KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_peed_pldi09_fig4_2_bb2_in

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ for location eval_peed_pldi09_fig4_2_bb1_in

Found invariant 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_peed_pldi09_fig4_2_bb3_in

Problem after Preprocessing

Start: eval_peed_pldi09_fig4_2_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_peed_pldi09_fig4_2_bb0_in, eval_peed_pldi09_fig4_2_bb1_in, eval_peed_pldi09_fig4_2_bb2_in, eval_peed_pldi09_fig4_2_bb3_in, eval_peed_pldi09_fig4_2_bb4_in, eval_peed_pldi09_fig4_2_start, eval_peed_pldi09_fig4_2_stop
Transitions:
t₂: eval_peed_pldi09_fig4_2_bb0_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₁, 0) :|: 1 ≤ X₀
t₁: eval_peed_pldi09_fig4_2_bb0_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₃: eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₄: eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb4_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₆: eval_peed_pldi09_fig4_2_bb2_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₂, 0) :|: X₀ ≤ X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₅: eval_peed_pldi09_fig4_2_bb2_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₇: eval_peed_pldi09_fig4_2_bb3_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₂-1, 1+X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₈: eval_peed_pldi09_fig4_2_bb4_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_stop(X₀, X₁, X₂, X₃)
t₀: eval_peed_pldi09_fig4_2_start(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₃: eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• eval_peed_pldi09_fig4_2_bb1_in: [2⋅X₂+X₃-1]
• eval_peed_pldi09_fig4_2_bb2_in: [2⋅X₂+X₃-2]
• eval_peed_pldi09_fig4_2_bb3_in: [2⋅X₂+X₃-2]

MPRF for transition t₅: eval_peed_pldi09_fig4_2_bb2_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• eval_peed_pldi09_fig4_2_bb1_in: [X₁+X₂-1]
• eval_peed_pldi09_fig4_2_bb2_in: [X₁+X₂-1]
• eval_peed_pldi09_fig4_2_bb3_in: [X₁+X₂-2]

MPRF for transition t₆: eval_peed_pldi09_fig4_2_bb2_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₂, 0) :|: X₀ ≤ X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_peed_pldi09_fig4_2_bb1_in: [X₂+X₃]
• eval_peed_pldi09_fig4_2_bb2_in: [X₂+X₃]
• eval_peed_pldi09_fig4_2_bb3_in: [X₂+X₃]

MPRF for transition t₇: eval_peed_pldi09_fig4_2_bb3_in(X₀, X₁, X₂, X₃) → eval_peed_pldi09_fig4_2_bb1_in(X₀, X₁, X₂-1, 1+X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

• eval_peed_pldi09_fig4_2_bb1_in: [X₀+X₂-1]
• eval_peed_pldi09_fig4_2_bb2_in: [X₀+X₂-1]
• eval_peed_pldi09_fig4_2_bb3_in: [X₀+X₂-1]

All Bounds

Timebounds

Overall timebound:6⋅X₁+X₀+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 2⋅X₁+1 {O(n)}
t₄: 1 {O(1)}
t₅: 2⋅X₁+1 {O(n)}
t₆: X₁ {O(n)}
t₇: X₀+X₁+1 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₁+X₀+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 2⋅X₁+1 {O(n)}
t₄: 1 {O(1)}
t₅: 2⋅X₁+1 {O(n)}
t₆: X₁ {O(n)}
t₇: X₀+X₁+1 {O(n)}
t₈: 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: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁ {O(n)}
t₂, X₃: 0 {O(1)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₁ {O(n)}
t₃, X₃: 2⋅X₁+2 {O(n)}
t₄, X₀: 2⋅X₀ {O(n)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: 2⋅X₁ {O(n)}
t₄, X₃: 2⋅X₁+2 {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₁ {O(n)}
t₅, X₃: 2⋅X₁+2 {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₁ {O(n)}
t₆, X₃: 0 {O(1)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁ {O(n)}
t₇, X₃: 2⋅X₁+2 {O(n)}
t₈, X₀: 3⋅X₀ {O(n)}
t₈, X₁: 3⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁+X₂ {O(n)}
t₈, X₃: 2⋅X₁+X₃+2 {O(n)}