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(X₂, X₃, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₄
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₄
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁-1, 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 ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

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

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ 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(X₂, X₃, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₄ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₄ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃, X₄) :|: 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₂ ∧ X₁ ≤ X₃
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+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁-X₄]
• eval_foo_bb2_in: [X₁-1-X₄]

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

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁-X₄]
• eval_foo_bb2_in: [X₁-X₄]

All Bounds

Timebounds

Overall timebound:2⋅X₃+2⋅X₄+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+X₄ {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₃+X₄ {O(n)}
t₆: 1 {O(1)}

Costbounds

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