WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: DIV(0(),s(z0)) -> c5() DIV(s(z0),s(z1)) -> c6(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ(z0,z1)) DIV(s(z0),s(z1)) -> c7(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),DIV(minus(z0,z1),s(z1)),MINUS(z0,z1)) GEQ(z0,0()) -> c2() GEQ(0(),s(z0)) -> c3() GEQ(s(z0),s(z1)) -> c4(GEQ(z0,z1)) IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() MINUS(0(),z0) -> c() MINUS(s(z0),s(z1)) -> c1(MINUS(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIV,GEQ,IF,MINUS,div,geq,if,minus} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DIV(0(),s(z0)) -> c5() DIV(s(z0),s(z1)) -> c6(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ(z0,z1)) DIV(s(z0),s(z1)) -> c7(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),DIV(minus(z0,z1),s(z1)),MINUS(z0,z1)) GEQ(z0,0()) -> c2() GEQ(0(),s(z0)) -> c3() GEQ(s(z0),s(z1)) -> c4(GEQ(z0,z1)) IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() MINUS(0(),z0) -> c() MINUS(s(z0),s(z1)) -> c1(MINUS(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIV,GEQ,IF,MINUS,div,geq,if,minus} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DIV(0(),s(z0)) -> c5() DIV(s(z0),s(z1)) -> c6(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ(z0,z1)) DIV(s(z0),s(z1)) -> c7(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),DIV(minus(z0,z1),s(z1)),MINUS(z0,z1)) GEQ(z0,0()) -> c2() GEQ(0(),s(z0)) -> c3() GEQ(s(z0),s(z1)) -> c4(GEQ(z0,z1)) IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() MINUS(0(),z0) -> c() MINUS(s(z0),s(z1)) -> c1(MINUS(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIV,GEQ,IF,MINUS,div,geq,if,minus} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: GEQ(x,y){x -> s(x),y -> s(y)} = GEQ(s(x),s(y)) ->^+ c4(GEQ(x,y)) = C[GEQ(x,y) = GEQ(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: DIV(0(),s(z0)) -> c5() DIV(s(z0),s(z1)) -> c6(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ(z0,z1)) DIV(s(z0),s(z1)) -> c7(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),DIV(minus(z0,z1),s(z1)),MINUS(z0,z1)) GEQ(z0,0()) -> c2() GEQ(0(),s(z0)) -> c3() GEQ(s(z0),s(z1)) -> c4(GEQ(z0,z1)) IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() MINUS(0(),z0) -> c() MINUS(s(z0),s(z1)) -> c1(MINUS(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIV,GEQ,IF,MINUS,div,geq,if,minus} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs DIV#(0(),s(z0)) -> c_1() DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) GEQ#(z0,0()) -> c_4() GEQ#(0(),s(z0)) -> c_5() GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() MINUS#(0(),z0) -> c_9() MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) Weak DPs div#(0(),s(z0)) -> c_11() div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) geq#(z0,0()) -> c_13() geq#(0(),s(z0)) -> c_14() geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) if#(false(),z0,z1) -> c_16() if#(true(),z0,z1) -> c_17() minus#(0(),z0) -> c_18() minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(0(),s(z0)) -> c_1() DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) GEQ#(z0,0()) -> c_4() GEQ#(0(),s(z0)) -> c_5() GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() MINUS#(0(),z0) -> c_9() MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Strict TRS: DIV(0(),s(z0)) -> c5() DIV(s(z0),s(z1)) -> c6(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ(z0,z1)) DIV(s(z0),s(z1)) -> c7(IF(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),DIV(minus(z0,z1),s(z1)),MINUS(z0,z1)) GEQ(z0,0()) -> c2() GEQ(0(),s(z0)) -> c3() GEQ(s(z0),s(z1)) -> c4(GEQ(z0,z1)) IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() MINUS(0(),z0) -> c() MINUS(s(z0),s(z1)) -> c1(MINUS(z0,z1)) - Weak DPs: div#(0(),s(z0)) -> c_11() div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) geq#(z0,0()) -> c_13() geq#(0(),s(z0)) -> c_14() geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) if#(false(),z0,z1) -> c_16() if#(true(),z0,z1) -> c_17() minus#(0(),z0) -> c_18() minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/3 