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_bb6_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₁, 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) :|: 0 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, 1, 1+X₀, X₃, X₄, X₅)
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₂
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₁
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃, X₄, X₅)
t₈: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂-2, X₁, X₂, X₃, X₄, X₅)
t₉: eval_foo_bb6_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; X₅] that do not contribute to the problem

Found invariant 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ X₁ ≤ 1+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location eval_foo_bb5_in

Found invariant X₀ ≤ X₃ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

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

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₃ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₃ ∧ 1+X₀ ≤ 0 for location eval_foo_bb6_in

Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ 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_bb6_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₉: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₁, X₂, X₃)
t₂₀: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ X₃
t₂₁: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb6_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₀ ≤ X₃
t₂₂: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, 1, 1+X₀, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃
t₂₃: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ X₂ ≤ 1+X₀ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₂₄: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₂ ≤ 1+X₀ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₂₅: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃) :|: X₂ ≤ 1+X₀ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃
t₂₆: eval_foo_bb5_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂-2, X₁, X₂, X₃) :|: X₁ ≤ 1+X₀ ∧ X₂ ≤ 1+X₀ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₂₇: eval_foo_bb6_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₀ ≤ X₃
t₂₈: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂₀: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀]
• eval_foo_bb4_in: [X₀]
• eval_foo_bb5_in: [X₀]

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

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [1+X₀]
• eval_foo_bb3_in: [X₂-1]
• eval_foo_bb4_in: [X₀]
• eval_foo_bb5_in: [X₂-1]

MPRF for transition t₂₄: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₂ ≤ 1+X₀ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀+X₃]
• eval_foo_bb2_in: [1+X₀+X₃]
• eval_foo_bb3_in: [1+X₀+X₃]
• eval_foo_bb4_in: [X₂+X₃]
• eval_foo_bb5_in: [X₀+X₃]

MPRF for transition t₂₆: eval_foo_bb5_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂-2, X₁, X₂, X₃) :|: X₁ ≤ 1+X₀ ∧ X₂ ≤ 1+X₀ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [1+X₀]
• eval_foo_bb3_in: [X₂]
• eval_foo_bb4_in: [X₂]
• eval_foo_bb5_in: [1+X₀]

MPRF for transition t₂₃: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ X₂ ≤ 1+X₀ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃⋅X₃+3⋅X₃+2 {O(n^2)}

MPRF:

• eval_foo_bb1_in: [1+X₃]
• eval_foo_bb2_in: [1+X₃]
• eval_foo_bb3_in: [2+X₃-X₁]
• eval_foo_bb4_in: [1+X₃-X₁]
• eval_foo_bb5_in: [X₃-X₁]

MPRF for transition t₂₅: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃) :|: X₂ ≤ 1+X₀ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃⋅X₃+3⋅X₃+1 {O(n^2)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₂-X₁]
• eval_foo_bb4_in: [1+X₀-X₁]
• eval_foo_bb5_in: [X₂-X₁]

Found invariant 2 ≤ X₃ ∧ 5 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1+X₀ ∧ 3 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 5 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb4_in_v2

Found invariant X₀ ≤ X₃ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

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

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₃ for location eval_foo_bb1_in

Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb4_in_v1

Found invariant X₀ ≤ X₃ ∧ 1+X₀ ≤ 0 for location eval_foo_bb6_in

Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 3 ≤ X₁+X₃ ∧ X₁ ≤ 1+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in_v1

All Bounds

Timebounds

Overall timebound:2⋅X₃⋅X₃+11⋅X₃+11 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: X₃+1 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: X₃+1 {O(n)}
t₂₃: X₃⋅X₃+3⋅X₃+2 {O(n^2)}
t₂₄: 2⋅X₃+1 {O(n)}
t₂₅: X₃⋅X₃+3⋅X₃+1 {O(n^2)}
t₂₆: X₃+1 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃⋅X₃+11⋅X₃+11 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: X₃+1 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: X₃+1 {O(n)}
t₂₃: X₃⋅X₃+3⋅X₃+2 {O(n^2)}
t₂₄: 2⋅X₃+1 {O(n)}
t₂₅: X₃⋅X₃+3⋅X₃+1 {O(n^2)}
t₂₆: X₃+1 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Sizebounds

t₁₉, X₀: X₃ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₂₀, X₀: X₃+1 {O(n)}
t₂₀, X₁: X₁+X₃+6 {O(n)}
t₂₀, X₂: 2⋅X₃+X₂+4 {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: 2⋅X₃+1 {O(n)}
t₂₁, X₁: X₁+X₃+6 {O(n)}
t₂₁, X₂: 2⋅X₃+X₂+4 {O(n)}
t₂₁, X₃: 2⋅X₃ {O(n)}
t₂₂, X₀: X₃+1 {O(n)}
t₂₂, X₁: 1 {O(1)}
t₂₂, X₂: X₃+2 {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: X₃+1 {O(n)}
t₂₃, X₁: X₃+6 {O(n)}
t₂₃, X₂: X₃+2 {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₄, X₀: X₃+1 {O(n)}
t₂₄, X₁: X₃+6 {O(n)}
t₂₄, X₂: 2⋅X₃+4 {O(n)}
t₂₄, X₃: X₃ {O(n)}
t₂₅, X₀: X₃+1 {O(n)}
t₂₅, X₁: X₃+5 {O(n)}
t₂₅, X₂: X₃+2 {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₆, X₀: X₃+1 {O(n)}
t₂₆, X₁: X₃+6 {O(n)}
t₂₆, X₂: 2⋅X₃+4 {O(n)}
t₂₆, X₃: X₃ {O(n)}
t₂₇, X₀: 2⋅X₃+1 {O(n)}
t₂₇, X₁: X₁+X₃+6 {O(n)}
t₂₇, X₂: 2⋅X₃+X₂+4 {O(n)}
t₂₇, X₃: 2⋅X₃ {O(n)}
t₂₈, X₀: X₀ {O(n)}
t₂₈, X₁: X₁ {O(n)}
t₂₈, X₂: X₂ {O(n)}
t₂₈, X₃: X₃ {O(n)}