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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₅, 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) :|: 1+X₀ ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₀, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, X₁, X₂, 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 ≤ X₃
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ 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_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 [X₂] that do not contribute to the problem

Found invariant X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₁ for location eval_foo_bb5_in

Found invariant 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1+X₀ ≤ X₃ for location eval_foo_bb2_in

Found invariant X₄ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₁ for location eval_foo_stop

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

Found invariant 1+X₄ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₀ ≤ 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_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₇: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₄, X₁, X₂, X₃, X₄)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₃ ∧ X₄ ≤ X₀
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₀, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₀
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀
t₂₁: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₁
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₁
t₂₃: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₁
t₂₄: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₁
t₂₅: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₃ ∧ X₄ ≤ X₀ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

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

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

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

MPRF for transition t₂₁: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₁ of depth 1:

new bound:

2⋅X₂+3⋅X₄+X₃ {O(n)}

MPRF:

• eval_foo_bb3_in: [X₂-X₁]
• eval_foo_bb4_in: [X₂-1-X₁]

MPRF for transition t₂₃: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, 1+X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₁ of depth 1:

new bound:

2⋅X₂+3⋅X₄+X₃ {O(n)}

MPRF:

• eval_foo_bb3_in: [X₂-X₁]
• eval_foo_bb4_in: [X₂-X₁]

All Bounds

Timebounds

Overall timebound:4⋅X₂+4⋅X₃+8⋅X₄+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₃+X₄ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₃+X₄ {O(n)}
t₂₁: 2⋅X₂+3⋅X₄+X₃ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 2⋅X₂+3⋅X₄+X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂+4⋅X₃+8⋅X₄+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₃+X₄ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₃+X₄ {O(n)}
t₂₁: 2⋅X₂+3⋅X₄+X₃ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 2⋅X₂+3⋅X₄+X₃ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₄ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₈, X₀: 2⋅X₄+X₃ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: 3⋅X₄+X₃ {O(n)}
t₁₉, X₁: 3⋅X₄+X₃ {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₁₉, X₄: 2⋅X₄ {O(n)}
t₂₀, X₀: 2⋅X₄+X₃ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₁, X₀: 3⋅X₄+X₃ {O(n)}
t₂₁, X₁: 2⋅X₂+2⋅X₃+6⋅X₄ {O(n)}
t₂₁, X₂: 2⋅X₂ {O(n)}
t₂₁, X₃: 2⋅X₃ {O(n)}
t₂₁, X₄: 2⋅X₄ {O(n)}
t₂₂, X₀: 2⋅X₃+6⋅X₄ {O(n)}
t₂₂, X₁: 2⋅X₂+3⋅X₃+9⋅X₄ {O(n)}
t₂₂, X₂: 4⋅X₂ {O(n)}
t₂₂, X₃: 4⋅X₃ {O(n)}
t₂₂, X₄: 4⋅X₄ {O(n)}
t₂₃, X₀: 3⋅X₄+X₃ {O(n)}
t₂₃, X₁: 2⋅X₂+2⋅X₃+6⋅X₄ {O(n)}
t₂₃, X₂: 2⋅X₂ {O(n)}
t₂₃, X₃: 2⋅X₃ {O(n)}
t₂₃, X₄: 2⋅X₄ {O(n)}
t₂₄, X₀: 2⋅X₃+6⋅X₄ {O(n)}
t₂₄, X₁: 2⋅X₂+3⋅X₃+9⋅X₄ {O(n)}
t₂₄, X₂: 4⋅X₂ {O(n)}
t₂₄, X₃: 4⋅X₃ {O(n)}
t₂₄, X₄: 4⋅X₄ {O(n)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₅, X₄: X₄ {O(n)}