Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅) :|: 0 < X₂
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, 3⋅X₁-4⋅X₂, 4⋅X₁-3⋅X₂, 5⋅X₃, 5⋅X₄-(X₀)², X₅) :|: 1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀
Preprocessing
Found invariant X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅) :|: 0 < X₂ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0 ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, 3⋅X₁-4⋅X₂, 4⋅X₁-3⋅X₂, 5⋅X₃, 5⋅X₄-(X₀)², X₅) :|: 1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀
TWN. Size Bound: t₃: l2→l2 for X₄
cycle: [t₃: l2→l2]
loop: (1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀,(X₀,X₄) -> (X₀,5⋅X₄-(X₀)²)
closed-form: X₄ * 5^n + [[n != 0]] * -1/4⋅(X₀)² * 5^n + [[n != 0]] * 1/4⋅(X₀)²
runtime bound: inf {Infinity}
TWN Size Bound - Lifting for t₃: l2→l2 and X₄: inf {Infinity}
Solv. Size Bound: t₂: l1→l2 for X₀
cycle: [t₂: l1→l2; t₄: l2→l1]
loop: (X₂ ≤ 0 ∧ 0 < X₅,(X₀,X₃) -> (X₃,X₃)
overappr. closed-form: 2⋅X₃ {O(n)}
runtime bound: X₅+1 {O(n)}
Solv. Size Bound - Lifting for t₂: l1→l2 and X₀: inf {Infinity}
Solv. Size Bound: t₂: l1→l2 for X₁
cycle: [t₂: l1→l2; t₄: l2→l1]
loop: (X₂ ≤ 0 ∧ 0 < X₅,(X₁,X₄) -> (X₄,X₄)
overappr. closed-form: 2⋅X₄ {O(n)}
runtime bound: X₅+1 {O(n)}
Solv. Size Bound - Lifting for t₂: l1→l2 and X₁: inf {Infinity}
Solv. Size Bound: t₃: l2→l2 for X₀
Solv. Size Bound: t₃: l2→l2 for X₁
cycle: [t₃: l2→l2]
loop: (1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀,(X₁,X₂) -> (3⋅X₁-4⋅X₂,4⋅X₁-3⋅X₂)
overappr. closed-form: 12⋅7^(n)⋅X₁+12⋅7^(n)⋅X₂ {O(EXP)}
runtime bound: inf {Infinity}
Solv. Size Bound - Lifting for t₃: l2→l2 and X₁: inf {Infinity}
Solv. Size Bound: t₃: l2→l2 for X₂
cycle: [t₃: l2→l2]
loop: (1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀,(X₁,X₂) -> (3⋅X₁-4⋅X₂,4⋅X₁-3⋅X₂)
overappr. closed-form: 13⋅7^(n)⋅X₂+6⋅7^(n)⋅X₁ {O(EXP)}
runtime bound: inf {Infinity}
Solv. Size Bound - Lifting for t₃: l2→l2 and X₂: inf {Infinity}
Solv. Size Bound: t₄: l2→l1 for X₀
cycle: [t₄: l2→l1; t₂: l1→l2]
loop: (0 < X₅ ∧ X₅ ≤ 0,(X₀,X₃) -> (X₃,X₃)
overappr. closed-form: 2⋅X₃ {O(n)}
runtime bound: 2 {O(1)}
Solv. Size Bound - Lifting for t₄: l2→l1 and X₀: inf {Infinity}
Solv. Size Bound: t₄: l2→l1 for X₁
cycle: [t₄: l2→l1; t₂: l1→l2]
loop: (0 < X₅ ∧ X₅ ≤ 0,(X₁,X₄) -> (X₄,X₄)
overappr. closed-form: 2⋅X₄ {O(n)}
runtime bound: 2 {O(1)}
Solv. Size Bound - Lifting for t₄: l2→l1 and X₁: inf {Infinity}
MPRF for transition t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ of depth 1:
new bound:
X₅ {O(n)}
TWN: t₁: l1→l1
cycle: [t₁: l1→l1]
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₄:
X₂: 2⋅X₅ {O(n)}
Runtime-bound of t₄: X₅ {O(n)}
Results in: 4⋅X₅⋅X₅+4⋅X₅ {O(n^2)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₀:
X₂: X₅ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₅+4 {O(n)}
knowledge_propagation leads to new time bound 4⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)} for transition t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0 ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
TWN Size Bound - Lifting for t₃: l2→l2 and X₄: inf {Infinity}
Solv. Size Bound - Lifting for t₂: l1→l2 and X₀: inf {Infinity}
