KoAT2 Proof Output

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) :|: 1 ≤ 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, 1+X₂, X₂+X₃) :|: 1+X₂ ≤ X₁
t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀

Preprocessing

Found invariant 0 ≤ X₀ for location l1

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 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) :|: 1 ≤ 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 ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 1+X₂, X₂+X₃) :|: 1+X₂ ≤ X₁ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 0

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 0, 0) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• l1: [X₁]
• l2: [X₁-1]

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l1(X₀+X₃, X₁-1, X₂, X₃) :|: X₁ ≤ X₂ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• l1: [1+X₁]
• l2: [1+X₁]

MPRF for transition t₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, 1+X₂, X₂+X₃) :|: 1+X₂ ≤ X₁ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁⋅X₁+3⋅X₁+2 {O(n^2)}

MPRF:

• l1: [1+X₁]
• l2: [1+X₁-X₂]

Cut unsatisfiable transition [t₃: l2→l1; t₃₅: 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 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 3 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2_v2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ 1+X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2_v1

Found invariant 0 ≤ X₀ for location l1

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3

MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l3(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 0 of depth 1:

new bound:

12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁+1 {O(EXP)}

MPRF:

• l3: [1+X₀]

Found invariant 0 ≤ X₀ for location l1

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l3_v1

All Bounds

Timebounds

Overall timebound:12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁+X₁⋅X₁+5⋅X₁+6 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₂: X₁⋅X₁+3⋅X₁+2 {O(n^2)}
t₃: X₁+1 {O(n)}
t₄: 1 {O(1)}
t₅: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁+1 {O(EXP)}

Costbounds

Overall costbound: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁+X₁⋅X₁+5⋅X₁+6 {O(EXP)}
t₀: 1 {O(1)}
t₁: X₁ {O(n)}
t₂: X₁⋅X₁+3⋅X₁+2 {O(n^2)}
t₃: X₁+1 {O(n)}
t₄: 1 {O(1)}
t₅: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅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: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 0 {O(1)}
t₁, X₃: 0 {O(1)}
t₂, X₀: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁⋅X₁+3⋅X₁+2 {O(n^2)}
t₂, X₃: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8 {O(EXP)}
t₃, X₀: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₁⋅X₁+3⋅X₁+2 {O(n^2)}
t₃, X₃: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8 {O(EXP)}
t₄, X₀: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: X₁⋅X₁+3⋅X₁+X₂+2 {O(n^2)}
t₄, X₃: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8+X₃ {O(EXP)}
t₅, X₀: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁⋅X₁+16⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)+24⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8⋅X₁⋅X₁ {O(EXP)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: X₁⋅X₁+3⋅X₁+X₂+2 {O(n^2)}
t₅, X₃: 12⋅2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅4⋅X₁⋅X₁+2^(3⋅X₁)⋅2^(X₁⋅X₁)⋅8+X₃ {O(EXP)}