KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
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, X₅, X₆) → eval_foo_bb1_in(X₆, 73, 74, X₃, X₄, X₅, X₆)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 0 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₀-X₁, 1+X₂, 1+X₁, X₃, X₄, X₅, X₆)
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Eliminate variables [X₃; X₄; X₅] that do not contribute to the problem

Found invariant 0 ≤ X₃ ∧ 74 ≤ X₂+X₃ ∧ 73 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ 74 ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 74 ≤ X₀+X₂ ∧ 73 ≤ X₁ ∧ 73 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ 74 ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 73 ≤ X₁ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ 74 ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 75+X₀ ≤ X₂ ∧ 73 ≤ X₁ ∧ 74+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 1+X₁ ∧ 74 ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 75+X₀ ≤ X₂ ∧ 73 ≤ X₁ ∧ 74+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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:
t₁₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, 73, 74, X₃)
t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₂ ≤ 1+X₁ ∧ X₁ ≤ 1+X₂ ∧ 73 ≤ X₁ ∧ 74 ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₀ ≤ X₃
t₁₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₂ ≤ 1+X₁ ∧ X₁ ≤ 1+X₂ ∧ 73 ≤ X₁ ∧ 74 ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₀ ≤ X₃
t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-X₁, 1+X₂, 1+X₁, X₃) :|: X₂ ≤ 1+X₁ ∧ X₁ ≤ 1+X₂ ∧ 73 ≤ X₀+X₁ ∧ 73 ≤ X₁ ∧ 73 ≤ X₁+X₃ ∧ 74 ≤ X₀+X₂ ∧ 74 ≤ X₂ ∧ 74 ≤ X₂+X₃ ∧ 147 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃
t₁₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1+X₁ ∧ X₁ ≤ 1+X₂ ∧ 1+X₀ ≤ 0 ∧ 73 ≤ X₁ ∧ 74+X₀ ≤ X₁ ∧ 74 ≤ X₂ ∧ 75+X₀ ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₀ ≤ X₃
t₁₆: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₂ ≤ 1+X₁ ∧ X₁ ≤ 1+X₂ ∧ 73 ≤ X₁ ∧ 74 ≤ X₂ ∧ 147 ≤ X₁+X₂ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

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

MPRF for transition t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-X₁, 1+X₂, 1+X₁, X₃) :|: X₂ ≤ 1+X₁ ∧ X₁ ≤ 1+X₂ ∧ 73 ≤ X₀+X₁ ∧ 73 ≤ X₁ ∧ 73 ≤ X₁+X₃ ∧ 74 ≤ X₀+X₂ ∧ 74 ≤ X₂ ∧ 74 ≤ X₂+X₃ ∧ 147 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+3 {O(n)}

MPRF:

• eval_foo_bb1_in: [3+X₀]
• eval_foo_bb2_in: [2+X₀+X₂-X₁]

All Bounds

Timebounds

Overall timebound:2⋅X₃+8 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₃+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃+3 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃+8 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₃+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃+3 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₃ {O(n)}
t₁₁, X₁: 73 {O(1)}
t₁₁, X₂: 74 {O(1)}
t₁₁, X₃: X₃ {O(n)}
t₁₂, X₀: 2⋅X₃+296 {O(n)}
t₁₂, X₁: X₃+296 {O(n)}
t₁₂, X₂: X₃+296 {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: 3⋅X₃+296 {O(n)}
t₁₃, X₁: X₃+5551 {O(n)}
t₁₃, X₂: X₃+371 {O(n)}
t₁₃, X₃: 2⋅X₃ {O(n)}
t₁₄, X₀: 2⋅X₃+296 {O(n)}
t₁₄, X₁: X₃+5478 {O(n)}
t₁₄, X₂: X₃+297 {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₅, X₀: 3⋅X₃+296 {O(n)}
t₁₅, X₁: X₃+5551 {O(n)}
t₁₅, X₂: X₃+371 {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}