MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: getGoto(Goto(int)) -> int getGotoFirst(IfGoto(i1,i2)) -> i1 getGotoSecond(IfGoto(i1,i2)) -> i2 getWrite(Write(int)) -> int instrConstrCheck(Goto(gtNat),Goto(gtNat2)) -> True() instrConstrCheck(Goto(gtNat),Halt()) -> False() instrConstrCheck(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Goto(gtNat),Left()) -> False() instrConstrCheck(Goto(gtNat),Right()) -> False() instrConstrCheck(Goto(gtNat),Write(wNat2)) -> False() instrConstrCheck(Halt(),Goto(gtNat2)) -> False() instrConstrCheck(Halt(),Halt()) -> True() instrConstrCheck(Halt(),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Halt(),Left()) -> False() instrConstrCheck(Halt(),Right()) -> False() instrConstrCheck(Halt(),Write(wNat2)) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Halt()) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> True() instrConstrCheck(IfGoto(igtNat1,igtNat2),Left()) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Right()) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> False() instrConstrCheck(Left(),Goto(gtNat2)) -> False() instrConstrCheck(Left(),Halt()) -> False() instrConstrCheck(Left(),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Left(),Left()) -> True() instrConstrCheck(Left(),Right()) -> False() instrConstrCheck(Left(),Write(wNat2)) -> False() instrConstrCheck(Right(),Goto(gtNat2)) -> False() instrConstrCheck(Right(),Halt()) -> False() instrConstrCheck(Right(),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Right(),Left()) -> False() instrConstrCheck(Right(),Right()) -> True() instrConstrCheck(Right(),Write(wNat2)) -> False() instrConstrCheck(Write(wNat),Goto(gtNat2)) -> False() instrConstrCheck(Write(wNat),Halt()) -> False() instrConstrCheck(Write(wNat),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Write(wNat),Left()) -> False() instrConstrCheck(Write(wNat),Right()) -> False() instrConstrCheck(Write(wNat),Write(wNat2)) -> True() instrsConstrCheck(Empty(),Empty()) -> True() instrsConstrCheck(Empty(),I(x,y)) -> False() instrsConstrCheck(I(l1,r1),Empty()) -> False() instrsConstrCheck(I(l1,r1),I(x,y)) -> True() instrsFirst(I(l,r)) -> l instrsSecond(I(l,r)) -> r lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() run(prog,tapeinput) -> turing(prog,Nil(),tapeinput,prog) turing(Empty(),revltape,rtape,prog) -> rtape turing(I(Goto(int),r),revltape,rtape,prog) -> turing(lookup(int,prog),revltape,rtape,prog) turing(I(Halt(),r),revltape,rtape,prog) -> rtape turing(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> turing[Ite](!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) turing(I(Left(),r),Cons(x,xs),rtape,prog) -> turing(r,xs,Cons(x,rtape),prog) turing(I(Left(),r),Nil(),rtape,prog) -> turing(r,Nil(),Cons(0(),rtape),prog) turing(I(Right(),r),revltape,Cons(x,xs),prog) -> turing(r,Cons(x,revltape),xs,prog) turing(I(Right(),r),revltape,Nil(),prog) -> turing(r,Cons(0(),revltape),Nil(),prog) turing(I(Write(int),r),revltape,Cons(x,xs),prog) -> turing(r,revltape,Cons(int,xs),prog) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) turing[Ite](False(),I(l,r),revltape,rtape,prog) -> turing(r,revltape,rtape,prog) turing[Ite](True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> turing(lookup(i2,prog),revltape,rtape,prog) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5} / {0/0,Cons/2,Empty/0 ,False/0,Goto/1,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,getGoto,getGotoFirst,getGotoSecond,getWrite ,instrConstrCheck,instrsConstrCheck,instrsFirst,instrsSecond,lookup,notEmpty,run,turing ,turing[Ite]} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs getGoto#(Goto(int)) -> c_1() getGotoFirst#(IfGoto(i1,i2)) -> c_2() getGotoSecond#(IfGoto(i1,i2)) -> c_3() getWrite#(Write(int)) -> c_4() instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() instrConstrCheck#(Goto(gtNat),Left()) -> c_8() instrConstrCheck#(Goto(gtNat),Right()) -> c_9() instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() instrConstrCheck#(Halt(),Halt()) -> c_12() instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() instrConstrCheck#(Halt(),Left()) -> c_14() instrConstrCheck#(Halt(),Right()) -> c_15() instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() instrConstrCheck#(Left(),Halt()) -> c_24() instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() instrConstrCheck#(Left(),Left()) -> c_26() instrConstrCheck#(Left(),Right()) -> c_27() instrConstrCheck#(Left(),Write(wNat2)) -> c_28() instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() instrConstrCheck#(Right(),Halt()) -> c_30() instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() instrConstrCheck#(Right(),Left()) -> c_32() instrConstrCheck#(Right(),Right()) -> c_33() instrConstrCheck#(Right(),Write(wNat2)) -> c_34() instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() instrConstrCheck#(Write(wNat),Halt()) -> c_36() instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() instrConstrCheck#(Write(wNat),Left()) -> c_38() instrConstrCheck#(Write(wNat),Right()) -> c_39() instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() instrsConstrCheck#(Empty(),Empty()) -> c_41() instrsConstrCheck#(Empty(),I(x,y)) -> c_42() instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() instrsFirst#(I(l,r)) -> c_45() instrsSecond#(I(l,r)) -> c_46() lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) notEmpty#(Cons(x,xs)) -> c_49() notEmpty#(Nil()) -> c_50() run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) Weak DPs !EQ#(0(),0()) -> c_61() !EQ#(0(),S(y)) -> c_62() !EQ#(S(x),0()) -> c_63() !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: getGoto#(Goto(int)) -> c_1() getGotoFirst#(IfGoto(i1,i2)) -> c_2() getGotoSecond#(IfGoto(i1,i2)) -> c_3() getWrite#(Write(int)) -> c_4() instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() instrConstrCheck#(Goto(gtNat),Left()) -> c_8() instrConstrCheck#(Goto(gtNat),Right()) -> c_9() instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() instrConstrCheck#(Halt(),Halt()) -> c_12() instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() instrConstrCheck#(Halt(),Left()) -> c_14() instrConstrCheck#(Halt(),Right()) -> c_15() instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() instrConstrCheck#(Left(),Halt()) -> c_24() instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() instrConstrCheck#(Left(),Left()) -> c_26() instrConstrCheck#(Left(),Right()) -> c_27() instrConstrCheck#(Left(),Write(wNat2)) -> c_28() instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() instrConstrCheck#(Right(),Halt()) -> c_30() instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() instrConstrCheck#(Right(),Left()) -> c_32() instrConstrCheck#(Right(),Right()) -> c_33() instrConstrCheck#(Right(),Write(wNat2)) -> c_34() instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() instrConstrCheck#(Write(wNat),Halt()) -> c_36() instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() instrConstrCheck#(Write(wNat),Left()) -> c_38() instrConstrCheck#(Write(wNat),Right()) -> c_39() instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() instrsConstrCheck#(Empty(),Empty()) -> c_41() instrsConstrCheck#(Empty(),I(x,y)) -> c_42() instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() instrsFirst#(I(l,r)) -> c_45() instrsSecond#(I(l,r)) -> c_46() lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) notEmpty#(Cons(x,xs)) -> c_49() notEmpty#(Nil()) -> c_50() run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: !EQ#(0(),0()) -> c_61() !EQ#(0(),S(y)) -> c_62() !EQ#(S(x),0()) -> c_63() !