Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, 1, 0)
t₁: l1(X₀, X₁, X₂) → l2(X₀-1, X₁, X₁) :|: 0 < X₀
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ ≤ 0
t₂: l2(X₀, X₁, X₂) → l1(X₀, X₂+X₁, X₂)
t₄: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: 0 < X₁
Preprocessing
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2
Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1
Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, 1, 0)
t₁: l1(X₀, X₁, X₂) → l2(X₀-1, X₁, X₁) :|: 0 < X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₂: l2(X₀, X₁, X₂) → l1(X₀, X₂+X₁, X₂) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₄: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0
Solv. Size Bound: t₁: l1→l2 for X₁
Solv. Size Bound: t₁: l1→l2 for X₂
cycle: [t₁: l1→l2; t₂: l2→l1]
loop: (0 < X₀,(X₁,X₂) -> (2⋅X₁,X₁)
overappr. closed-form: 2^(n)⋅X₁+X₁ {O(EXP)}
runtime bound: X₀+1 {O(n)}
Solv. Size Bound - Lifting for t₁: l1→l2 and X₂: 2^(X₀+1)+1 {O(EXP)}
Solv. Size Bound: t₂: l2→l1 for X₁
cycle: [t₂: l2→l1; t₁: l1→l2]
loop: (0 < X₀,(X₁,X₂) -> (X₂+X₁,X₂+X₁)
overappr. closed-form: 2^(n)⋅X₁+2^(n)⋅X₂+X₁+X₂ {O(EXP)}
runtime bound: X₀+1 {O(n)}
Solv. Size Bound - Lifting for t₂: l2→l1 and X₁: 2^(X₀+1)+1 {O(EXP)}
Solv. Size Bound: t₂: l2→l1 for X₂
cycle: [t₂: l2→l1; t₁: l1→l2]
loop: (0 < X₀,(X₁,X₂) -> (X₂+X₁,X₂+X₁)
overappr. closed-form: 2^(n)⋅X₁+2^(n)⋅X₂+X₁+X₂ {O(EXP)}
runtime bound: X₀+1 {O(n)}
Solv. Size Bound - Lifting for t₂: l2→l1 and X₂: 2^(X₀+1)+1 {O(EXP)}
MPRF for transition t₁: l1(X₀, X₁, X₂) → l2(X₀-1, X₁, X₁) :|: 0 < X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₀ {O(n)}
MPRF for transition t₂: l2(X₀, X₁, X₂) → l1(X₀, X₂+X₁, X₂) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
Solv. Size Bound: t₁: l1→l2 for X₁
Solv. Size Bound - Lifting for t₁: l1→l2 and X₂: 2^(X₀+1)+1 {O(EXP)}
Solv. Size Bound - Lifting for t₂: l2→l1 and X₁: 2^(X₀+1)+1 {O(EXP)}
Solv. Size Bound - Lifting for t₂: l2→l1 and X₂: 2^(X₀+1)+1 {O(EXP)}
MPRF for transition t₄: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
2^(X₀)+2 {O(EXP)}
Analysing control-flow refined program
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2
Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l3___2
Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1
Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l3___1
Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l3
MPRF for transition t₅₃: n_l3___1(X₀, X₁, X₂) → n_l3___1(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₂ ∧ 0 ≤ X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
2^(X₀)+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^(X₀)+2⋅X₀+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: 2^(X₀)+2 {O(EXP)}
Costbounds
Overall costbound: 2^(X₀)+2⋅X₀+4 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: 2^(X₀)+2 {O(EXP)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: 1 {O(1)}
t₀, X₂: 0 {O(1)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2^(X₀) {O(EXP)}
t₁, X₂: 2^(X₀)+1 {O(EXP)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: 2^(X₀) {O(EXP)}
t₂, X₂: 2^(X₀)+1 {O(EXP)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2^(X₀)+1 {O(EXP)}
t₃, X₂: 2^(X₀)+1 {O(EXP)}
t₄, X₀: 2⋅X₀ {O(n)}
t₄, X₁: 2^(X₀)+1 {O(EXP)}
t₄, X₂: 2^(X₀)+1 {O(EXP)}