KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
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_bb6_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₅, X₆, X₂, X₃, X₄, X₅, X₆)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀+X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀+X₁ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb5_in(X₀, X₁, X₀-1, X₁, X₄, X₅, X₆)
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb5_in(X₀, X₁, X₀, X₁-1, X₄, X₅, X₆) :|: 1 ≤ X₁
t₈: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb5_in(X₀, X₁, X₀, X₁, X₄, X₅, X₆) :|: X₁ ≤ 0
t₉: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅, X₆)
t₁₀: eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Eliminate variables [X₄] that do not contribute to the problem

Found invariant 1 ≤ X₄+X₅ ∧ X₃ ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 0 ≤ X₃+X₄ ∧ X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂+X₃ ∧ X₁ ≤ 1+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₀+X₁ for location eval_foo_bb5_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ ∧ X₀+X₁ ≤ 0 for location eval_foo_stop

Found invariant 1 ≤ X₄+X₅ ∧ X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant 1 ≤ X₄+X₅ ∧ X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ X₀ ≤ X₄ ∧ 1 ≤ X₀+X₁ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ ∧ X₀+X₁ ≤ 0 for location eval_foo_bb6_in

Found invariant 1 ≤ X₅ ∧ 1 ≤ X₄+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1+X₀ ≤ X₅ ∧ 1 ≤ X₁+X₄ ∧ X₀ ≤ X₄ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_foo_bb4_in

Cut unsatisfiable transition [t₂₈: eval_foo_bb4_in→eval_foo_bb5_in]

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
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_bb6_in, eval_foo_start, eval_foo_stop
Transitions:
t₂₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀+X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀-1, X₁, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄+X₅ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀, X₁-1, X₄, X₅) :|: 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₉: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁+X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₅
t₃₀: eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₃₁: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀+X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₄+X₅ {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]
• eval_foo_bb5_in: [X₂+X₃]

MPRF for transition t₂₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₅]
• eval_foo_bb2_in: [X₀+X₅]
• eval_foo_bb3_in: [X₀+X₅-1]
• eval_foo_bb4_in: [X₀+X₅]
• eval_foo_bb5_in: [X₂+X₅]

MPRF for transition t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁+X₄]
• eval_foo_bb2_in: [X₁+X₄]
• eval_foo_bb3_in: [X₁+X₄]
• eval_foo_bb4_in: [X₁+X₄-1]
• eval_foo_bb5_in: [X₃+X₄]

MPRF for transition t₂₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀-1, X₁, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄+X₅ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₅]
• eval_foo_bb2_in: [X₀+X₅]
• eval_foo_bb3_in: [X₀+X₅]
• eval_foo_bb4_in: [X₀+X₅]
• eval_foo_bb5_in: [X₂+X₅]

MPRF for transition t₂₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀, X₁-1, X₄, X₅) :|: 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁]
• eval_foo_bb4_in: [X₁]
• eval_foo_bb5_in: [X₃]

MPRF for transition t₂₉: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁+X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁]
• eval_foo_bb2_in: [X₀+X₁]
• eval_foo_bb3_in: [X₀+X₁]
• eval_foo_bb4_in: [X₀+X₁]
• eval_foo_bb5_in: [1+X₂+X₃]

All Bounds

Timebounds

Overall timebound:5⋅X₄+6⋅X₅+4 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: X₄+X₅ {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₄+X₅ {O(n)}
t₂₅: X₄+X₅ {O(n)}
t₂₆: X₄+X₅ {O(n)}
t₂₇: X₅ {O(n)}
t₂₉: X₄+X₅ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₄+6⋅X₅+4 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: X₄+X₅ {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₄+X₅ {O(n)}
t₂₅: X₄+X₅ {O(n)}
t₂₆: X₄+X₅ {O(n)}
t₂₇: X₅ {O(n)}
t₂₉: X₄+X₅ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Sizebounds