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) getGoto(Goto(int)) -> int getGotoFirst(IfGoto(i1,i2)) -> i1 getGotoSecond(IfGoto(i1,i2)) -> i2 getWrite(Write(int)) -> int instrConstrCheck(Goto(gtNat),Goto(gtNat2)) -> True() instrConstrCheck(Goto(gtNat),Halt()) -> False() instrConstrCheck(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Goto(gtNat),Left()) -> False() instrConstrCheck(Goto(gtNat),Right()) -> False() instrConstrCheck(Goto(gtNat),Write(wNat2)) -> False() instrConstrCheck(Halt(),Goto(gtNat2)) -> False() instrConstrCheck(Halt(),Halt()) -> True() instrConstrCheck(Halt(),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Halt(),Left()) -> False() instrConstrCheck(Halt(),Right()) -> False() instrConstrCheck(Halt(),Write(wNat2)) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Halt()) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> True() instrConstrCheck(IfGoto(igtNat1,igtNat2),Left()) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Right()) -> False() instrConstrCheck(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> False() instrConstrCheck(Left(),Goto(gtNat2)) -> False() instrConstrCheck(Left(),Halt()) -> False() instrConstrCheck(Left(),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Left(),Left()) -> True() instrConstrCheck(Left(),Right()) -> False() instrConstrCheck(Left(),Write(wNat2)) -> False() instrConstrCheck(Right(),Goto(gtNat2)) -> False() instrConstrCheck(Right(),Halt()) -> False() instrConstrCheck(Right(),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Right(),Left()) -> False() instrConstrCheck(Right(),Right()) -> True() instrConstrCheck(Right(),Write(wNat2)) -> False() instrConstrCheck(Write(wNat),Goto(gtNat2)) -> False() instrConstrCheck(Write(wNat),Halt()) -> False() instrConstrCheck(Write(wNat),IfGoto(igtNat12,igtNat22)) -> False() instrConstrCheck(Write(wNat),Left()) -> False() instrConstrCheck(Write(wNat),Right()) -> False() instrConstrCheck(Write(wNat),Write(wNat2)) -> True() instrsConstrCheck(Empty(),Empty()) -> True() instrsConstrCheck(Empty(),I(x,y)) -> False() instrsConstrCheck(I(l1,r1),Empty()) -> False() instrsConstrCheck(I(l1,r1),I(x,y)) -> True() instrsFirst(I(l,r)) -> l instrsSecond(I(l,r)) -> r lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() run(prog,tapeinput) -> turing(prog,Nil(),tapeinput,prog) turing(Empty(),revltape,rtape,prog) -> rtape turing(I(Goto(int),r),revltape,rtape,prog) -> turing(lookup(int,prog),revltape,rtape,prog) turing(I(Halt(),r),revltape,rtape,prog) -> rtape turing(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> turing[Ite](!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) turing(I(Left(),r),Cons(x,xs),rtape,prog) -> turing(r,xs,Cons(x,rtape),prog) turing(I(Left(),r),Nil(),rtape,prog) -> turing(r,Nil(),Cons(0(),rtape),prog) turing(I(Right(),r),revltape,Cons(x,xs),prog) -> turing(r,Cons(x,revltape),xs,prog) turing(I(Right(),r),revltape,Nil(),prog) -> turing(r,Cons(0(),revltape),Nil(),prog) turing(I(Write(int),r),revltape,Cons(x,xs),prog) -> turing(r,revltape,Cons(int,xs),prog) turing[Ite](False(),I(l,r),revltape,rtape,prog) -> turing(r,revltape,rtape,prog) turing[Ite](True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> turing(lookup(i2,prog),revltape,rtape,prog) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/2,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) !EQ#(0(),0()) -> c_61() !EQ#(0(),S(y)) -> c_62() !EQ#(S(x),0()) -> c_63() !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) getGoto#(Goto(int)) -> c_1() getGotoFirst#(IfGoto(i1,i2)) -> c_2() getGotoSecond#(IfGoto(i1,i2)) -> c_3() getWrite#(Write(int)) -> c_4() instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() instrConstrCheck#(Goto(gtNat),Left()) -> c_8() instrConstrCheck#(Goto(gtNat),Right()) -> c_9() instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() instrConstrCheck#(Halt(),Halt()) -> c_12() instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() instrConstrCheck#(Halt(),Left()) -> c_14() instrConstrCheck#(Halt(),Right()) -> c_15() instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() instrConstrCheck#(Left(),Halt()) -> c_24() instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() instrConstrCheck#(Left(),Left()) -> c_26() instrConstrCheck#(Left(),Right()) -> c_27() instrConstrCheck#(Left(),Write(wNat2)) -> c_28() instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() instrConstrCheck#(Right(),Halt()) -> c_30() instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() instrConstrCheck#(Right(),Left()) -> c_32() instrConstrCheck#(Right(),Right()) -> c_33() instrConstrCheck#(Right(),Write(wNat2)) -> c_34() instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() instrConstrCheck#(Write(wNat),Halt()) -> c_36() instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() instrConstrCheck#(Write(wNat),Left()) -> c_38() instrConstrCheck#(Write(wNat),Right()) -> c_39() instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() instrsConstrCheck#(Empty(),Empty()) -> c_41() instrsConstrCheck#(Empty(),I(x,y)) -> c_42() instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() instrsFirst#(I(l,r)) -> c_45() instrsSecond#(I(l,r)) -> c_46() lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) notEmpty#(Cons(x,xs)) -> c_49() notEmpty#(Nil()) -> c_50() run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: