KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_foo_.critedge_in, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_start, eval_foo_stop
Transitions:
t₆: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀+X₁ ≤ 0
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀+X₁+X₄, -1-X₄, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_foo_.critedge_in, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_start, eval_foo_stop
Transitions:
t₆: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀+X₁ ≤ 0
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀+X₁+X₄, -1-X₄, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+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) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀+X₁]
• eval_foo_bb2_in: [X₀+X₁]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀+X₁+X₄, -1-X₄, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀+X₁]
• eval_foo_bb2_in: [1+X₀+X₁]

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₂+X₃+1 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₂+X₃+1 {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: 2⋅X₂⋅X₄+2⋅X₃⋅X₄+2⋅X₂+2⋅X₃+5⋅X₄+3 {O(n^2)}
t₂, X₁: X₃+X₄+1 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2⋅X₂⋅X₄+2⋅X₃⋅X₄+2⋅X₃+3⋅X₂+5⋅X₄+3 {O(n^2)}
t₃, X₁: X₃+X₄+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 2⋅X₂⋅X₄+2⋅X₃⋅X₄+2⋅X₃+3⋅X₂+5⋅X₄+3 {O(n^2)}
t₄, X₁: X₃+X₄+1 {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₀: 2⋅X₂⋅X₄+2⋅X₃⋅X₄+2⋅X₂+2⋅X₃+5⋅X₄+3 {O(n^2)}
t₅, X₁: X₄+1 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 4⋅X₂⋅X₄+4⋅X₃⋅X₄+10⋅X₄+4⋅X₃+6⋅X₂+6 {O(n^2)}
t₆, X₁: 2⋅X₃+2⋅X₄+2 {O(n)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₆, X₄: 4⋅X₄ {O(n)}