KoAT2 Proof Output

Initial Problem

Start: eval_gcd_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_gcd_bb0_in, eval_gcd_bb1_in, eval_gcd_bb2_in, eval_gcd_bb3_in, eval_gcd_start, eval_gcd_stop
Transitions:
t₁: eval_gcd_bb0_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_gcd_bb1_in(X₀, X₁, X₂, X₃) → eval_gcd_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁
t₃: eval_gcd_bb1_in(X₀, X₁, X₂, X₃) → eval_gcd_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₄: eval_gcd_bb1_in(X₀, X₁, X₂, X₃) → eval_gcd_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0
t₅: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀-X₁, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₆: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₇: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀-X₁, X₁-X₀, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₈: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀, X₁-X₀, X₂, X₃) :|: X₀ ≤ X₁
t₉: eval_gcd_bb3_in(X₀, X₁, X₂, X₃) → eval_gcd_stop(X₀, X₁, X₂, X₃)
t₀: eval_gcd_start(X₀, X₁, X₂, X₃) → eval_gcd_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Cut unsatisfiable transition [t₆: eval_gcd_bb2_in→eval_gcd_bb1_in; t₇: eval_gcd_bb2_in→eval_gcd_bb1_in]

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_gcd_bb2_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_gcd_bb1_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_gcd_bb3_in

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

Problem after Preprocessing

Start: eval_gcd_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_gcd_bb0_in, eval_gcd_bb1_in, eval_gcd_bb2_in, eval_gcd_bb3_in, eval_gcd_start, eval_gcd_stop
Transitions:
t₁: eval_gcd_bb0_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_gcd_bb1_in(X₀, X₁, X₂, X₃) → eval_gcd_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₃: eval_gcd_bb1_in(X₀, X₁, X₂, X₃) → eval_gcd_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₄: eval_gcd_bb1_in(X₀, X₁, X₂, X₃) → eval_gcd_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₅: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀-X₁, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₈: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀, X₁-X₀, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₉: eval_gcd_bb3_in(X₀, X₁, X₂, X₃) → eval_gcd_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₀: eval_gcd_start(X₀, X₁, X₂, X₃) → eval_gcd_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_gcd_bb1_in(X₀, X₁, X₂, X₃) → eval_gcd_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_gcd_bb1_in: [X₀+X₁-1]
• eval_gcd_bb2_in: [X₀+X₁-2]

MPRF for transition t₅: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀-X₁, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_gcd_bb1_in: [X₀+X₁-1]
• eval_gcd_bb2_in: [X₀+X₁-1]

MPRF for transition t₈: eval_gcd_bb2_in(X₀, X₁, X₂, X₃) → eval_gcd_bb1_in(X₀, X₁-X₀, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_gcd_bb1_in: [X₁]
• eval_gcd_bb2_in: [X₁]

All Bounds

Timebounds

Overall timebound:2⋅X₂+3⋅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)}
t₈: X₃ {O(n)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+3⋅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)}
t₈: X₃ {O(n)}
t₉: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: X₃ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: 2⋅X₂ {O(n)}
t₄, X₁: 2⋅X₃ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₃ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: X₃ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₀: 3⋅X₂ {O(n)}
t₉, X₁: 3⋅X₃ {O(n)}
t₉, X₂: 3⋅X₂ {O(n)}
t₉, X₃: 3⋅X₃ {O(n)}