getGoto#(Goto(int)) -> c_1() getGotoFirst#(IfGoto(i1,i2)) -> c_2() getGotoSecond#(IfGoto(i1,i2)) -> c_3() getWrite#(Write(int)) -> c_4() instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() instrConstrCheck#(Goto(gtNat),Left()) -> c_8() instrConstrCheck#(Goto(gtNat),Right()) -> c_9() instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() instrConstrCheck#(Halt(),Halt()) -> c_12() instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() instrConstrCheck#(Halt(),Left()) -> c_14() instrConstrCheck#(Halt(),Right()) -> c_15() instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() instrConstrCheck#(Left(),Halt()) -> c_24() instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() instrConstrCheck#(Left(),Left()) -> c_26() instrConstrCheck#(Left(),Right()) -> c_27() instrConstrCheck#(Left(),Write(wNat2)) -> c_28() instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() instrConstrCheck#(Right(),Halt()) -> c_30() instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() instrConstrCheck#(Right(),Left()) -> c_32() instrConstrCheck#(Right(),Right()) -> c_33() instrConstrCheck#(Right(),Write(wNat2)) -> c_34() instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() instrConstrCheck#(Write(wNat),Halt()) -> c_36() instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() instrConstrCheck#(Write(wNat),Left()) -> c_38() instrConstrCheck#(Write(wNat),Right()) -> c_39() instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() instrsConstrCheck#(Empty(),Empty()) -> c_41() instrsConstrCheck#(Empty(),I(x,y)) -> c_42() instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() instrsFirst#(I(l,r)) -> c_45() instrsSecond#(I(l,r)) -> c_46() lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) notEmpty#(Cons(x,xs)) -> c_49() notEmpty#(Nil()) -> c_50() run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: !EQ#(0(),0()) -> c_61() !EQ#(0(),S(y)) -> c_62() !EQ#(S(x),0()) -> c_63() !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/2,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38 ,39,40,41,42,43,44,45,46,49,50} by application of Pre({1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37 ,38,39,40,41,42,43,44,45,46,49,50}) = {}. Here rules are labelled as follows: 1: getGoto#(Goto(int)) -> c_1() 2: getGotoFirst#(IfGoto(i1,i2)) -> c_2() 3: getGotoSecond#(IfGoto(i1,i2)) -> c_3() 4: getWrite#(Write(int)) -> c_4() 5: instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() 6: instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() 7: instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() 8: instrConstrCheck#(Goto(gtNat),Left()) -> c_8() 9: instrConstrCheck#(Goto(gtNat),Right()) -> c_9() 10: instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() 11: instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() 12: instrConstrCheck#(Halt(),Halt()) -> c_12() 13: instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() 14: instrConstrCheck#(Halt(),Left()) -> c_14() 15: instrConstrCheck#(Halt(),Right()) -> c_15() 16: instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() 17: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() 18: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() 19: instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() 20: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() 21: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() 22: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() 23: instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() 24: instrConstrCheck#(Left(),Halt()) -> c_24() 25: instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() 26: instrConstrCheck#(Left(),Left()) -> c_26() 27: instrConstrCheck#(Left(),Right()) -> c_27() 28: instrConstrCheck#(Left(),Write(wNat2)) -> c_28() 29: instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() 30: instrConstrCheck#(Right(),Halt()) -> c_30() 31: instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() 32: instrConstrCheck#(Right(),Left()) -> c_32() 33: instrConstrCheck#(Right(),Right()) -> c_33() 34: instrConstrCheck#(Right(),Write(wNat2)) -> c_34() 35: instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() 36: instrConstrCheck#(Write(wNat),Halt()) -> c_36() 37: instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() 38: instrConstrCheck#(Write(wNat),Left()) -> c_38() 39: instrConstrCheck#(Write(wNat),Right()) -> c_39() 40: instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() 41: instrsConstrCheck#(Empty(),Empty()) -> c_41() 42: instrsConstrCheck#(Empty(),I(x,y)) -> c_42() 43: instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() 44: instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() 45: instrsFirst#(I(l,r)) -> c_45() 46: instrsSecond#(I(l,r)) -> c_46() 47: lookup#(0(),instrs) -> c_47() 48: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) 49: notEmpty#(Cons(x,xs)) -> c_49() 50: notEmpty#(Nil()) -> c_50() 51: run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) 52: turing#(Empty(),revltape,rtape,prog) -> c_52() 53: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) 54: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 55: turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) 56: turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) 57: turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) 58: turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) 59: turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) 60: turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) 61: !EQ#(0(),0()) -> c_61() 62: !EQ#(0(),S(y)) -> c_62() 63: !EQ#(S(x),0()) -> c_63() 64: !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) 65: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) 66: turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: !EQ#(0(),0()) -> c_61() !EQ#(0(),S(y)) -> c_62() !EQ#(S(x),0()) -> c_63() !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) getGoto#(Goto(int)) -> c_1() getGotoFirst#(IfGoto(i1,i2)) -> c_2() getGotoSecond#(IfGoto(i1,i2)) -> c_3() getWrite#(Write(int)) -> c_4() instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() instrConstrCheck#(Goto(gtNat),Left()) -> c_8() instrConstrCheck#(Goto(gtNat),Right()) -> c_9() instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() instrConstrCheck#(Halt(),Halt()) -> c_12() instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() instrConstrCheck#(Halt(),Left()) -> c_14() instrConstrCheck#(Halt(),Right()) -> c_15() instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() instrConstrCheck#(Left(),Halt()) -> c_24() instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() instrConstrCheck#(Left(),Left()) -> c_26() instrConstrCheck#(Left(),Right()) -> c_27() instrConstrCheck#(Left(),Write(wNat2)) -> c_28() instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() instrConstrCheck#(Right(),Halt()) -> c_30() instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() instrConstrCheck#(Right(),Left()) -> c_32() instrConstrCheck#(Right(),Right()) -> c_33() instrConstrCheck#(Right(),Write(wNat2)) -> c_34() instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() instrConstrCheck#(Write(wNat),Halt()) -> c_36() instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() instrConstrCheck#(Write(wNat),Left()) -> c_38() instrConstrCheck#(Write(wNat),Right()) -> c_39() instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() instrsConstrCheck#(Empty(),Empty()) -> c_41() instrsConstrCheck#(Empty(),I(x,y)) -> c_42() instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() instrsFirst#(I(l,r)) -> c_45() instrsSecond#(I(l,r)) -> c_46() notEmpty#(Cons(x,xs)) -> c_49() notEmpty#(Nil()) -> c_50() turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/2,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:lookup#(0(),instrs) -> c_47() 