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(X₀, -X₄, -X₅, X₃, X₄, X₅) :|: 1+X₄ ≤ 0 ∧ 1+X₅ ≤ 0
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, -X₄, X₅, X₃, X₄, X₅) :|: 1+X₄ ≤ 0 ∧ 0 ≤ X₅
t₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, X₄, -X₅, X₃, X₄, X₅) :|: 1+X₅ ≤ 0 ∧ 0 ≤ X₄
t₄: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, X₄, X₅, X₃, X₄, X₅) :|: 0 ≤ X₄ ∧ 0 ≤ X₅
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₁, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂
t₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ 0
t₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, X₂, X₀, X₃, X₄, X₅) :|: 1+X₀ ≤ X₂
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₀
t₉: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀-X₂, X₁, X₂, X₃, X₄, X₅)
t₁₀: 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 [X₃] that do not contribute to the problem

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

Found invariant 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_foo_bb1_in

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_foo_stop

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

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_foo_bb4_in

Problem after Preprocessing

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_start, eval_foo_stop
Transitions:
t₂₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, -X₃, -X₄, X₃, X₄) :|: 1+X₃ ≤ 0 ∧ 1+X₄ ≤ 0
t₂₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, -X₃, X₄, X₃, X₄) :|: 1+X₃ ≤ 0 ∧ 0 ≤ X₄
t₂₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₃, -X₄, X₃, X₄) :|: 1+X₄ ≤ 0 ∧ 0 ≤ X₃
t₂₄: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₃, X₄, X₃, X₄) :|: 0 ≤ X₃ ∧ 0 ≤ X₄
t₂₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₁, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂
t₂₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂
t₂₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₂, X₀, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₂₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₂₉: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀-X₂, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁
t₃₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 0
t₃₁: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₁, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ of depth 1:

new bound:

4⋅X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₂]
• eval_foo_bb2_in: [X₂-1]
• eval_foo_bb3_in: [X₂-1]

MPRF for transition t₂₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₂, X₀, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ of depth 1:

new bound:

8⋅X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [2⋅X₂]
• eval_foo_bb2_in: [2⋅X₂-1]
• eval_foo_bb3_in: [2⋅X₂-1]

MPRF for transition t₂₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ of depth 1:

new bound:

4⋅X₃+4⋅X₄+4 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁+X₂-1]
• eval_foo_bb2_in: [X₀+X₂-1]
• eval_foo_bb3_in: [X₀+X₂-2]

MPRF for transition t₂₉: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀-X₂, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ of depth 1:

new bound:

4⋅X₃+4⋅X₄+4 {O(n)}

MPRF:

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

All Bounds

Timebounds

Overall timebound:20⋅X₄+8⋅X₃+15 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 4⋅X₄ {O(n)}
t₂₆: 1 {O(1)}
t₂₇: 8⋅X₄ {O(n)}
t₂₈: 4⋅X₃+4⋅X₄+4 {O(n)}
t₂₉: 4⋅X₃+4⋅X₄+4 {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Costbounds

Overall costbound: 20⋅X₄+8⋅X₃+15 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 4⋅X₄ {O(n)}
t₂₆: 1 {O(1)}
t₂₇: 8⋅X₄ {O(n)}
t₂₈: 4⋅X₃+4⋅X₄+4 {O(n)}
t₂₉: 4⋅X₃+4⋅X₄+4 {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}

Sizebounds

t₂₁, X₀: X₀ {O(n)}
t₂₁, X₁: X₃ {O(n)}
t₂₁, X₂: X₄ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₂, X₀: X₀ {O(n)}
t₂₂, X₁: X₃ {O(n)}
t₂₂, X₂: X₄ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₃, X₀: X₀ {O(n)}
t₂₃, X₁: X₃ {O(n)}
t₂₃, X₂: X₄ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₄, X₀: X₀ {O(n)}
t₂₄, X₁: X₃ {O(n)}
t₂₄, X₂: X₄ {O(n)}
t₂₄, X₃: X₃ {O(n)}
t₂₄, X₄: X₄ {O(n)}
t₂₅, X₀: 4⋅X₃+4⋅X₄ {O(n)}
t₂₅, X₁: 4⋅X₄+8⋅X₃ {O(n)}
t₂₅, X₂: 4⋅X₃+4⋅X₄ {O(n)}
t₂₅, X₃: 4⋅X₃ {O(n)}
t₂₅, X₄: 4⋅X₄ {O(n)}
t₂₆, X₀: 2⋅X₀+8⋅X₃+8⋅X₄ {O(n)}
t₂₆, X₁: 4⋅X₄+6⋅X₃ {O(n)}
t₂₆, X₂: 4⋅X₃+6⋅X₄ {O(n)}
t₂₆, X₃: 6⋅X₃ {O(n)}
t₂₆, X₄: 6⋅X₄ {O(n)}
t₂₇, X₀: 8⋅X₃+8⋅X₄ {O(n)}
t₂₇, X₁: 4⋅X₃+4⋅X₄ {O(n)}
t₂₇, X₂: 4⋅X₃+4⋅X₄ {O(n)}
t₂₇, X₃: 4⋅X₃ {O(n)}
t₂₇, X₄: 4⋅X₄ {O(n)}
t₂₈, X₀: 4⋅X₃+4⋅X₄ {O(n)}
t₂₈, X₁: 4⋅X₄+8⋅X₃ {O(n)}
t₂₈, X₂: 4⋅X₃+4⋅X₄ {O(n)}
t₂₈, X₃: 4⋅X₃ {O(n)}
t₂₈, X₄: 4⋅X₄ {O(n)}
t₂₉, X₀: 4⋅X₃+4⋅X₄ {O(n)}
t₂₉, X₁: 4⋅X₄+8⋅X₃ {O(n)}
t₂₉, X₂: 4⋅X₃+4⋅X₄ {O(n)}
t₂₉, X₃: 4⋅X₃ {O(n)}
t₂₉, X₄: 4⋅X₄ {O(n)}
t₃₀, X₀: 2⋅X₀+8⋅X₃+8⋅X₄ {O(n)}
t₃₀, X₁: 4⋅X₄+6⋅X₃ {O(n)}
t₃₀, X₂: 4⋅X₃+6⋅X₄ {O(n)}
t₃₀, X₃: 6⋅X₃ {O(n)}
t₃₀, X₄: 6⋅X₄ {O(n)}
t₃₁, X₀: X₀ {O(n)}
t₃₁, X₁: X₁ {O(n)}
t₃₁, X₂: X₂ {O(n)}
t₃₁, X₃: X₃ {O(n)}
t₃₁, X₄: X₄ {O(n)}