Solv. Size Bound - Lifting for t₂: l1→l2 and X₁: inf {Infinity}
Solv. Size Bound: t₃: l2→l2 for X₀
Solv. Size Bound - Lifting for t₃: l2→l2 and X₁: inf {Infinity}
Solv. Size Bound - Lifting for t₃: l2→l2 and X₂: inf {Infinity}
Solv. Size Bound - Lifting for t₄: l2→l1 and X₀: inf {Infinity}
Solv. Size Bound - Lifting for t₄: l2→l1 and X₁: inf {Infinity}
Analysing control-flow refined program
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location n_l2___6
Found invariant 1+X₅ ≤ X₂ ∧ 0 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ for location n_l1___4
Found invariant X₅ ≤ X₂ ∧ 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ for location n_l2___5
Found invariant 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___7
Found invariant 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___2
Found invariant 1 ≤ X₅ for location n_l2___3
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1
MPRF for transition t₁₃₉: n_l1___7(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___7(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: 0 < X₅ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:
new bound:
X₅+1 {O(n)}
Solv. Size Bound: t₁₃₆: n_l1___2→n_l1___2 for X₀
Solv. Size Bound: t₁₃₆: n_l1___2→n_l1___2 for X₁
Solv. Size Bound: t₁₃₆: n_l1___2→n_l1___2 for X₃
Solv. Size Bound: t₁₃₆: n_l1___2→n_l1___2 for X₄
cycle: [t₁₃₆: n_l1___2→n_l1___2]
loop: (0 < X₂,(X₁,X₄) -> (X₁,X₁)
overappr. closed-form: 2⋅X₁ {O(n)}
runtime bound: X₂+1 {O(n)}
Solv. Size Bound - Lifting for t₁₃₆: n_l1___2→n_l1___2 and X₄: inf {Infinity}
Solv. Size Bound: t₁₃₇: n_l1___2→n_l2___6 for X₀
Solv. Size Bound: t₁₃₇: n_l1___2→n_l2___6 for X₁
Solv. Size Bound: t₁₃₇: n_l1___2→n_l2___6 for X₃
Solv. Size Bound: t₁₃₇: n_l1___2→n_l2___6 for X₄
cycle: [t₁₃₇: n_l1___2→n_l2___6; t₁₄₉: n_l2___6→n_l1___4; t₁₃₈: n_l1___4→n_l1___2]
loop: (X₂ ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ X₃ ≤ X₀ ∧ 0 < X₅ ∧ 0 < X₅ ∧ 0 < X₅,(X₁,X₄) -> (X₁,X₁)
overappr. closed-form: 2⋅X₁ {O(n)}
runtime bound: X₅+1 {O(n)}
Solv. Size Bound - Lifting for t₁₃₇: n_l1___2→n_l2___6 and X₄: inf {Infinity}
Solv. Size Bound: t₁₃₈: n_l1___4→n_l1___2 for X₀
cycle: [t₁₃₈: n_l1___4→n_l1___2; t₁₃₇: n_l1___2→n_l2___6; t₁₄₉: n_l2___6→n_l1___4]
loop: (0 < X₂ ∧ 0 < X₂ ∧ X₂ ≤ 1 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ X₃ ≤ X₀+X₂ ∧ 0 < X₅,(X₀,X₃) -> (X₃,X₃)
overappr. closed-form: 2⋅X₃ {O(n)}
runtime bound: X₅+1 {O(n)}
Solv. Size Bound - Lifting for t₁₃₈: n_l1___4→n_l1___2 and X₀: inf {Infinity}
Solv. Size Bound: t₁₃₈: n_l1___4→n_l1___2 for X₁
Solv. Size Bound: t₁₃₈: n_l1___4→n_l1___2 for X₃
Solv. Size Bound: t₁₃₈: n_l1___4→n_l1___2 for X₄
cycle: [t₁₃₈: n_l1___4→n_l1___2; t₁₃₇: n_l1___2→n_l2___6; t₁₄₉: n_l2___6→n_l1___4]
loop: (0 < X₂ ∧ 0 < X₂ ∧ X₂ ≤ 1 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ X₃ ≤ X₀+X₂ ∧ 0 < X₅,(X₁,X₄) -> (X₁,X₁)
overappr. closed-form: 2⋅X₁ {O(n)}
runtime bound: X₅+1 {O(n)}
Solv. Size Bound - Lifting for t₁₃₈: n_l1___4→n_l1___2 and X₄: inf {Infinity}
Solv. Size Bound: t₁₄₉: n_l2___6→n_l1___4 for X₀
Solv. Size Bound: t₁₄₉: n_l2___6→n_l1___4 for X₁
cycle: [t₁₄₉: n_l2___6→n_l1___4; t₁₃₈: n_l1___4→n_l1___2; t₁₃₇: n_l1___2→n_l2___6]
