KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀+2⋅X₁, 3⋅X₀+4⋅X₁, X₂-1) :|: 1 ≤ X₂
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₃: l2(X₀, X₁, X₂) → l2(X₀-1, X₁, X₂) :|: 1 ≤ X₀

Preprocessing

Problem after Preprocessing

MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀+2⋅X₁, 3⋅X₀+4⋅X₁, X₂-1) :|: 1 ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• l1: [X₂]

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

new bound:

8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₀ {O(EXP)}

MPRF:

• l2: [X₀]

Found invariant 0 ≤ X₀ for location l2_v1

All Bounds

Timebounds

Overall timebound:8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₀+X₂+2 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₀ {O(EXP)}

Costbounds

Overall costbound: 8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₀+X₂+2 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₀ {O(EXP)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 8^(X₂)⋅X₀+8^(X₂)⋅X₁ {O(EXP)}
t₁, X₁: 8^(X₂)⋅X₀+8^(X₂)⋅X₁ {O(EXP)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₀ {O(EXP)}
t₂, X₁: 8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₁ {O(EXP)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₀: 8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₀ {O(EXP)}
t₃, X₁: 8^(X₂)⋅X₀+8^(X₂)⋅X₁+X₁ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}