.56/4.09 EOF