Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₂ ≤ X₁
t₁: l1(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂-1) :|: 1 ≤ X₀ ∧ X₂ ≤ X₁+1
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₂-1, X₂-1) :|: 0 ≤ X₀ ∧ 0 ≤ X₁

Preprocessing

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₂ ≤ X₁
t₁: l1(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂-1) :|: 1 ≤ X₀ ∧ X₂ ≤ X₁+1 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₂-1, X₂-1) :|: 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀

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

new bound:

X₀+1 {O(n)}

MPRF:

l1 [X₀+1 ]

Found invariant 1 ≤ 0 for location l1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Analysing control-flow refined program

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___2

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___3

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___1

MPRF for transition t₄₇: n_l1___3(X₀, X₁, X₂) → n_l1___3(X₀-1, X₁, X₂-1) :|: X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

n_l1___3 [X₀+1 ]

MPRF for transition t₄₂: n_l1___1(X₀, X₁, X₂) → n_l1___1(X₀-1, X₁, X₂-1) :|: X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+2⋅X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

3⋅X₀ {O(n)}

MPRF:

n_l1___1 [X₀+1 ]
n_l1___2 [X₀ ]

MPRF for transition t₄₃: n_l1___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁+X₂-1, X₂-1) :|: X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+2⋅X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

3⋅X₀ {O(n)}

MPRF:

n_l1___1 [X₀+1 ]
n_l1___2 [X₀ ]

MPRF for transition t₄₄: n_l1___2(X₀, X₁, X₂) → n_l1___1(X₀-1, X₁, X₂-1) :|: X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+2⋅X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

3⋅X₀+2 {O(n)}

MPRF:

n_l1___1 [X₀+1 ]
n_l1___2 [X₀+1 ]

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___2

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___3

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___1

Found invariant 1 ≤ 0 for location n_l1___2

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___3

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 1 ≤ 0 for location n_l1___1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___2

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___3

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___1

CFR did not improve the program. Rolling back

All Bounds

Timebounds

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

Costbounds

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

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 2⋅X₀ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}