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(X₃, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₃ ∧ 1 ≤ X₄
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ 0
t₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₀, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃, X₄, X₅)
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₄
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₁
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃, X₄, X₅)
t₁₀: 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 [X₂; X₅] that do not contribute to the problem

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

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb1_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location eval_foo_bb3_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ 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:
t₂₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ 1 ≤ X₃
t₂₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₂₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0
t₂₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₂
t₂₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₀, X₂, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₂
t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂
t₂₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₃ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₃
t₂₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₃
t₂₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁
t₃₀: eval_foo_bb5_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₃₁: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₂₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₃ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_bb3_in: [1+X₃-X₁]
• eval_foo_bb4_in: [X₃-X₁]

MPRF for transition t₂₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_foo_bb3_in: [X₃-X₁]
• eval_foo_bb4_in: [X₃-X₁]

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+9 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: X₂+1 {O(n)}
t₂₅: 1 {O(1)}
t₂₆: X₂ {O(n)}
t₂₇: X₃+1 {O(n)}
t₂₈: 1 {O(1)}
t₂₉: X₃ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+9 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: X₂+1 {O(n)}
t₂₅: 1 {O(1)}
t₂₆: X₂ {O(n)}
t₂₇: X₃+1 {O(n)}
t₂₈: 1 {O(1)}
t₂₉: X₃ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Sizebounds

t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: X₀ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: X₀ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₄, X₀: X₂ {O(n)}
t₂₄, X₁: X₁ {O(n)}
t₂₄, X₂: X₂ {O(n)}
t₂₄, X₃: X₃ {O(n)}
t₂₅, X₀: 0 {O(1)}
t₂₅, X₁: 0 {O(1)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₆, X₀: X₂ {O(n)}
t₂₆, X₁: X₁ {O(n)}
t₂₆, X₂: X₂ {O(n)}
t₂₆, X₃: X₃ {O(n)}
t₂₇, X₀: 0 {O(1)}
t₂₇, X₁: X₃ {O(n)}
t₂₇, X₂: X₂ {O(n)}
t₂₇, X₃: X₃ {O(n)}
t₂₈, X₀: 0 {O(1)}
t₂₈, X₁: X₃ {O(n)}
t₂₈, X₂: X₂ {O(n)}
t₂₈, X₃: X₃ {O(n)}
t₂₉, X₀: 0 {O(1)}
t₂₉, X₁: X₃ {O(n)}
t₂₉, X₂: X₂ {O(n)}
t₂₉, X₃: X₃ {O(n)}
t₃₀, X₀: 2⋅X₀ {O(n)}
t₃₀, X₁: 2⋅X₁+X₃ {O(n)}
t₃₀, X₂: 3⋅X₂ {O(n)}
t₃₀, X₃: 3⋅X₃ {O(n)}
t₃₁, X₀: X₀ {O(n)}
t₃₁, X₁: X₁ {O(n)}
t₃₁, X₂: X₂ {O(n)}
t₃₁, X₃: X₃ {O(n)}