Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 0, 0) :|: 0 < X₁
t₄: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂
t₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂+1, X₃+X₂) :|: X₂ < X₁
t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 0 < X₀
Preprocessing
Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2
Found invariant 0 ≤ X₀ for location l1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 0, 0) :|: 0 < X₁ ∧ 0 ≤ X₀
t₄: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 0 ≤ X₀
t₃: l2(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂+1, X₃+X₂) :|: X₂ < X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀
MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 0, 0) :|: 0 < X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
MPRF:
l2 [X₁-1 ]
l1 [X₁ ]
MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
MPRF:
l2 [X₁ ]
l1 [X₁ ]
MPRF for transition t₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂+1, X₃+X₂) :|: X₂ < X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₁⋅X₁+2⋅X₁+1 {O(n^2)}
MPRF:
l1 [X₁+1 ]
l2 [X₁+1-X₂ ]
Analysing control-flow refined program
Cut unsatisfiable transition t₃: l2→l1
Found invariant X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ 1+X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2
Found invariant 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l2___1
Found invariant 0 ≤ X₀ for location l1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₁ {O(n)} for transition t₄₇: l2(X₀, X₁, X₂, X₃) → n_l2___1(X₀, X₁, X₂+1, X₂+X₃) :|: X₂ ≤ 1+X₃ ∧ 0 ≤ X₂ ∧ X₂ < X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ < X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₃ ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ 1+X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
MPRF for transition t₄₆: n_l2___1(X₀, X₁, X₂, X₃) → n_l2___1(X₀, X₁, X₂+1, X₂+X₃) :|: X₂ ≤ 1+X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ < X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₃ ∧ X₂ ≤ 1+X₃ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₁⋅X₁+2⋅X₁ {O(n^2)}
MPRF:
l2 [0 ]
n_l2___1 [X₁+1-X₂ ]
l1 [0 ]
MPRF for transition t₅₀: n_l2___1(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
MPRF:
l2 [X₁ ]
n_l2___1 [X₁ ]
l1 [X₁ ]
CFR did not improve the program. Rolling back
MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ of depth 1:
new bound:
2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁+1 {O(EXP)}
MPRF:
l3 [X₀+1 ]
Analysing control-flow refined program
Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2
Found invariant 0 ≤ X₀ for location l1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location n_l3___1
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3
MPRF for transition t₉₆: n_l3___1(X₀, X₁, X₂, X₃) → n_l3___1(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 0 < X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ of depth 1:
new bound:
2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁+1 {O(EXP)}
MPRF:
n_l3___1 [X₀+1 ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁+X₁⋅X₁+4⋅X₁+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₄: 1 {O(1)}
t₂: X₁⋅X₁+2⋅X₁+1 {O(n^2)}
t₃: X₁ {O(n)}
t₅: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁+1 {O(EXP)}
Costbounds
Overall costbound: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁+X₁⋅X₁+4⋅X₁+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₄: 1 {O(1)}
t₂: X₁⋅X₁+2⋅X₁+1 {O(n^2)}
t₃: X₁ {O(n)}
t₅: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁+1 {O(EXP)}
Sizebounds
t₀, X₀: 0 {O(1)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁ {O(EXP)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 0 {O(1)}
t₁, X₃: 0 {O(1)}
t₄, X₀: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁ {O(EXP)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: X₁⋅X₁+2⋅X₁+X₂+1 {O(n^2)}
t₄, X₃: 2⋅2^(X₁⋅X₁+2⋅X₁+1)⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁+X₃ {O(EXP)}
t₂, X₀: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁ {O(EXP)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁⋅X₁+2⋅X₁+1 {O(n^2)}
t₂, X₃: 2⋅2^(X₁⋅X₁+2⋅X₁+1)⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁ {O(EXP)}
t₃, X₀: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁ {O(EXP)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₁⋅X₁+2⋅X₁+1 {O(n^2)}
t₃, X₃: 2⋅2^(X₁⋅X₁+2⋅X₁+1)⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁ {O(EXP)}
t₅, X₀: 2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅3⋅X₁⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁⋅X₁ {O(EXP)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: X₁⋅X₁+2⋅X₁+X₂+1 {O(n^2)}
t₅, X₃: 2⋅2^(X₁⋅X₁+2⋅X₁+1)⋅X₁+2^(X₁⋅X₁+2⋅X₁+1)+2^(X₁⋅X₁+2⋅X₁+1)⋅X₁⋅X₁+X₃ {O(EXP)}