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) DIV#(0(),s(z0)) -> c_1() DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) GEQ#(z0,0()) -> c_4() GEQ#(0(),s(z0)) -> c_5() GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() MINUS#(0(),z0) -> c_9() MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) div#(0(),s(z0)) -> c_11() div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) geq#(z0,0()) -> c_13() geq#(0(),s(z0)) -> c_14() geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) if#(false(),z0,z1) -> c_16() if#(true(),z0,z1) -> c_17() minus#(0(),z0) -> c_18() minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(0(),s(z0)) -> c_1() DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) GEQ#(z0,0()) -> c_4() GEQ#(0(),s(z0)) -> c_5() GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() MINUS#(0(),z0) -> c_9() MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak DPs: div#(0(),s(z0)) -> c_11() div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) geq#(z0,0()) -> c_13() geq#(0(),s(z0)) -> c_14() geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) if#(false(),z0,z1) -> c_16() if#(true(),z0,z1) -> c_17() minus#(0(),z0) -> c_18() minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/3 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,7,8,9} by application of Pre({1,4,5,7,8,9}) = {2,3,6,10}. Here rules are labelled as follows: 1: DIV#(0(),s(z0)) -> c_1() 2: DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) 3: DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) 4: GEQ#(z0,0()) -> c_4() 5: GEQ#(0(),s(z0)) -> c_5() 6: GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) 7: IF#(false(),z0,z1) -> c_7() 8: IF#(true(),z0,z1) -> c_8() 9: MINUS#(0(),z0) -> c_9() 10: MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) 11: div#(0(),s(z0)) -> c_11() 12: div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) 13: geq#(z0,0()) -> c_13() 14: geq#(0(),s(z0)) -> c_14() 15: geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) 16: if#(false(),z0,z1) -> c_16() 17: if#(true(),z0,z1) -> c_17() 18: minus#(0(),z0) -> c_18() 19: minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) ** Step 1.b:4: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak DPs: DIV#(0(),s(z0)) -> c_1() GEQ#(z0,0()) -> c_4() GEQ#(0(),s(z0)) -> c_5() IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() MINUS#(0(),z0) -> c_9() div#(0(),s(z0)) -> c_11() div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) geq#(z0,0()) -> c_13() geq#(0(),s(z0)) -> c_14() geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) if#(false(),z0,z1) -> c_16() if#(true(),z0,z1) -> c_17() minus#(0(),z0) -> c_18() minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/3 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) -->_2 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 -->_1 IF#(true(),z0,z1) -> c_8():9 -->_1 IF#(false(),z0,z1) -> c_7():8 -->_2 GEQ#(0(),s(z0)) -> c_5():7 -->_2 GEQ#(z0,0()) -> c_4():6 2:S:DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) -->_3 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):4 -->_3 MINUS#(0(),z0) -> c_9():10 -->_1 IF#(true(),z0,z1) -> c_8():9 -->_1 IF#(false(),z0,z1) -> c_7():8 -->_2 DIV#(0(),s(z0)) -> c_1():5 -->_2 DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)):2 -->_2 DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)):1 3:S:GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) -->_1 GEQ#(0(),s(z0)) -> c_5():7 -->_1 GEQ#(z0,0()) -> c_4():6 -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 4:S:MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) -->_1 MINUS#(0(),z0) -> c_9():10 -->_1 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):4 5:W:DIV#(0(),s(z0)) -> c_1() 6:W:GEQ#(z0,0()) -> c_4() 7:W:GEQ#(0(),s(z0)) -> c_5() 8:W:IF#(false(),z0,z1) -> c_7() 9:W:IF#(true(),z0,z1) -> c_8() 10:W:MINUS#(0(),z0) -> c_9() 11:W:div#(0(),s(z0)) -> c_11() 12:W:div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) -->_1 if#(true(),z0,z1) -> c_17():17 -->_1 if#(false(),z0,z1) -> c_16():16 13:W:geq#(z0,0()) -> c_13() 14:W:geq#(0(),s(z0)) -> c_14() 15:W:geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) -->_1 geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)):15 -->_1 geq#(0(),s(z0)) -> c_14():14 -->_1 geq#(z0,0()) -> c_13():13 16:W:if#(false(),z0,z1) -> c_16() 17:W:if#(true(),z0,z1) -> c_17() 18:W:minus#(0(),z0) -> c_18() 19:W:minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) -->_1 minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)):19 -->_1 minus#(0(),z0) -> c_18():18 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 19: minus#(s(z0),s(z1)) -> c_19(minus#(z0,z1)) 18: minus#(0(),z0) -> c_18() 15: geq#(s(z0),s(z1)) -> c_15(geq#(z0,z1)) 14: geq#(0(),s(z0)) -> c_14() 13: geq#(z0,0()) -> c_13() 12: div#(s(z0),s(z1)) -> c_12(if#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0())) 16: if#(false(),z0,z1) -> c_16() 17: if#(true(),z0,z1) -> c_17() 11: div#(0(),s(z0)) -> c_11() 5: DIV#(0(),s(z0)) -> c_1() 10: MINUS#(0(),z0) -> c_9() 8: IF#(false(),z0,z1) -> c_7() 9: IF#(true(),z0,z1) -> c_8() 6: GEQ#(z0,0()) -> c_4() 7: GEQ#(0(),s(z0)) -> c_5() ** Step 1.b:5: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/3 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)) -->_2 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 2:S:DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)) -->_3 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):4 -->_2 DIV#(s(z0),s(z1)) -> c_3(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) ,DIV#(minus(z0,z1),s(z1)) ,MINUS#(z0,z1)):2 -->_2 DIV#(s(z0),s(z1)) -> c_2(IF#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()),GEQ#(z0,z1)):1 3:S:GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 4:S:MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) ** Step 1.b:6: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: div(0(),s(z0)) -> 0() div(s(z0),s(z1)) -> if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0()) geq(z0,0()) -> true() geq(0(),s(z0)) -> false() geq(s(z0),s(z1)) -> geq(z0,z1) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) ** Step 1.b:7: Decompose. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + 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: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0 ,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} Problem (S) - Strict DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0 ,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} *** Step 1.b:7.a:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 2:W:DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) -->_1 DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)):1 -->_2 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):4 -->_1 DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)):2 3:S:GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 4:W:MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) *** Step 1.b:7.a:2: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 2:W:DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) -->_1 DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)):1 -->_1 DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)):2 3:S:GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))) *** Step 1.b:7.a:3: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + 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: 1: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) 2: GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) The strictly oriented rules are moved into the weak component. **** Step 1.b:7.a:3.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + 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_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if#,minus#} TcT has computed the following interpretation: p(0) = [0] p(DIV) = [0] p(GEQ) = [0] p(IF) = [0] p(MINUS) = [0] p(c) = [0] p(c1) = [1] x1 + [0] p(c2) = [0] p(c3) = [0] p(c4) = [1] x1 + [0] p(c5) = [0] p(c6) = [1] x1 + [1] x2 + [0] p(c7) = [1] x2 + [1] x3 + [0] p(c8) = [0] p(c9) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(minus) = [0] p(s) = [1] x1 + [2] p(true) = [0] p(DIV#) = [8] x2 + [13] p(GEQ#) = [2] x2 + [0] p(IF#) = [0] p(MINUS#) = [0] p(div#) = [0] p(geq#) = [0] p(if#) = [0] p(minus#) = [0] p(c_1) = [0] p(c_2) = [4] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [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] Following rules are strictly oriented: DIV#(s(z0),s(z1)) = [8] z1 + [29] > [8] z1 + [0] = c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) = [2] z1 + [4] > [2] z1 + [0] = c_6(GEQ#(z0,z1)) Following rules are (at-least) weakly oriented: DIV#(s(z0),s(z1)) = [8] z1 + [29] >= [8] z1 + [29] = c_3(DIV#(minus(z0,z1),s(z1))) **** Step 1.b:7.a:3.