WORST_CASE(?,O(n^1)) * Step 1: RestrictVarsProcessor WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C) -> f2(1 + A,1 + B,C) [1 >= A] (?,1) 1. f2(A,B,C) -> f2(1 + A,1 + B,C) [2 >= B && A >= 2] (?,1) 2. f2(A,B,C) -> f300(A,B,D) [B >= 3 && A >= 2] (?,1) 3. f1(A,B,C) -> f2(A,B,C) True (1,1) Signature: {(f1,3);(f2,3);(f300,3)} Flow Graph: [0->{0,1,2},1->{0,1,2},2->{},3->{0,1,2}] + Applied Processor: RestrictVarsProcessor + Details: We removed the arguments [C] . * Step 2: LocalSizeboundsProc WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B) -> f2(1 + A,1 + B) [1 >= A] (?,1) 1. f2(A,B) -> f2(1 + A,1 + B) [2 >= B && A >= 2] (?,1) 2. f2(A,B) -> f300(A,B) [B >= 3 && A >= 2] (?,1) 3. f1(A,B) -> f2(A,B) True (1,1) Signature: {(f1,2);(f2,2);(f300,2)} Flow Graph: [0->{0,1,2},1->{0,1,2},2->{},3->{0,1,2}] + Applied Processor: LocalSizeboundsProc + Details: LocalSizebounds generated; rvgraph (<0,0,A>, 1 + A, .+ 1) (<0,0,B>, 1 + B, .+ 1) (<1,0,A>, 1 + A, .+ 1) (<1,0,B>, 1 + B, .+ 1) (<2,0,A>, A, .= 0) (<2,0,B>, B, .= 0) (<3,0,A>, A, .= 0) (<3,0,B>, B, .= 0) * Step 3: SizeboundsProc WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B) -> f2(1 + A,1 + B) [1 >= A] (?,1) 1. f2(A,B) -> f2(1 + A,1 + B) [2 >= B && A >= 2] (?,1) 2. f2(A,B) -> f300(A,B) [B >= 3 && A >= 2] (?,1) 3. f1(A,B) -> f2(A,B) True (1,1) Signature: {(f1,2);(f2,2);(f300,2)} Flow Graph: [0->{0,1,2},1->{0,1,2},2->{},3->{0,1,2}] Sizebounds: (<0,0,A>, ?) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, ?) (<2,0,A>, ?) (<2,0,B>, ?) (<3,0,A>, ?) (<3,0,B>, ?) + Applied Processor: SizeboundsProc + Details: Sizebounds computed: (<0,0,A>, 2) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, 3) (<2,0,A>, ?) (<2,0,B>, ?) (<3,0,A>, A) (<3,0,B>, B) * Step 4: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B) -> f2(1 + A,1 + B) [1 >= A] (?,1) 1. f2(A,B) -> f2(1 + A,1 + B) [2 >= B && A >= 2] (?,1) 2. f2(A,B) -> f300(A,B) [B >= 3 && A >= 2] (?,1) 3. f1(A,B) -> f2(A,B) True (1,1) Signature: {(f1,2);(f2,2);(f300,2)} Flow Graph: [0->{0,1,2},1->{0,1,2},2->{},3->{0,1,2}] Sizebounds: (<0,0,A>, 2) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, 3) (<2,0,A>, ?) (<2,0,B>, ?) (<3,0,A>, A) (<3,0,B>, B) + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(1,0)] * Step 5: LeafRules WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B) -> f2(1 + A,1 + B) [1 >= A] (?,1) 1. f2(A,B) -> f2(1 + A,1 + B) [2 >= B && A >= 2] (?,1) 2. f2(A,B) -> f300(A,B) [B >= 3 && A >= 2] (?,1) 3. f1(A,B) -> f2(A,B) True (1,1) Signature: {(f1,2);(f2,2);(f300,2)} Flow Graph: [0->{0,1,2},1->{1,2},2->{},3->{0,1,2}] Sizebounds: (<0,0,A>, 2) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, 3) (<2,0,A>, ?) (<2,0,B>, ?) (<3,0,A>, A) (<3,0,B>, B) + Applied Processor: LeafRules + Details: The following transitions are estimated by its predecessors and are removed [2] * Step 6: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B) -> f2(1 + A,1 + B) [1 >= A] (?,1) 1. f2(A,B) -> f2(1 + A,1 + B) [2 >= B && A >= 2] (?,1) 3. f1(A,B) -> f2(A,B) True (1,1) Signature: {(f1,2);(f2,2);(f300,2)} Flow Graph: [0->{0,1},1->{1},3->{0,1}] Sizebounds: (<0,0,A>, 2) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, 3) (<3,0,A>, A) (<3,0,B>, B) + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = 3 + -1*x2 p(f2) = 3 + -1*x2 The following rules are strictly oriented: [2 >= B && A >= 2] ==> f2(A,B) = 3 + -1*B > 2 + -1*B = f2(1 + A,1 + B) The following rules are weakly oriented: [1 >= A] ==> f2(A,B) = 3 + -1*B >= 2 + -1*B = f2(1 + A,1 + B) True ==> f1(A,B) = 3 + -1*B >= 3 + -1*B = f2(A,B) * Step 7: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B) -> f2(1 + A,1 + B) [1 >= A] (?,1) 1. f2(A,B) -> f2(1 + A,1 + B) [2 >= B && A >= 2] (3 + B,1) 3. f1(A,B) -> f2(A,B) True (1,1) Signature: {(f1,2);(f2,2);(f300,2)} Flow Graph: [0->{0,1},1->{1},3->{0,1}] Sizebounds: (<0,0,A>, 2) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, 3) (<3,0,A>, A) (<3,0,B>, B) + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = 2 + -1*x1 p(f2) = 2 + -1*x1 The following rules are strictly oriented: [1 >= A] ==> f2(A,B) = 2 + -1*A > 1 + -1*A = f2(1 + A,1 + B) The following rules are weakly oriented: [2 >= B && A >= 2] ==> f2(A,B) = 2 + -1*A >= 1 + -1*A = f2(1 + A,1 + B) True ==> f1(A,B) = 2 + -1*A >= 2 + -1*A = f2(A,B) * Step 8: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B) -> f2(1 + A,1 + B) [1 >= A] (2 + A,1) 1. f2(A,B) -> f2(1 + A,1 + B) [2 >= B && A >= 2] (3 + B,1) 3. f1(A,B) -> f2(A,B) True (1,1) Signature: {(f1,2);(f2,2);(f300,2)} Flow Graph: [0->{0,1},1->{1},3->{0,1}] Sizebounds: (<0,0,A>, 2) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, 3) (<3,0,A>, A) (<3,0,B>, B) + Applied Processor: KnowledgePropagation + Details: The problem is already solved. WORST_CASE(?,O(n^1))