2:S:lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) -->_1 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_1 lookup#(0(),instrs) -> c_47():1 3:S:run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 4:S:turing#(Empty(),revltape,rtape,prog) -> c_52() 5:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_2 lookup#(0(),instrs) -> c_47():1 6:S:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 7:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):66 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):65 -->_2 !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)):16 -->_2 !EQ#(S(x),0()) -> c_63():15 -->_2 !EQ#(0(),S(y)) -> c_62():14 -->_2 !EQ#(0(),0()) -> c_61():13 8:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 9:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 10:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 11:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 12:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 13:W:!EQ#(0(),0()) -> c_61() 14:W:!EQ#(0(),S(y)) -> c_62() 15:W:!EQ#(S(x),0()) -> c_63() 16:W:!EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) -->_1 !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)):16 -->_1 !EQ#(S(x),0()) -> c_63():15 -->_1 !EQ#(0(),S(y)) -> c_62():14 -->_1 !EQ#(0(),0()) -> c_61():13 17:W:getGoto#(Goto(int)) -> c_1() 18:W:getGotoFirst#(IfGoto(i1,i2)) -> c_2() 19:W:getGotoSecond#(IfGoto(i1,i2)) -> c_3() 20:W:getWrite#(Write(int)) -> c_4() 21:W:instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() 22:W:instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() 23:W:instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() 24:W:instrConstrCheck#(Goto(gtNat),Left()) -> c_8() 25:W:instrConstrCheck#(Goto(gtNat),Right()) -> c_9() 26:W:instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() 27:W:instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() 28:W:instrConstrCheck#(Halt(),Halt()) -> c_12() 29:W:instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() 30:W:instrConstrCheck#(Halt(),Left()) -> c_14() 31:W:instrConstrCheck#(Halt(),Right()) -> c_15() 32:W:instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() 33:W:instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() 34:W:instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() 35:W:instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() 36:W:instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() 37:W:instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() 38:W:instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() 39:W:instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() 40:W:instrConstrCheck#(Left(),Halt()) -> c_24() 41:W:instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() 42:W:instrConstrCheck#(Left(),Left()) -> c_26() 43:W:instrConstrCheck#(Left(),Right()) -> c_27() 44:W:instrConstrCheck#(Left(),Write(wNat2)) -> c_28() 45:W:instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() 46:W:instrConstrCheck#(Right(),Halt()) -> c_30() 47:W:instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() 48:W:instrConstrCheck#(Right(),Left()) -> c_32() 49:W:instrConstrCheck#(Right(),Right()) -> c_33() 50:W:instrConstrCheck#(Right(),Write(wNat2)) -> c_34() 51:W:instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() 52:W:instrConstrCheck#(Write(wNat),Halt()) -> c_36() 53:W:instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() 54:W:instrConstrCheck#(Write(wNat),Left()) -> c_38() 55:W:instrConstrCheck#(Write(wNat),Right()) -> c_39() 56:W:instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() 57:W:instrsConstrCheck#(Empty(),Empty()) -> c_41() 58:W:instrsConstrCheck#(Empty(),I(x,y)) -> c_42() 59:W:instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() 60:W:instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() 61:W:instrsFirst#(I(l,r)) -> c_45() 62:W:instrsSecond#(I(l,r)) -> c_46() 63:W:notEmpty#(Cons(x,xs)) -> c_49() 64:W:notEmpty#(Nil()) -> c_50() 65:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 66:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_2 lookup#(0(),instrs) -> c_47():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 64: notEmpty#(Nil()) -> c_50() 63: notEmpty#(Cons(x,xs)) -> c_49() 62: instrsSecond#(I(l,r)) -> c_46() 61: instrsFirst#(I(l,r)) -> c_45() 60: instrsConstrCheck#(I(l1,r1),I(x,y)) -> c_44() 59: instrsConstrCheck#(I(l1,r1),Empty()) -> c_43() 58: instrsConstrCheck#(Empty(),I(x,y)) -> c_42() 57: instrsConstrCheck#(Empty(),Empty()) -> c_41() 56: instrConstrCheck#(Write(wNat),Write(wNat2)) -> c_40() 55: instrConstrCheck#(Write(wNat),Right()) -> c_39() 54: instrConstrCheck#(Write(wNat),Left()) -> c_38() 53: instrConstrCheck#(Write(wNat),IfGoto(igtNat12,igtNat22)) -> c_37() 52: instrConstrCheck#(Write(wNat),Halt()) -> c_36() 51: instrConstrCheck#(Write(wNat),Goto(gtNat2)) -> c_35() 50: instrConstrCheck#(Right(),Write(wNat2)) -> c_34() 49: instrConstrCheck#(Right(),Right()) -> c_33() 48: instrConstrCheck#(Right(),Left()) -> c_32() 47: instrConstrCheck#(Right(),IfGoto(igtNat12,igtNat22)) -> c_31() 46: instrConstrCheck#(Right(),Halt()) -> c_30() 45: instrConstrCheck#(Right(),Goto(gtNat2)) -> c_29() 44: instrConstrCheck#(Left(),Write(wNat2)) -> c_28() 43: instrConstrCheck#(Left(),Right()) -> c_27() 42: instrConstrCheck#(Left(),Left()) -> c_26() 41: instrConstrCheck#(Left(),IfGoto(igtNat12,igtNat22)) -> c_25() 40: instrConstrCheck#(Left(),Halt()) -> c_24() 39: instrConstrCheck#(Left(),Goto(gtNat2)) -> c_23() 38: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Write(wNat2)) -> c_22() 37: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Right()) -> c_21() 36: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Left()) -> c_20() 35: instrConstrCheck#(IfGoto(igtNat1,igtNat2),IfGoto(igtNat12,igtNat22)) -> c_19() 34: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Halt()) -> c_18() 33: instrConstrCheck#(IfGoto(igtNat1,igtNat2),Goto(gtNat2)) -> c_17() 32: instrConstrCheck#(Halt(),Write(wNat2)) -> c_16() 31: instrConstrCheck#(Halt(),Right()) -> c_15() 