KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀+X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃)
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₈: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₁: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₂: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₀ ≤ X₁
t₁₃: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₁₄: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₁₅: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₁ ≤ X₀
t₁₆: eval_foo_bb5_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Cut unsatisfiable transition [t₈: eval_foo_bb4_in→eval_foo_bb1_in; t₉: eval_foo_bb4_in→eval_foo_bb1_in; t₁₀: eval_foo_bb4_in→eval_foo_bb1_in; t₁₁: eval_foo_bb4_in→eval_foo_bb1_in; t₁₃: eval_foo_bb4_in→eval_foo_bb1_in; t₁₄: eval_foo_bb4_in→eval_foo_bb1_in; t₁₅: eval_foo_bb4_in→eval_foo_bb1_in]

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 for location eval_foo_bb5_in

Found invariant 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 for location eval_foo_stop

Found invariant 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀+X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃
t₁₂: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃
t₁₆: eval_foo_bb5_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀+X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁]
• eval_foo_bb2_in: [X₀+X₁-1]
• eval_foo_bb3_in: [X₀+X₁-1]
• eval_foo_bb4_in: [X₀+X₁-1]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₂-1]
• eval_foo_bb2_in: [X₀+X₂-1]
• eval_foo_bb3_in: [X₀+X₂-2]
• eval_foo_bb4_in: [X₀+X₂-1]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

2⋅X₂+2⋅X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [2⋅X₀+X₁+X₃-1]
• eval_foo_bb2_in: [2⋅X₀+X₁+X₃-1]
• eval_foo_bb3_in: [2⋅X₀+X₁+X₃-1]
• eval_foo_bb4_in: [2⋅X₀+X₁+X₃-2]

MPRF for transition t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀]
• eval_foo_bb4_in: [X₀]

MPRF for transition t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+2⋅X₀]
• eval_foo_bb2_in: [1+2⋅X₀]
• eval_foo_bb3_in: [2⋅X₀]
• eval_foo_bb4_in: [1+2⋅X₀]

MPRF for transition t₁₂: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

2⋅X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁+X₃]
• eval_foo_bb2_in: [X₁+X₃]
• eval_foo_bb3_in: [X₁+X₃]
• eval_foo_bb4_in: [X₁+X₃]

All Bounds

Timebounds

Overall timebound:5⋅X₃+8⋅X₂+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₂+1 {O(n)}
t₅: 2⋅X₂+2⋅X₃+1 {O(n)}
t₆: X₂ {O(n)}
t₇: 2⋅X₂+1 {O(n)}
t₁₂: 2⋅X₃ {O(n)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₃+8⋅X₂+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₂+1 {O(n)}
t₅: 2⋅X₂+2⋅X₃+1 {O(n)}
t₆: X₂ {O(n)}
t₇: 2⋅X₂+1 {O(n)}
t₁₂: 2⋅X₃ {O(n)}
t₁₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: X₃ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 3⋅X₂ {O(n)}
t₃, X₁: 3⋅X₃ {O(n)}
t₃, X₂: 3⋅X₂ {O(n)}
t₃, X₃: 3⋅X₃ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: X₃ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₃ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₂ {O(n)}
t₆, X₁: X₃ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: X₃ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₁₂, X₀: X₂ {O(n)}
t₁₂, X₁: X₃ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₆, X₀: 3⋅X₂ {O(n)}
t₁₆, X₁: 3⋅X₃ {O(n)}
t₁₆, X₂: 3⋅X₂ {O(n)}
t₁₆, X₃: 3⋅X₃ {O(n)}