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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₃, 0, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ 0 ≤ X₃
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₃ ≤ 0
t₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ 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_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, 0, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₁
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₂
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, 1+X₁, X₂, X₃, X₄, X₅)
t₉: eval_foo_bb4_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 [X₄; X₅] that do not contribute to the problem

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_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_start, eval_foo_stop
Transitions:
t₁₉: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, 0, X₂, X₃) :|: 1 ≤ X₂ ∧ 0 ≤ X₃
t₂₀: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0
t₂₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0
t₂₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₂₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₂₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, 0, X₂, X₃) :|: X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂
t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂
t₂₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁
t₂₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₂₈: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₃+1 {O(n)}

MPRF:

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

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

new bound:

X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁]
• eval_foo_bb2_in: [X₀+X₁]
• eval_foo_bb3_in: [X₀+X₁]

MPRF for transition t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

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

MPRF for transition t₂₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

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

All Bounds

Timebounds

Overall timebound:5⋅X₃+7 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₃+1 {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₃ {O(n)}
t₂₅: X₃ {O(n)}
t₂₆: X₃ {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₃+7 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 2⋅X₃+1 {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₃ {O(n)}
t₂₅: X₃ {O(n)}
t₂₆: X₃ {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Sizebounds

t₁₉, X₀: X₃ {O(n)}
t₁₉, X₁: 0 {O(1)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₂₀, X₀: X₀ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: X₀ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: X₃ {O(n)}
t₂₂, X₁: X₃ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: 0 {O(1)}
t₂₃, X₁: X₃ {O(n)}
t₂₃, X₂: 2⋅X₂ {O(n)}
t₂₃, X₃: 2⋅X₃ {O(n)}
t₂₄, X₀: X₃ {O(n)}
t₂₄, X₁: 0 {O(1)}
t₂₄, X₂: X₂ {O(n)}
t₂₄, X₃: X₃ {O(n)}
t₂₅, X₀: X₃ {O(n)}
t₂₅, X₁: X₃ {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₆, X₀: X₃ {O(n)}
t₂₆, X₁: X₃ {O(n)}
t₂₆, X₂: X₂ {O(n)}
t₂₆, X₃: X₃ {O(n)}
t₂₇, X₀: 2⋅X₀ {O(n)}
t₂₇, X₁: 2⋅X₁+X₃ {O(n)}
t₂₇, X₂: 4⋅X₂ {O(n)}
t₂₇, X₃: 4⋅X₃ {O(n)}
t₂₈, X₀: X₀ {O(n)}
t₂₈, X₁: X₁ {O(n)}
t₂₈, X₂: X₂ {O(n)}
t₂₈, X₃: X₃ {O(n)}