Initial Problem

Start: f
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f, g, h
Transitions:
t₀: f(X₀, X₁) → g(X₀, X₁)
t₁: g(X₀, X₁) → g(2⋅X₀, X₁-1) :|: 1 ≤ X₁
t₂: g(X₀, X₁) → h(X₀, X₁) :|: X₁ ≤ 0
t₃: h(X₀, X₁) → h(X₀-1, X₁) :|: 1 ≤ X₀

Preprocessing

Found invariant X₁ ≤ 0 for location h

Problem after Preprocessing

Start: f
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f, g, h
Transitions:
t₀: f(X₀, X₁) → g(X₀, X₁)
t₁: g(X₀, X₁) → g(2⋅X₀, X₁-1) :|: 1 ≤ X₁
t₂: g(X₀, X₁) → h(X₀, X₁) :|: X₁ ≤ 0
t₃: h(X₀, X₁) → h(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 0

MPRF for transition t₁: g(X₀, X₁) → g(2⋅X₀, X₁-1) :|: 1 ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

• g: [X₁]

MPRF for transition t₃: h(X₀, X₁) → h(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 0 of depth 1:

new bound:

2^(X₁)⋅X₀+X₀ {O(EXP)}

• h: [X₀]

Found invariant X₁ ≤ 0 for location h

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location h_v1

All Bounds

Timebounds

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

Costbounds

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

Sizebounds

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