30: instrConstrCheck#(Halt(),Left()) -> c_14() 29: instrConstrCheck#(Halt(),IfGoto(igtNat12,igtNat22)) -> c_13() 28: instrConstrCheck#(Halt(),Halt()) -> c_12() 27: instrConstrCheck#(Halt(),Goto(gtNat2)) -> c_11() 26: instrConstrCheck#(Goto(gtNat),Write(wNat2)) -> c_10() 25: instrConstrCheck#(Goto(gtNat),Right()) -> c_9() 24: instrConstrCheck#(Goto(gtNat),Left()) -> c_8() 23: instrConstrCheck#(Goto(gtNat),IfGoto(igtNat12,igtNat22)) -> c_7() 22: instrConstrCheck#(Goto(gtNat),Halt()) -> c_6() 21: instrConstrCheck#(Goto(gtNat),Goto(gtNat2)) -> c_5() 20: getWrite#(Write(int)) -> c_4() 19: getGotoSecond#(IfGoto(i1,i2)) -> c_3() 18: getGotoFirst#(IfGoto(i1,i2)) -> c_2() 17: getGoto#(Goto(int)) -> c_1() 16: !EQ#(S(x),S(y)) -> c_64(!EQ#(x,y)) 13: !EQ#(0(),0()) -> c_61() 14: !EQ#(0(),S(y)) -> c_62() 15: !EQ#(S(x),0()) -> c_63() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/2,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:lookup#(0(),instrs) -> c_47() 2:S:lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) -->_1 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_1 lookup#(0(),instrs) -> c_47():1 3:S:run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 4:S:turing#(Empty(),revltape,rtape,prog) -> c_52() 5:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_2 lookup#(0(),instrs) -> c_47():1 6:S:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 7:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):66 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):65 8:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 9:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 10:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 11:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 12:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 65:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 66:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog) ,!EQ#(x,i1)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_2 lookup#(0(),instrs) -> c_47():1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) * Step 6: RemoveHeads MAYBE + Considered Problem: - Strict DPs: lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:lookup#(0(),instrs) -> c_47() 2:S:lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) -->_1 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_1 lookup#(0(),instrs) -> c_47():1 3:S:run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 4:S:turing#(Empty(),revltape,rtape,prog) -> c_52() 5:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_2 lookup#(0(),instrs) -> c_47():1 6:S:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 7:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):14 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):13 8:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 9:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 10:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 11:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 12:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 13:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 14:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_2 lookup#(0(),instrs) -> c_47():1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(3,run#(prog,tapeinput) -> c_51(turing#(prog,Nil(),tapeinput,prog)))] * Step 7: Decompose MAYBE + Considered Problem: - Strict DPs: lookup#(0(),instrs) -> c_47() lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: lookup#(0(),instrs) -> c_47() - Weak DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0 ,Goto/1,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0 ,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0 ,c_37/0,c_38/0,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1 ,c_52/0,c_53/2,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} Problem (S) - Strict DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: lookup#(0(),instrs) -> c_47() turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0 ,Goto/1,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0 ,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0 ,c_37/0,c_38/0,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1 ,c_52/0,c_53/2,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} ** Step 7.a:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: lookup#(0(),instrs) -> c_47() - Weak DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:lookup#(0(),instrs) -> c_47() 2:W:lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) -->_1 lookup#(0(),instrs) -> c_47():1 -->_1 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 4:W:turing#(Empty(),revltape,rtape,prog) -> c_52() 5:W:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_2 lookup#(0(),instrs) -> c_47():1 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 6:W:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 7:W:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):14 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):13 8:W:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 9:W:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 10:W:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 11:W:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 12:W:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 13:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 14:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_2 lookup#(0(),instrs) -> c_47():1 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):11 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):12 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 4: turing#(Empty(),revltape,rtape,prog) -> c_52() ** Step 7.a:2: Failure MAYBE + Considered Problem: - Strict DPs: lookup#(0(),instrs) -> c_47() - Weak DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 7.