t₂₁, X₀: X₄ {O(n)}
t₂₁, X₁: X₅ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₁, X₅: X₅ {O(n)}
t₂₂, X₀: 2⋅X₄+X₅ {O(n)}
t₂₂, X₁: 2⋅X₅+X₄ {O(n)}
t₂₂, X₂: 2⋅X₅+4⋅X₄+X₂ {O(n)}
t₂₂, X₃: 2⋅X₄+4⋅X₅+X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₂, X₅: X₅ {O(n)}
t₂₃, X₀: 3⋅X₄+X₅ {O(n)}
t₂₃, X₁: 3⋅X₅+X₄ {O(n)}
t₂₃, X₂: 2⋅X₅+4⋅X₄+X₂ {O(n)}
t₂₃, X₃: 2⋅X₄+4⋅X₅+X₃ {O(n)}
t₂₃, X₄: 2⋅X₄ {O(n)}
t₂₃, X₅: 2⋅X₅ {O(n)}
t₂₄, X₀: 2⋅X₄+X₅ {O(n)}
t₂₄, X₁: 2⋅X₅+X₄ {O(n)}
t₂₄, X₂: 2⋅X₅+4⋅X₄+X₂ {O(n)}
t₂₄, X₃: 2⋅X₄+4⋅X₅+X₃ {O(n)}
t₂₄, X₄: X₄ {O(n)}
t₂₄, X₅: X₅ {O(n)}
t₂₅, X₀: 2⋅X₄+X₅ {O(n)}
t₂₅, X₁: 2⋅X₅+X₄ {O(n)}
t₂₅, X₂: 2⋅X₅+4⋅X₄+X₂ {O(n)}
t₂₅, X₃: 2⋅X₄+4⋅X₅+X₃ {O(n)}
t₂₅, X₄: X₄ {O(n)}
t₂₅, X₅: X₅ {O(n)}
t₂₆, X₀: 2⋅X₄+X₅ {O(n)}
t₂₆, X₁: 2⋅X₅+X₄ {O(n)}
t₂₆, X₂: 2⋅X₄+X₅ {O(n)}
t₂₆, X₃: 2⋅X₅+X₄ {O(n)}
t₂₆, X₄: X₄ {O(n)}
t₂₆, X₅: X₅ {O(n)}
t₂₇, X₀: 2⋅X₄+X₅ {O(n)}
t₂₇, X₁: 2⋅X₅+X₄ {O(n)}
t₂₇, X₂: 2⋅X₄+X₅ {O(n)}
t₂₇, X₃: 2⋅X₅+X₄ {O(n)}
t₂₇, X₄: X₄ {O(n)}
t₂₇, X₅: X₅ {O(n)}
t₂₉, X₀: 2⋅X₄+X₅ {O(n)}
t₂₉, X₁: 2⋅X₅+X₄ {O(n)}
t₂₉, X₂: 2⋅X₅+4⋅X₄ {O(n)}
t₂₉, X₃: 2⋅X₄+4⋅X₅ {O(n)}
t₂₉, X₄: X₄ {O(n)}
t₂₉, X₅: X₅ {O(n)}
t₃₀, X₀: 3⋅X₄+X₅ {O(n)}
t₃₀, X₁: 3⋅X₅+X₄ {O(n)}
t₃₀, X₂: 2⋅X₅+4⋅X₄+X₂ {O(n)}
t₃₀, X₃: 2⋅X₄+4⋅X₅+X₃ {O(n)}
t₃₀, X₄: 2⋅X₄ {O(n)}
t₃₀, X₅: 2⋅X₅ {O(n)}
t₃₁, X₀: X₀ {O(n)}
t₃₁, X₁: X₁ {O(n)}
t₃₁, X₂: X₂ {O(n)}
t₃₁, X₃: X₃ {O(n)}
t₃₁, X₄: X₄ {O(n)}
t₃₁, X₅: X₅ {O(n)}