KoAT2 Proof Output

Initial Problem

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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₁, X₃, X₄, X₅) :|: 1 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ 0
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃, X₄, X₅)
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄, X₅)
t₈: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Eliminate variables [X₃] that do not contribute to the problem

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 for location eval_foo_bb5_in

Found invariant X₂ ≤ X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₂ ≤ X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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:
t₁₇: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₁, X₃, X₄) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₄
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₄
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄
t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄
t₂₃: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₄
t₂₄: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₄
t₂₅: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₁, X₃, X₄) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

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

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₂]
• eval_foo_bb3_in: [X₂-1]
• eval_foo_bb4_in: [X₂]

MPRF for transition t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

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

MPRF for transition t₂₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₃+X₄ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₂]
• eval_foo_bb3_in: [X₂]
• eval_foo_bb4_in: [X₂]

MPRF for transition t₂₃: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₄ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

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

All Bounds

Timebounds

Overall timebound:2⋅X₄+3⋅X₃+4 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₄ {O(n)}
t₂₁: X₃ {O(n)}
t₂₂: X₄ {O(n)}
t₂₃: X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₄+3⋅X₃+4 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₄ {O(n)}
t₂₁: X₃ {O(n)}
t₂₂: X₄ {O(n)}
t₂₃: X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₃ {O(n)}
t₁₇, X₁: X₄ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₈, X₀: X₃ {O(n)}
t₁₈, X₁: 2⋅X₄ {O(n)}
t₁₈, X₂: X₄ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: 2⋅X₃ {O(n)}
t₁₉, X₁: 2⋅X₄ {O(n)}
t₁₉, X₂: X₂+X₄ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₁₉, X₄: 2⋅X₄ {O(n)}
t₂₀, X₀: X₃ {O(n)}
t₂₀, X₁: 2⋅X₄ {O(n)}
t₂₀, X₂: X₄ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₁, X₀: X₃ {O(n)}
t₂₁, X₁: 4⋅X₄ {O(n)}
t₂₁, X₂: X₄ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₂, X₀: X₃ {O(n)}
t₂₂, X₁: 2⋅X₄ {O(n)}
t₂₂, X₂: X₄ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₃, X₀: X₃ {O(n)}
t₂₃, X₁: X₄ {O(n)}
t₂₃, X₂: X₄ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₄, X₀: 2⋅X₃ {O(n)}
t₂₄, X₁: 2⋅X₄ {O(n)}
t₂₄, X₂: X₂+X₄ {O(n)}
t₂₄, X₃: 2⋅X₃ {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₅, X₄: X₄ {O(n)}