KoAT2 Proof Output

Initial Problem

Start: eval_peed_pldi09_fig4_4_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_peed_pldi09_fig4_4_bb0_in, eval_peed_pldi09_fig4_4_bb1_in, eval_peed_pldi09_fig4_4_bb2_in, eval_peed_pldi09_fig4_4_bb3_in, eval_peed_pldi09_fig4_4_start, eval_peed_pldi09_fig4_4_stop
Transitions:
t₂: eval_peed_pldi09_fig4_4_bb0_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₂, X₁, X₂) :|: 1 ≤ X₁
t₁: eval_peed_pldi09_fig4_4_bb0_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 0
t₃: eval_peed_pldi09_fig4_4_bb1_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀
t₄: eval_peed_pldi09_fig4_4_bb1_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb3_in(X₀, X₁, X₂) :|: X₀ ≤ 0
t₅: eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₀-1, X₁, X₂) :|: 1+X₀ ≤ X₁
t₆: eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₀-X₁, X₁, X₂) :|: X₁ ≤ X₀
t₇: eval_peed_pldi09_fig4_4_bb3_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_stop(X₀, X₁, X₂)
t₀: eval_peed_pldi09_fig4_4_start(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb0_in(X₀, X₁, X₂)

Preprocessing

Found invariant X₀ ≤ X₂ ∧ 1 ≤ X₁ for location eval_peed_pldi09_fig4_4_bb1_in

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_peed_pldi09_fig4_4_bb2_in

Problem after Preprocessing

Start: eval_peed_pldi09_fig4_4_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_peed_pldi09_fig4_4_bb0_in, eval_peed_pldi09_fig4_4_bb1_in, eval_peed_pldi09_fig4_4_bb2_in, eval_peed_pldi09_fig4_4_bb3_in, eval_peed_pldi09_fig4_4_start, eval_peed_pldi09_fig4_4_stop
Transitions:
t₂: eval_peed_pldi09_fig4_4_bb0_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₂, X₁, X₂) :|: 1 ≤ X₁
t₁: eval_peed_pldi09_fig4_4_bb0_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 0
t₃: eval_peed_pldi09_fig4_4_bb1_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂
t₄: eval_peed_pldi09_fig4_4_bb1_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb3_in(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂
t₅: eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₀-1, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₂
t₆: eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₀-X₁, X₁, X₂) :|: X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₂
t₇: eval_peed_pldi09_fig4_4_bb3_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_stop(X₀, X₁, X₂)
t₀: eval_peed_pldi09_fig4_4_start(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb0_in(X₀, X₁, X₂)

MPRF for transition t₃: eval_peed_pldi09_fig4_4_bb1_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_peed_pldi09_fig4_4_bb1_in: [X₀]
• eval_peed_pldi09_fig4_4_bb2_in: [X₀-1]

MPRF for transition t₅: eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₀-1, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_peed_pldi09_fig4_4_bb1_in: [X₀]
• eval_peed_pldi09_fig4_4_bb2_in: [X₀]

MPRF for transition t₆: eval_peed_pldi09_fig4_4_bb2_in(X₀, X₁, X₂) → eval_peed_pldi09_fig4_4_bb1_in(X₀-X₁, X₁, X₂) :|: X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_peed_pldi09_fig4_4_bb1_in: [X₀]
• eval_peed_pldi09_fig4_4_bb2_in: [X₀]

All Bounds

Timebounds

Overall timebound:3⋅X₂+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₂ {O(n)}
t₄: 1 {O(1)}
t₅: X₂ {O(n)}
t₆: X₂ {O(n)}
t₇: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₂ {O(n)}
t₄: 1 {O(1)}
t₅: X₂ {O(n)}
t₆: X₂ {O(n)}
t₇: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: 3⋅X₂ {O(n)}
t₄, X₁: 3⋅X₁ {O(n)}
t₄, X₂: 3⋅X₂ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₂ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₇, X₀: 3⋅X₂+X₀ {O(n)}
t₇, X₁: 4⋅X₁ {O(n)}
t₇, X₂: 4⋅X₂ {O(n)}