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: lookup#(0(),instrs) -> c_47() turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) -->_1 lookup#(0(),instrs) -> c_47():11 -->_1 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):1 2:S:turing#(Empty(),revltape,rtape,prog) -> c_52() 3:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_2 lookup#(0(),instrs) -> c_47():11 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):1 4:S:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 5:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):13 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):12 6:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 7:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 8:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 9:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 10:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 11:W:lookup#(0(),instrs) -> c_47() 12:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 13:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_2 lookup#(0(),instrs) -> c_47():11 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: lookup#(0(),instrs) -> c_47() ** Step 7.b:2: Decompose MAYBE + Considered Problem: - Strict DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) - Weak DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0 ,Goto/1,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0 ,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0 ,c_37/0,c_38/0,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1 ,c_52/0,c_53/2,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} Problem (S) - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0 ,Goto/1,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0 ,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0 ,c_37/0,c_38/0,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1 ,c_52/0,c_53/2,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} *** Step 7.b:2.a:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) - Weak DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) -->_1 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):1 2:W:turing#(Empty(),revltape,rtape,prog) -> c_52() 3:W:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):1 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 4:W:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 5:W:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):13 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):12 6:W:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 7:W:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 8:W:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 9:W:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 10:W:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 12:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 13:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):1 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):9 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 2: turing#(Empty(),revltape,rtape,prog) -> c_52() *** Step 7.b:2.a:2: Failure MAYBE + Considered Problem: - Strict DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) - Weak DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 7.b:2.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:turing#(Empty(),revltape,rtape,prog) -> c_52() 2:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):10 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 3:S:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 4:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):12 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):11 5:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 6:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 7:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 8:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 9:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 10:W:lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) -->_1 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):10 11:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 12:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_2 lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)):10 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: lookup#(S(x),I(l,r)) -> c_48(lookup#(x,r)) *** Step 7.b:2.b:2: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/2 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/2} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:turing#(Empty(),revltape,rtape,prog) -> c_52() 2:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 3:S:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 4:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)):12 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):11 5:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 6:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 7:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 8:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 9:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 11:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 12:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog) ,lookup#(i2,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog) ,lookup#(int,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) *** Step 7.b:2.b:3: Decompose MAYBE + Considered Problem: - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() - Weak DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0 ,Goto/1,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0 ,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0 ,c_37/0,c_38/0,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1 ,c_52/0,c_53/1,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} Problem (S) - Strict DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0 ,Goto/1,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0 ,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0 ,c_37/0,c_38/0,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1 ,c_52/0,c_53/1,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} **** Step 7.b:2.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() - Weak DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:turing#(Empty(),revltape,rtape,prog) -> c_52() 2:W:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 3:W:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 4:W:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)):11 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):10 5:W:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 6:W:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 7:W:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 8:W:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 9:W:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 10:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 11:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):6 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):7 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):8 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() **** Step 7.b:2.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() - Weak DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: turing#(Empty(),revltape,rtape,prog) -> c_52() The strictly oriented rules are moved into the weak component. ***** Step 7.b:2.b:3.a:2.