KoAT2 Proof Output

Initial Problem

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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄)
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant 1+X₄ ≤ X₂ ∧ X₄ ≤ X₁ ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ for location eval_foo_bb2_in

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb1_in

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

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb3_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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

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

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₂-X₀]
• eval_foo_bb2_in: [X₂-1-X₀]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₂-X₀]
• eval_foo_bb2_in: [X₂-X₀]

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₂+X₃ {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₂+X₃ {O(n)}
t₆: 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)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₃+X₂ {O(n)}
t₂, X₁: 2⋅X₄+X₂+1 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 3⋅X₃+X₂ {O(n)}
t₃, X₁: 3⋅X₄+X₂+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 3⋅X₃+X₂ {O(n)}
t₄, X₁: 3⋅X₄+X₂+1 {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₀: 2⋅X₃+X₂ {O(n)}
t₅, X₁: 2⋅X₄+X₂+1 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 2⋅X₂+6⋅X₃ {O(n)}
t₆, X₁: 2⋅X₂+6⋅X₄+2 {O(n)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₆, X₄: 4⋅X₄ {O(n)}