Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(1, 1, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₁, X₀+X₁, X₂-1) :|: 0 < X₂
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0
t₃: l2(X₀, X₁, X₂) → l2(X₀, X₁-1, X₂) :|: 0 < X₁

Preprocessing

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

Found invariant 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(1, 1, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₁, X₀+X₁, X₂-1) :|: 0 < X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₃: l2(X₀, X₁, X₂) → l2(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀

Solv. Size Bound: t₁: l1→l1 for X₀

cycle: [t₁: l1→l1]
loop: (0 < X₂,(X₀,X₁) -> (X₀+X₁,X₀+X₁)
overappr. closed-form: 2^(n)⋅X₀+2^(n)⋅X₁+X₀+X₁ {O(EXP)}
runtime bound: X₂+1 {O(n)}

Solv. Size Bound - Lifting for t₁: l1→l1 and X₀: 2⋅2^(X₂+1)+2 {O(EXP)}

Solv. Size Bound: t₁: l1→l1 for X₁

cycle: [t₁: l1→l1]
loop: (0 < X₂,(X₀,X₁) -> (X₀+X₁,X₀+X₁)
overappr. closed-form: 2^(n)⋅X₀+2^(n)⋅X₁+X₀+X₁ {O(EXP)}
runtime bound: X₂+1 {O(n)}

Solv. Size Bound - Lifting for t₁: l1→l1 and X₁: 2⋅2^(X₂+1)+2 {O(EXP)}

MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀+X₁, X₀+X₁, X₂-1) :|: 0 < X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

Solv. Size Bound - Lifting for t₁: l1→l1 and X₀: 2⋅2^(X₂+1)+2 {O(EXP)}

Solv. Size Bound - Lifting for t₁: l1→l1 and X₁: 2⋅2^(X₂+1)+2 {O(EXP)}

MPRF for transition t₃: l2(X₀, X₁, X₂) → l2(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅2^(X₂)+2 {O(EXP)}

Analysing control-flow refined program

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l2___1

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l2___2

Found invariant 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

MPRF for transition t₃₉: n_l2___1(X₀, X₁, X₂) → n_l2___1(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅2^(X₂)+2 {O(EXP)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅2^(X₂)+X₂+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅2^(X₂)+2 {O(EXP)}

Costbounds

Overall costbound: 2⋅2^(X₂)+X₂+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅2^(X₂)+2 {O(EXP)}

Sizebounds

t₀, X₀: 1 {O(1)}
t₀, X₁: 1 {O(1)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 2⋅2^(X₂) {O(EXP)}
t₁, X₁: 2⋅2^(X₂) {O(EXP)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅2^(X₂)+1 {O(EXP)}
t₂, X₁: 2⋅2^(X₂)+1 {O(EXP)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₀: 2⋅2^(X₂)+1 {O(EXP)}
t₃, X₁: 2⋅2^(X₂)+1 {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}