KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₅ ≤ X₁ ∧ X₀ ≤ X₄ for location eval_foo_bb5_in

Found invariant X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₁ ≤ X₀ for location eval_foo_0

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

Found invariant X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₁ ≤ X₀ for location eval_foo_bb3_in

Found invariant X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₁ ≤ X₀ for location eval_foo_bb2_in

Found invariant X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₁ ≤ X₀ for location eval_foo_1

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

Found invariant X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ X₁ ≤ X₀ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: nondef.0
Locations: eval_foo_0, eval_foo_1, 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:
t₇: eval_foo_0(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_1(X₀, X₁, nondef.0, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄
t₈: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄
t₉: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄
t₁₀: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄
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, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ 0 ≤ X₃ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₃ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_0(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄
t₁₁: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1-X₃, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄
t₁₂: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, 1+X₁+X₃, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄
t₁₃: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₄ ∧ X₅ ≤ X₁
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ 0 ≤ X₃ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [X₀-X₁]
• eval_foo_1: [X₀-X₁]
• eval_foo_bb1_in: [1+X₀-X₁]
• eval_foo_bb2_in: [X₀-X₁]
• eval_foo_bb3_in: [X₀-X₁]
• eval_foo_bb4_in: [X₀-X₁]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_0(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [X₀-X₁]
• eval_foo_1: [X₀-X₁]
• eval_foo_bb1_in: [1+X₀-X₁]
• eval_foo_bb2_in: [1+X₀-X₁]
• eval_foo_bb3_in: [X₀-X₁]
• eval_foo_bb4_in: [X₀-X₁]

MPRF for transition t₇: eval_foo_0(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_1(X₀, X₁, nondef.0, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₀-X₁]
• eval_foo_1: [X₀-X₁]
• eval_foo_bb1_in: [1+X₀-X₁]
• eval_foo_bb2_in: [1+X₀-X₁]
• eval_foo_bb3_in: [X₀-X₁]
• eval_foo_bb4_in: [X₀-X₁]

MPRF for transition t₈: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₀-X₅]
• eval_foo_1: [1+X₀-X₅]
• eval_foo_bb1_in: [1+X₀-X₅]
• eval_foo_bb2_in: [1+X₀-X₅]
• eval_foo_bb3_in: [X₀-X₅]
• eval_foo_bb4_in: [1+X₀-X₅]

MPRF for transition t₉: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₀-X₅]
• eval_foo_1: [1+X₀-X₅]
• eval_foo_bb1_in: [1+X₀-X₅]
• eval_foo_bb2_in: [1+X₀-X₅]
• eval_foo_bb3_in: [X₀-X₅]
• eval_foo_bb4_in: [1+X₀-X₅]

MPRF for transition t₁₀: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₄-X₁]
• eval_foo_1: [1+X₄-X₁]
• eval_foo_bb1_in: [1+X₄-X₁]
• eval_foo_bb2_in: [1+X₄-X₁]
• eval_foo_bb3_in: [1+X₄-X₁]
• eval_foo_bb4_in: [X₄-X₁]

MPRF for transition t₁₁: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1-X₃, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₀-X₅]
• eval_foo_1: [1+X₀-X₅]
• eval_foo_bb1_in: [1+X₀-X₅]
• eval_foo_bb2_in: [1+X₀-X₅]
• eval_foo_bb3_in: [1+X₀-X₅]
• eval_foo_bb4_in: [1+X₀-X₅]

MPRF for transition t₁₂: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, 1+X₁+X₃, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃ ∧ X₅ ≤ X₄ of depth 1:

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₄-X₁]
• eval_foo_1: [1+X₄-X₁]
• eval_foo_bb1_in: [1+X₄-X₁]
• eval_foo_bb2_in: [1+X₄-X₁]
• eval_foo_bb3_in: [1+X₄-X₁]
• eval_foo_bb4_in: [1+X₄-X₁]

All Bounds

Timebounds

Overall timebound:8⋅X₄+8⋅X₅+13 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₄+X₅+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₄+X₅+1 {O(n)}
t₇: X₄+X₅+1 {O(n)}
t₈: X₄+X₅+1 {O(n)}
t₉: X₄+X₅+1 {O(n)}
t₁₀: X₄+X₅+1 {O(n)}
t₁₁: X₄+X₅+1 {O(n)}
t₁₂: X₄+X₅+1 {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₄+8⋅X₅+13 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₄+X₅+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₄+X₅+1 {O(n)}
t₇: X₄+X₅+1 {O(n)}
t₈: X₄+X₅+1 {O(n)}
t₉: X₄+X₅+1 {O(n)}
t₁₀: X₄+X₅+1 {O(n)}
t₁₁: X₄+X₅+1 {O(n)}
t₁₂: X₄+X₅+1 {O(n)}
t₁₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₂, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: 4⋅X₃⋅X₄+4⋅X₃⋅X₅+2⋅X₅+5⋅X₄+8⋅X₃+2 {O(n^2)}
t₃, X₁: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+5⋅X₅+2 {O(n^2)}
t₃, X₃: 3⋅X₃ {O(n)}
t₃, X₄: 3⋅X₄ {O(n)}
t₃, X₅: 3⋅X₅ {O(n)}
t₄, X₀: X₄ {O(n)}
t₄, X₁: X₅ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₅, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₇, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₇, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₈, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₅ {O(n)}
t₉, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₉, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: X₅ {O(n)}
t₁₀, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₁₀, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₁₀, X₂: 0 {O(1)}
t₁₀, X₃: X₃ {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₀, X₅: X₅ {O(n)}
t₁₁, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₁₁, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₁₁, X₃: X₃ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₁, X₅: X₅ {O(n)}
t₁₂, X₀: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+X₅+1 {O(n^2)}
t₁₂, X₁: X₃⋅X₄+X₃⋅X₅+2⋅X₃+2⋅X₅+X₄+1 {O(n^2)}
t₁₂, X₂: 0 {O(1)}
t₁₂, X₃: X₃ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₂, X₅: X₅ {O(n)}
t₁₃, X₀: 4⋅X₃⋅X₄+4⋅X₃⋅X₅+2⋅X₅+6⋅X₄+8⋅X₃+2 {O(n^2)}
t₁₃, X₁: 2⋅X₃⋅X₄+2⋅X₃⋅X₅+2⋅X₄+4⋅X₃+6⋅X₅+2 {O(n^2)}
t₁₃, X₃: 4⋅X₃ {O(n)}
t₁₃, X₄: 4⋅X₄ {O(n)}
t₁₃, X₅: 4⋅X₅ {O(n)}