WORST_CASE(?,O(n^2)) * Step 1: LocalSizeboundsProc WORST_CASE(?,O(n^2)) + Considered Problem: Rules: 0. f1(A,B) -> f3(A,A) [A >= 1] (1,1) 1. f3(A,B) -> f3(A,-1 + B) [B >= 1] (?,1) 2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1) Signature: {(f1,2);(f3,2)} Flow Graph: [0->{1,2},1->{1,2},2->{1,2}] + Applied Processor: LocalSizeboundsProc + Details: LocalSizebounds generated; rvgraph (<0,0,A>, A, .= 0) (<0,0,B>, A, .= 0) (<1,0,A>, A, .= 0) (<1,0,B>, 1 + B, .+ 1) (<2,0,A>, 1 + A, .+ 1) (<2,0,B>, 1 + A, .+ 1) * Step 2: SizeboundsProc WORST_CASE(?,O(n^2)) + Considered Problem: Rules: 0. f1(A,B) -> f3(A,A) [A >= 1] (1,1) 1. f3(A,B) -> f3(A,-1 + B) [B >= 1] (?,1) 2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1) Signature: {(f1,2);(f3,2)} Flow Graph: [0->{1,2},1->{1,2},2->{1,2}] Sizebounds: (<0,0,A>, ?) (<0,0,B>, ?) (<1,0,A>, ?) (<1,0,B>, ?) (<2,0,A>, ?) (<2,0,B>, ?) + Applied Processor: SizeboundsProc + Details: Sizebounds computed: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, ?) (<1,0,B>, ?) (<2,0,A>, ?) (<2,0,B>, ?) * Step 3: UnsatPaths WORST_CASE(?,O(n^2)) + Considered Problem: Rules: 0. f1(A,B) -> f3(A,A) [A >= 1] (1,1) 1. f3(A,B) -> f3(A,-1 + B) [B >= 1] (?,1) 2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1) Signature: {(f1,2);(f3,2)} Flow Graph: [0->{1,2},1->{1,2},2->{1,2}] Sizebounds: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, ?) (<1,0,B>, ?) (<2,0,A>, ?) (<2,0,B>, ?) + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,2),(2,2)] * Step 4: PolyRank WORST_CASE(?,O(n^2)) + Considered Problem: Rules: 0. f1(A,B) -> f3(A,A) [A >= 1] (1,1) 1. f3(A,B) -> f3(A,-1 + B) [B >= 1] (?,1) 2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1) Signature: {(f1,2);(f3,2)} Flow Graph: [0->{1},1->{1,2},2->{1}] Sizebounds: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, ?) (<1,0,B>, ?) (<2,0,A>, ?) (<2,0,B>, ?) + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 p(f3) = -1 + x1 The following rules are strictly oriented: [A >= 1] ==> f1(A,B) = A > -1 + A = f3(A,A) [0 >= B && A >= 2] ==> f3(A,B) = -1 + A > -2 + A = f3(-1 + A,-1 + A) The following rules are weakly oriented: [B >= 1] ==> f3(A,B) = -1 + A >= -1 + A = f3(A,-1 + B) * Step 5: SizeboundsProc WORST_CASE(?,O(n^2)) + Considered Problem: Rules: 0. f1(A,B) -> f3(A,A) [A >= 1] (1,1) 1. f3(A,B) -> f3(A,-1 + B) [B >= 1] (?,1) 2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (A,1) Signature: {(f1,2);(f3,2)} Flow Graph: [0->{1},1->{1,2},2->{1}] Sizebounds: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, ?) (<1,0,B>, ?) (<2,0,A>, ?) (<2,0,B>, ?) + Applied Processor: SizeboundsProc + Details: Sizebounds computed: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, 2*A) (<1,0,B>, ?) (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) * Step 6: PolyRank WORST_CASE(?,O(n^2)) + Considered Problem: Rules: 0. f1(A,B) -> f3(A,A) [A >= 1] (1,1) 1. f3(A,B) -> f3(A,-1 + B) [B >= 1] (?,1) 2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (A,1) Signature: {(f1,2);(f3,2)} Flow Graph: [0->{1},1->{1,2},2->{1}] Sizebounds: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, 2*A) (<1,0,B>, ?) (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [1], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f3) = x2 The following rules are strictly oriented: [B >= 1] ==> f3(A,B) = B > -1 + B = f3(A,-1 + B) The following rules are weakly oriented: We use the following global sizebounds: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, 2*A) (<1,0,B>, ?) (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) * Step 7: KnowledgePropagation WORST_CASE(?,O(n^2)) + Considered Problem: Rules: 0. f1(A,B) -> f3(A,A) [A >= 1] (1,1) 1. f3(A,B) -> f3(A,-1 + B) [B >= 1] (2*A + 2*A^2,1) 2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (A,1) Signature: {(f1,2);(f3,2)} Flow Graph: [0->{1},1->{1,2},2->{1}] Sizebounds: (<0,0,A>, A) (<0,0,B>, A) (<1,0,A>, 2*A) (<1,0,B>, ?) (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) + Applied Processor: KnowledgePropagation + Details: The problem is already solved. WORST_CASE(?,O(n^2))