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:7.a:3.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 2:W:DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))) -->_1 DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))):2 -->_1 DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)):1 3:W:GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1))) 1: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) 3: GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) **** Step 1.b:7.a:3.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:7.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak DPs: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) -->_1 DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)):3 -->_2 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):2 -->_1 DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)):1 2:S:MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):2 3:W:DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):4 4:W:GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) -->_1 GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: DIV#(s(z0),s(z1)) -> c_2(GEQ#(z0,z1)) 4: GEQ#(s(z0),s(z1)) -> c_6(GEQ#(z0,z1)) *** Step 1.b:7.b:2: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + 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: 1: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) 2: MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) The strictly oriented rules are moved into the weak component. **** Step 1.b:7.b:2.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + 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_3) = {1,2}, uargs(c_10) = {1} Following symbols are considered usable: {minus,DIV#,GEQ#,IF#,MINUS#,div#,geq#,if#,minus#} TcT has computed the following interpretation: p(0) = [0] p(DIV) = [1] p(GEQ) = [1] x1 + [1] x2 + [1] p(IF) = [1] x2 + [1] p(MINUS) = [1] x1 + [1] p(c) = [4] p(c1) = [0] p(c2) = [1] p(c3) = [2] p(c4) = [0] p(c5) = [2] p(c6) = [1] x2 + [1] p(c7) = [8] p(c8) = [1] p(c9) = [0] p(div) = [1] x1 + [1] p(false) = [0] p(geq) = [1] x1 + [1] x2 + [2] p(if) = [4] x1 + [1] x3 + [0] p(minus) = [0] p(s) = [1] x1 + [4] p(true) = [1] p(DIV#) = [4] x1 + [2] p(GEQ#) = [1] x1 + [1] x2 + [1] p(IF#) = [1] x2 + [2] x3 + [1] p(MINUS#) = [4] x1 + [4] p(div#) = [1] x1 + [1] x2 + [0] p(geq#) = [0] p(if#) = [1] x1 + [1] p(minus#) = [4] x1 + [4] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [14] p(c_11) = [2] p(c_12) = [4] p(c_13) = [0] p(c_14) = [2] p(c_15) = [2] x1 + [0] p(c_16) = [1] p(c_17) = [1] p(c_18) = [1] p(c_19) = [1] x1 + [1] Following rules are strictly oriented: DIV#(s(z0),s(z1)) = [4] z0 + [18] > [4] z0 + [6] = c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) = [4] z0 + [20] > [4] z0 + [18] = c_10(MINUS#(z0,z1)) Following rules are (at-least) weakly oriented: minus(0(),z0) = [0] >= [0] = 0() minus(s(z0),s(z1)) = [0] >= [0] = minus(z0,z1) **** Step 1.b:7.b:2.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:7.b:2.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) -->_2 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):2 -->_1 DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)):1 2:W:MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: DIV#(s(z0),s(z1)) -> c_3(DIV#(minus(z0,z1),s(z1)),MINUS#(z0,z1)) 2: MINUS#(s(z0),s(z1)) -> c_10(MINUS#(z0,z1)) **** Step 1.b:7.b:2.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {DIV/2,GEQ/2,IF/3,MINUS/2,div/2,geq/2,if/3,minus/2,DIV#/2,GEQ#/2,IF#/3,MINUS#/2,div#/2,geq#/2,if#/3 ,minus#/2} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/0,c6/2,c7/3,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {DIV#,GEQ#,IF#,MINUS#,div#,geq#,if# ,minus#} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))