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() - Weak DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_53) = {1}, uargs(c_55) = {1}, uargs(c_56) = {1}, uargs(c_57) = {1}, uargs(c_58) = {1}, uargs(c_59) = {1}, uargs(c_60) = {1}, uargs(c_65) = {1}, uargs(c_66) = {1} Following symbols are considered usable: {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite#,instrConstrCheck#,instrsConstrCheck#,instrsFirst# ,instrsSecond#,lookup#,notEmpty#,run#,turing#,turing[Ite]#} TcT has computed the following interpretation: p(!EQ) = [0] p(0) = [0] p(Cons) = [0] p(Empty) = [0] p(False) = [0] p(Goto) = [0] p(Halt) = [0] p(I) = [1] p(IfGoto) = [0] p(Left) = [0] p(Nil) = [0] p(Right) = [0] p(S) = [0] p(True) = [0] p(Write) = [0] p(getGoto) = [0] p(getGotoFirst) = [0] p(getGotoSecond) = [0] p(getWrite) = [0] p(instrConstrCheck) = [0] p(instrsConstrCheck) = [0] p(instrsFirst) = [0] p(instrsSecond) = [1] x1 + [0] p(lookup) = [1] x1 + [0] p(notEmpty) = [0] p(run) = [0] p(turing) = [0] p(turing[Ite]) = [0] p(!EQ#) = [0] p(getGoto#) = [0] p(getGotoFirst#) = [0] p(getGotoSecond#) = [0] p(getWrite#) = [0] p(instrConstrCheck#) = [0] p(instrsConstrCheck#) = [0] p(instrsFirst#) = [0] p(instrsSecond#) = [0] p(lookup#) = [0] p(notEmpty#) = [0] p(run#) = [0] p(turing#) = [2] p(turing[Ite]#) = [2] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [0] p(c_38) = [0] p(c_39) = [0] p(c_40) = [0] p(c_41) = [0] p(c_42) = [0] p(c_43) = [0] p(c_44) = [0] p(c_45) = [0] p(c_46) = [0] p(c_47) = [0] p(c_48) = [0] p(c_49) = [0] p(c_50) = [0] p(c_51) = [0] p(c_52) = [0] p(c_53) = [1] x1 + [0] p(c_54) = [0] p(c_55) = [1] x1 + [0] p(c_56) = [1] x1 + [0] p(c_57) = [1] x1 + [0] p(c_58) = [1] x1 + [0] p(c_59) = [1] x1 + [0] p(c_60) = [1] x1 + [0] p(c_61) = [0] p(c_62) = [0] p(c_63) = [0] p(c_64) = [1] x1 + [0] p(c_65) = [1] x1 + [0] p(c_66) = [1] x1 + [0] Following rules are strictly oriented: turing#(Empty(),revltape,rtape,prog) = [2] > [0] = c_52() Following rules are (at-least) weakly oriented: turing#(I(Goto(int),r),revltape,rtape,prog) = [2] >= [2] = c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) = [2] >= [2] = c_55(turing[Ite]#(!EQ(x,i1),I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) = [2] >= [2] = c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) = [2] >= [2] = c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) = [2] >= [2] = c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) = [2] >= [2] = c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) = [2] >= [2] = c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) = [2] >= [2] = c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) = [2] >= [2] = c_66(turing#(lookup(i2,prog),revltape,rtape,prog)) ***** Step 7.b:2.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.b:2.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:turing#(Empty(),revltape,rtape,prog) -> c_52() 2:W:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 3:W:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)):10 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):9 4:W:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 5:W:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 6:W:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 7:W:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 8:W:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 9:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 10:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):2 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) 10: turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) 3: turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) 9: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) 8: turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) 6: turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) 5: turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) 4: turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) 7: turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) 1: turing#(Empty(),revltape,rtape,prog) -> c_52() ***** Step 7.b:2.b:3.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 7.b:2.b:3.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing#(Empty(),revltape,rtape,prog) -> c_52() turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 2:S:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 3:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)):11 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):10 4:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 5:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 6:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 7:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 8:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 9:W:turing#(Empty(),revltape,rtape,prog) -> c_52() 10:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 11:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) -->_1 turing#(Empty(),revltape,rtape,prog) -> c_52():9 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):6 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):3 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: turing#(Empty(),revltape,rtape,prog) -> c_52() **** Step 7.b:2.b:3.b:2: PredecessorEstimationCP MAYBE + Considered Problem: - Strict DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() The strictly oriented rules are moved into the weak component. ***** Step 7.b:2.b:3.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_53) = {1}, uargs(c_55) = {1}, uargs(c_56) = {1}, uargs(c_57) = {1}, uargs(c_58) = {1}, uargs(c_59) = {1}, uargs(c_60) = {1}, uargs(c_65) = {1}, uargs(c_66) = {1} Following symbols are considered usable: {!EQ,!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite#,instrConstrCheck#,instrsConstrCheck# ,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing#,turing[Ite]#} TcT has computed the following interpretation: p(!EQ) = [2] p(0) = [0] p(Cons) = [1] x2 + [0] p(Empty) = [4] p(False) = [2] p(Goto) = [4] p(Halt) = [1] p(I) = [0] p(IfGoto) = [0] p(Left) = [0] p(Nil) = [2] p(Right) = [4] p(S) = [1] x1 + [4] p(True) = [2] p(Write) = [0] p(getGoto) = [4] x1 + [2] p(getGotoFirst) = [1] x1 + [0] p(getGotoSecond) = [1] x1 + [1] p(getWrite) = [1] p(instrConstrCheck) = [0] p(instrsConstrCheck) = [1] p(instrsFirst) = [0] p(instrsSecond) = [0] p(lookup) = [1] x1 + [6] p(notEmpty) = [1] x1 + [0] p(run) = [1] p(turing) = [0] p(turing[Ite]) = [1] x1 + [2] x2 + [1] x4 + [4] x5 + [1] p(!EQ#) = [1] p(getGoto#) = [1] x1 + [4] p(getGotoFirst#) = [4] x1 + [0] p(getGotoSecond#) = [1] p(getWrite#) = [0] p(instrConstrCheck#) = [4] x2 + [0] p(instrsConstrCheck#) = [0] p(instrsFirst#) = [1] p(instrsSecond#) = [1] x1 + [0] p(lookup#) = [0] p(notEmpty#) = [0] p(run#) = [2] p(turing#) = [4] x3 + [4] p(turing[Ite]#) = [2] x1 + [4] x4 + [0] p(c_1) = [4] p(c_2) = [1] p(c_3) = [2] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [2] p(c_11) = [0] p(c_12) = [2] p(c_13) = [0] p(c_14) = [1] p(c_15) = [1] p(c_16) = [2] p(c_17) = [0] p(c_18) = [0] p(c_19) = [2] p(c_20) = [0] p(c_21) = [1] p(c_22) = [1] p(c_23) = [2] p(c_24) = [1] p(c_25) = [1] p(c_26) = [1] p(c_27) = [1] p(c_28) = [1] p(c_29) = [1] p(c_30) = [1] p(c_31) = [4] p(c_32) = [4] p(c_33) = [0] p(c_34) = [4] p(c_35) = [0] p(c_36) = [1] p(c_37) = [4] p(c_38) = [0] p(c_39) = [2] p(c_40) = [1] p(c_41) = [2] p(c_42) = [0] p(c_43) = [0] p(c_44) = [0] p(c_45) = [0] p(c_46) = [0] p(c_47) = [1] p(c_48) = [1] p(c_49) = [1] p(c_50) = [1] p(c_51) = [1] x1 + [4] p(c_52) = [4] p(c_53) = [1] x1 + [0] p(c_54) = [2] p(c_55) = [1] x1 + [0] p(c_56) = [1] x1 + [0] p(c_57) = [1] x1 + [0] p(c_58) = [1] x1 + [0] p(c_59) = [1] x1 + [0] p(c_60) = [1] x1 + [0] p(c_61) = [2] p(c_62) = [1] p(c_63) = [1] p(c_64) = [0] p(c_65) = [1] x1 + [0] p(c_66) = [1] x1 + [0] Following rules are strictly oriented: turing#(I(Halt(),r),revltape,rtape,prog) = [4] rtape + [4] > [2] = c_54() Following rules are (at-least) weakly oriented: turing#(I(Goto(int),r),revltape,rtape,prog) = [4] rtape + [4] >= [4] rtape + [4] = c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) = [4] xs + [4] >= [4] xs + [4] = c_55(turing[Ite]#(!EQ(x,i1),I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) = [4] rtape + [4] >= [4] rtape + [4] = c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) = [4] rtape + [4] >= [4] rtape + [4] = c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) = [4] xs + [4] >= [4] xs + [4] = c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) = [12] >= [12] = c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) = [4] xs + [4] >= [4] xs + [4] = c_60(turing#(r,revltape,Cons(int,xs),prog)) turing[Ite]#(False(),I(l,r),revltape,rtape,prog) = [4] rtape + [4] >= [4] rtape + [4] = c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) = [4] rtape + [4] >= [4] rtape + [4] = c_66(turing#(lookup(i2,prog),revltape,rtape,prog)) !EQ(0(),0()) = [2] >= [2] = True() !EQ(0(),S(y)) = [2] >= [2] = False() !EQ(S(x),0()) = [2] >= [2] = False() !EQ(S(x),S(y)) = [2] >= [2] = !EQ(x,y) ***** Step 7.b:2.b:3.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.b:2.b:3.b:2.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):7 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):6 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):2 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 2:S:turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) -->_1 turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)):10 -->_1 turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)):9 3:S:turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):4 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 4:S:turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):5 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):4 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 5:S:turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):7 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):6 -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):5 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 6:S:turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):6 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):3 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 7:S:turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):7 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 8:W:turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() 9:W:turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):7 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):6 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 10:W:turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) -->_1 turing#(I(Halt(),r),revltape,rtape,prog) -> c_54():8 -->_1 turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)):7 -->_1 turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)):6 -->_1 turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)):5 -->_1 turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)):4 -->_1 turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)):3 -->_1 turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)):2 -->_1 turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: turing#(I(Halt(),r),revltape,rtape,prog) -> c_54() ***** Step 7.b:2.b:3.b:2.b:2: Failure MAYBE + Considered Problem: - Strict DPs: turing#(I(Goto(int),r),revltape,rtape,prog) -> c_53(turing#(lookup(int,prog),revltape,rtape,prog)) turing#(I(IfGoto(i1,i2),r),revltape,Cons(x,xs),prog) -> c_55(turing[Ite]#(!EQ(x,i1) ,I(IfGoto(i1,i2),r) ,revltape ,Cons(x,xs) ,prog)) turing#(I(Left(),r),Cons(x,xs),rtape,prog) -> c_56(turing#(r,xs,Cons(x,rtape),prog)) turing#(I(Left(),r),Nil(),rtape,prog) -> c_57(turing#(r,Nil(),Cons(0(),rtape),prog)) turing#(I(Right(),r),revltape,Cons(x,xs),prog) -> c_58(turing#(r,Cons(x,revltape),xs,prog)) turing#(I(Right(),r),revltape,Nil(),prog) -> c_59(turing#(r,Cons(0(),revltape),Nil(),prog)) turing#(I(Write(int),r),revltape,Cons(x,xs),prog) -> c_60(turing#(r,revltape,Cons(int,xs),prog)) - Weak DPs: turing[Ite]#(False(),I(l,r),revltape,rtape,prog) -> c_65(turing#(r,revltape,rtape,prog)) turing[Ite]#(True(),I(IfGoto(i1,i2),r),revltape,rtape,prog) -> c_66(turing#(lookup(i2,prog) ,revltape ,rtape ,prog)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) lookup(0(),instrs) -> instrs lookup(S(x),I(l,r)) -> lookup(x,r) - Signature: {!EQ/2,getGoto/1,getGotoFirst/1,getGotoSecond/1,getWrite/1,instrConstrCheck/2,instrsConstrCheck/2 ,instrsFirst/1,instrsSecond/1,lookup/2,notEmpty/1,run/2,turing/4,turing[Ite]/5,!EQ#/2,getGoto#/1 ,getGotoFirst#/1,getGotoSecond#/1,getWrite#/1,instrConstrCheck#/2,instrsConstrCheck#/2,instrsFirst#/1 ,instrsSecond#/1,lookup#/2,notEmpty#/1,run#/2,turing#/4,turing[Ite]#/5} / {0/0,Cons/2,Empty/0,False/0,Goto/1 ,Halt/0,I/2,IfGoto/2,Left/0,Nil/0,Right/0,S/1,True/0,Write/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0 ,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0,c_31/0,c_32/0,c_33/0,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/0,c_40/0,c_41/0,c_42/0,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/1,c_49/0,c_50/0,c_51/1,c_52/0,c_53/1 ,c_54/0,c_55/1,c_56/1,c_57/1,c_58/1,c_59/1,c_60/1,c_61/0,c_62/0,c_63/0,c_64/1,c_65/1,c_66/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,getGoto#,getGotoFirst#,getGotoSecond#,getWrite# ,instrConstrCheck#,instrsConstrCheck#,instrsFirst#,instrsSecond#,lookup#,notEmpty#,run#,turing# ,turing[Ite]#} and constructors {0,Cons,Empty,False,Goto,Halt,I,IfGoto,Left,Nil,Right,S,True,Write} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE