KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₄ ≤ X₀ for location eval_t27_bb1_in

Found invariant 1+X₄ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ X₄ ≤ X₀ ∧ 2+X₀+X₄ ≤ 0 ∧ X₃ ≤ 0 ∧ X₃ ≤ 1+X₀ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ 1000+X₁ ∧ 1+X₀ ≤ 0 for location eval_t27_bb3_in

Found invariant 1+X₄ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ 101+X₄ ≤ X₂ ∧ X₄ ≤ 899+X₁ ∧ X₄ ≤ X₀ ∧ 2+X₀+X₄ ≤ 0 ∧ X₃ ≤ 0 ∧ 100+X₃ ≤ X₂ ∧ X₃ ≤ 900+X₁ ∧ X₃ ≤ 1+X₀ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ 1000+X₁ ∧ 100 ≤ X₂ ∧ 0 ≤ 800+X₁+X₂ ∧ 101+X₀ ≤ X₂ ∧ 0 ≤ 900+X₁ ∧ X₀ ≤ 899+X₁ ∧ 1+X₀ ≤ 0 for location eval_t27_bb4_in

Found invariant X₄ ≤ X₀ ∧ 0 ≤ X₀ for location eval_t27_bb5_in

Found invariant X₄ ≤ X₀ ∧ 0 ≤ X₀ for location eval_t27_stop

Found invariant 1+X₄ ≤ 0 ∧ X₄ ≤ X₀ ∧ 2+X₀+X₄ ≤ 0 ∧ 1+X₀ ≤ 0 for location eval_t27_bb2_in

Problem after Preprocessing

Start: eval_t27_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_t27_bb0_in, eval_t27_bb1_in, eval_t27_bb2_in, eval_t27_bb3_in, eval_t27_bb4_in, eval_t27_bb5_in, eval_t27_start, eval_t27_stop
Transitions:
t₁: eval_t27_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂: eval_t27_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ X₄ ≤ X₀
t₃: eval_t27_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₄ ≤ X₀
t₄: eval_t27_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb3_in(X₀, X₁, 1000+X₁, 1+X₀, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ X₄ ≤ X₀
t₆: eval_t27_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb1_in(X₃, X₂, X₂, X₃, X₄, X₅) :|: X₂ ≤ 99 ∧ X₂ ≤ 1000+X₁ ∧ X₃ ≤ 1+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ X₄ ≤ X₀ ∧ X₃ ≤ 0
t₅: eval_t27_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 100 ≤ X₂ ∧ X₂ ≤ 1000+X₁ ∧ X₃ ≤ 1+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ X₄ ≤ X₀ ∧ X₃ ≤ 0
t₇: eval_t27_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb3_in(X₀, X₁, X₂-100, X₃, X₄, X₅) :|: X₂ ≤ 1000+X₁ ∧ 0 ≤ 900+X₁ ∧ X₃ ≤ 900+X₁ ∧ X₀ ≤ 899+X₁ ∧ X₄ ≤ 899+X₁ ∧ 0 ≤ 800+X₁+X₂ ∧ X₃ ≤ 1+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ 100 ≤ X₂ ∧ 100+X₃ ≤ X₂ ∧ 101+X₀ ≤ X₂ ∧ 101+X₄ ≤ X₂ ∧ X₄ ≤ X₀ ∧ X₃ ≤ 0
t₈: eval_t27_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₄ ≤ X₀
t₀: eval_t27_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂: eval_t27_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ X₄ ≤ X₀ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_t27_bb1_in: [-X₀]
• eval_t27_bb2_in: [-1-X₀]
• eval_t27_bb3_in: [-X₃]
• eval_t27_bb4_in: [-X₃]

MPRF for transition t₄: eval_t27_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb3_in(X₀, X₁, 1000+X₁, 1+X₀, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ X₄ ≤ X₀ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_t27_bb1_in: [-X₀]
• eval_t27_bb2_in: [-X₀]
• eval_t27_bb3_in: [-1-X₀]
• eval_t27_bb4_in: [-1-X₀]

MPRF for transition t₅: eval_t27_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 100 ≤ X₂ ∧ X₂ ≤ 1000+X₁ ∧ X₃ ≤ 1+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ X₄ ≤ X₀ ∧ X₃ ≤ 0 of depth 1:

new bound:

1000⋅X₄+X₅+99 {O(n)}

MPRF:

• eval_t27_bb1_in: [X₁-99-1000⋅X₀]
• eval_t27_bb2_in: [X₁-99-1000⋅X₀]
• eval_t27_bb3_in: [X₂-99-1000⋅X₃]
• eval_t27_bb4_in: [X₂-100-1000⋅X₃]

MPRF for transition t₆: eval_t27_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb1_in(X₃, X₂, X₂, X₃, X₄, X₅) :|: X₂ ≤ 99 ∧ X₂ ≤ 1000+X₁ ∧ X₃ ≤ 1+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ X₄ ≤ X₀ ∧ X₃ ≤ 0 of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_t27_bb1_in: [-X₀]
• eval_t27_bb2_in: [-X₀]
• eval_t27_bb3_in: [1-X₃]
• eval_t27_bb4_in: [1-X₃]

MPRF for transition t₇: eval_t27_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_t27_bb3_in(X₀, X₁, X₂-100, X₃, X₄, X₅) :|: X₂ ≤ 1000+X₁ ∧ 0 ≤ 900+X₁ ∧ X₃ ≤ 900+X₁ ∧ X₀ ≤ 899+X₁ ∧ X₄ ≤ 899+X₁ ∧ 0 ≤ 800+X₁+X₂ ∧ X₃ ≤ 1+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₀ ≤ X₃ ∧ 1+X₀+X₃ ≤ 0 ∧ 1+X₄ ≤ X₃ ∧ 1+X₃+X₄ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 2+X₀+X₄ ≤ 0 ∧ 100 ≤ X₂ ∧ 100+X₃ ≤ X₂ ∧ 101+X₀ ≤ X₂ ∧ 101+X₄ ≤ X₂ ∧ X₄ ≤ X₀ ∧ X₃ ≤ 0 of depth 1:

new bound:

1000⋅X₄+X₅ {O(n)}

MPRF:

• eval_t27_bb1_in: [X₁-1000⋅X₀]
• eval_t27_bb2_in: [X₁-1000⋅X₀]
• eval_t27_bb3_in: [X₂-1000-1000⋅X₀]
• eval_t27_bb4_in: [X₂-99-1000⋅X₃]

All Bounds

Timebounds

Overall timebound:2⋅X₅+2003⋅X₄+103 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₄ {O(n)}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: 1000⋅X₄+X₅+99 {O(n)}
t₆: X₄ {O(n)}
t₇: 1000⋅X₄+X₅ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₅+2003⋅X₄+103 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₄ {O(n)}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: 1000⋅X₄+X₅+99 {O(n)}
t₆: X₄ {O(n)}
t₇: 1000⋅X₄+X₅ {O(n)}
t₈: 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: X₄ {O(n)}
t₂, X₁: 1000⋅X₄+X₅ {O(n)}
t₂, X₂: 2⋅X₅+2000⋅X₄+X₂ {O(n)}
t₂, X₃: 2⋅X₄+X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: 2⋅X₄ {O(n)}
t₃, X₁: 1000⋅X₄+2⋅X₅ {O(n)}
t₃, X₂: 2⋅X₅+2000⋅X₄+X₂ {O(n)}
t₃, X₃: 2⋅X₄+X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₄, X₀: X₄ {O(n)}
t₄, X₁: 1000⋅X₄+X₅ {O(n)}
t₄, X₂: 1000⋅X₄+X₅ {O(n)}
t₄, X₃: X₄ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: X₄ {O(n)}
t₅, X₁: 1000⋅X₄+X₅ {O(n)}
t₅, X₂: 1000⋅X₄+X₅ {O(n)}
t₅, X₃: X₄ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₆, X₀: X₄ {O(n)}
t₆, X₁: 1000⋅X₄+X₅ {O(n)}
t₆, X₂: 2⋅X₅+2000⋅X₄ {O(n)}
t₆, X₃: 2⋅X₄ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₇, X₀: X₄ {O(n)}
t₇, X₁: 1000⋅X₄+X₅ {O(n)}
t₇, X₂: 1000⋅X₄+X₅ {O(n)}
t₇, X₃: X₄ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₀: 2⋅X₄ {O(n)}
t₈, X₁: 1000⋅X₄+2⋅X₅ {O(n)}
t₈, X₂: 2⋅X₅+2000⋅X₄+X₂ {O(n)}
t₈, X₃: 2⋅X₄+X₃ {O(n)}
t₈, X₄: 2⋅X₄ {O(n)}
t₈, X₅: 2⋅X₅ {O(n)}