KoAT2 Proof Output

Initial Problem

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

Preprocessing

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

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location eval_foo_bb5_in

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

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location eval_foo_stop

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

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

MPRF for transition t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, 0, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₃-X₀]
• eval_foo_bb2_in: [X₃-1-X₀]
• eval_foo_bb3_in: [X₃-1-X₀]
• eval_foo_bb4_in: [X₃-1-X₀]

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

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₃-X₀]
• eval_foo_bb2_in: [X₃-X₀]
• eval_foo_bb3_in: [X₃-X₀]
• eval_foo_bb4_in: [X₃-1-X₀]

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

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₃-X₀]
• eval_foo_bb2_in: [X₃-X₀]
• eval_foo_bb3_in: [X₃-X₀]
• eval_foo_bb4_in: [X₃-X₀]

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₃+X₃⋅X₃+X₃ {O(n^2)}

MPRF:

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

MPRF for transition t₂₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

2⋅X₂⋅X₂+3⋅X₂⋅X₃+X₃⋅X₃+2⋅X₂+X₃+1 {O(n^2)}

MPRF:

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

Found invariant 2 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in_v2

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location eval_foo_bb5_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location eval_foo_stop

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

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb3_in_v1

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

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+2⋅X₃⋅X₃+4⋅X₂⋅X₃+5⋅X₂+5⋅X₃+5 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: X₂+X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂⋅X₃+X₃⋅X₃+X₃ {O(n^2)}
t₂₁: X₂+X₃ {O(n)}
t₂₂: 2⋅X₂⋅X₂+3⋅X₂⋅X₃+X₃⋅X₃+2⋅X₂+X₃+1 {O(n^2)}
t₂₃: X₂+X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+2⋅X₃⋅X₃+4⋅X₂⋅X₃+5⋅X₂+5⋅X₃+5 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: X₂+X₃ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂⋅X₃+X₃⋅X₃+X₃ {O(n^2)}
t₂₁: X₂+X₃ {O(n)}
t₂₂: 2⋅X₂⋅X₂+3⋅X₂⋅X₃+X₃⋅X₃+2⋅X₂+X₃+1 {O(n^2)}
t₂₃: X₂+X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₂ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: 2⋅X₂+X₃ {O(n)}
t₁₈, X₁: 0 {O(1)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: 3⋅X₂+X₃ {O(n)}
t₁₉, X₁: 2⋅X₂+X₁+X₃+2 {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₂₀, X₀: 2⋅X₂+X₃ {O(n)}
t₂₀, X₁: 2⋅X₂+X₃+2 {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: 2⋅X₂+X₃ {O(n)}
t₂₁, X₁: 2⋅X₂+X₃+2 {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: 2⋅X₂+X₃ {O(n)}
t₂₂, X₁: 2⋅X₂+X₃+2 {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: 2⋅X₂+X₃ {O(n)}
t₂₃, X₁: 2⋅X₂+X₃+2 {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₄, X₀: 3⋅X₂+X₃ {O(n)}
t₂₄, X₁: 2⋅X₂+X₁+X₃+2 {O(n)}
t₂₄, X₂: 2⋅X₂ {O(n)}
t₂₄, X₃: 2⋅X₃ {O(n)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}