loop: (X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 < X₅ ∧ 0 < X₅ ∧ 0 < X₅ ∧ X₅ ≤ 1,(X₁,X₄) -> (X₄,X₄)
overappr. closed-form: 2⋅X₄ {O(n)}
runtime bound: X₅+1 {O(n)}
Solv. Size Bound - Lifting for t₁₄₉: n_l2___6→n_l1___4 and X₁: inf {Infinity}
MPRF for transition t₁₃₇: n_l1___2(X₀, X₁, X₂, X₃, X₄, X₅) → n_l2___6(X₀, X₁, X₅, X₃, X₁, X₅) :|: X₂ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF for transition t₁₃₈: n_l1___4(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___2(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: 0 < X₂ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 1+X₅ ≤ X₂ ∧ 0 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF for transition t₁₄₃: n_l2___1(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___4(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF for transition t₁₄₉: n_l2___6(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___4(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 < X₅ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
TWN: t₁₃₆: n_l1___2→n_l1___2
cycle: [t₁₃₆: n_l1___2→n_l1___2]
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
TWN - Lifting for t₁₃₆: n_l1___2→n_l1___2 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₁₃₈:
X₂: 34⋅X₅ {O(n)}
Runtime-bound of t₁₃₈: 7⋅X₅+2 {O(n)}
Results in: 476⋅X₅⋅X₅+164⋅X₅+8 {O(n^2)}
knowledge_propagation leads to new time bound 7⋅X₅+3 {O(n)} for transition t₁₅₀: n_l2___6(X₀, X₁, X₂, X₃, X₄, X₅) → n_l2___1(X₀, Arg1_P, Arg2_P, 5⋅X₃, NoDet0, X₅) :|: X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 4⋅X₁ ≤ 3⋅X₂+Arg2_P ∧ 3⋅X₂+Arg2_P ≤ 4⋅X₁ ∧ 7⋅X₁+3⋅Arg1_P ≤ 4⋅Arg2_P ∧ 4⋅Arg2_P ≤ 7⋅X₁+3⋅Arg1_P ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
Solv. Size Bound: t₁₃₆: n_l1___2→n_l1___2 for X₀
Solv. Size Bound: t₁₃₆: n_l1___2→n_l1___2 for X₁
Solv. Size Bound: t₁₃₆: n_l1___2→n_l1___2 for X₃
Solv. Size Bound - Lifting for t₁₃₆: n_l1___2→n_l1___2 and X₄: inf {Infinity}
Solv. Size Bound: t₁₃₇: n_l1___2→n_l2___6 for X₀
Solv. Size Bound: t₁₃₇: n_l1___2→n_l2___6 for X₁
Solv. Size Bound: t₁₃₇: n_l1___2→n_l2___6 for X₃
Solv. Size Bound - Lifting for t₁₃₇: n_l1___2→n_l2___6 and X₄: inf {Infinity}
Solv. Size Bound - Lifting for t₁₃₈: n_l1___4→n_l1___2 and X₀: inf {Infinity}
Solv. Size Bound: t₁₃₈: n_l1___4→n_l1___2 for X₁
Solv. Size Bound: t₁₃₈: n_l1___4→n_l1___2 for X₃
Solv. Size Bound - Lifting for t₁₃₈: n_l1___4→n_l1___2 and X₄: inf {Infinity}
Solv. Size Bound: t₁₄₉: n_l2___6→n_l1___4 for X₀
Solv. Size Bound - Lifting for t₁₄₉: n_l2___6→n_l1___4 and X₁: inf {Infinity}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 4⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
t₂: 4⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)}
t₃: inf {Infinity}
t₄: X₅ {O(n)}
Costbounds
Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 4⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
t₂: 4⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)}
t₃: inf {Infinity}
t₄: X₅ {O(n)}
Sizebounds
t₀, X₀: X₃ {O(n)}
t₀, X₁: X₄ {O(n)}
t₀, X₂: X₅ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₁, X₂: 3⋅X₅ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₂: 2⋅X₅ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₂: 2⋅X₅ {O(n)}
t₄, X₅: X₅ {O(n)}