KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars: C
Locations: l0, l1
Transitions:
t₂: l0(X₀, X₁) → l1(X₀, X₁)
t₀: l1(X₀, X₁) → l1(X₀-1, C) :|: 0 ≤ X₀
t₁: l1(X₀, X₁) → l1(X₀, X₁-1) :|: 0 ≤ X₁

Preprocessing

Problem after Preprocessing

MPRF for transition t₀: l1(X₀, X₁) → l1(X₀-1, C) :|: 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l1 [X₀+1 ]

Found invariant 1 ≤ 0 for location l1

Analysing control-flow refined program

Found invariant 0 ≤ 1+X₁ for location n_l1___2

Found invariant 0 ≤ 1+X₀ for location n_l1___3

Found invariant 0 ≤ 1+X₁ ∧ 0 ≤ 2+X₀+X₁ ∧ 0 ≤ 1+X₀ for location n_l1___1

MPRF for transition t₄₁: n_l1___2(X₀, X₁) → n_l1___2(X₀, X₁-1) :|: 0 ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ 1+X₁ of depth 1:

new bound:

X₁+2 {O(n)}

MPRF:

n_l1___2 [X₁+1 ]

MPRF for transition t₄₀: n_l1___1(X₀, X₁) → n_l1___3(Arg0_P, NoDet0) :|: 0 ≤ 1+X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+Arg0_P ∧ X₀ ≤ Arg0_P+1 ∧ 1+Arg0_P ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 2+X₀+X₁ ∧ 0 ≤ 1+X₀ of depth 1:

new bound:

3⋅X₀+7 {O(n)}

MPRF:

n_l1___1 [X₀+2 ]
n_l1___3 [X₀+2 ]

MPRF for transition t₄₃: n_l1___3(X₀, X₁) → n_l1___1(X₀, X₁-1) :|: 0 ≤ 1+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ 1+X₀ of depth 1:

new bound:

3⋅X₀+7 {O(n)}

MPRF:

n_l1___1 [X₀+1 ]
n_l1___3 [X₀+2 ]

MPRF for transition t₄₄: n_l1___3(X₀, X₁) → n_l1___3(Arg0_P, NoDet0) :|: 0 ≤ 1+X₀ ∧ 0 ≤ 1+Arg0_P ∧ X₀ ≤ Arg0_P+1 ∧ 1+Arg0_P ≤ X₀ ∧ 0 ≤ 1+X₀ of depth 1:

new bound:

3⋅X₀+7 {O(n)}

MPRF:

n_l1___1 [X₀+1 ]
n_l1___3 [X₀+2 ]

Found invariant 0 ≤ 1+X₁ for location n_l1___2

Found invariant 0 ≤ 1+X₀ for location n_l1___3

Found invariant 0 ≤ 1+X₁ ∧ 0 ≤ 2+X₀+X₁ ∧ 0 ≤ 1+X₀ for location n_l1___1

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₂: 1 {O(1)}
t₀: X₀+1 {O(n)}
t₁: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₂: 1 {O(1)}
t₀: X₀+1 {O(n)}
t₁: inf {Infinity}

Sizebounds

t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₀, X₀: 2⋅X₀+1 {O(n)}
t₁, X₀: 2⋅X₀+1 {O(n)}