KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_6, eval_start_7, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃) → eval_start_1(X₀, X₁, X₂, X₃)
t₃: eval_start_1(X₀, X₁, X₂, X₃) → eval_start_2(X₀, X₁, X₂, X₃)
t₄: eval_start_2(X₀, X₁, X₂, X₃) → eval_start_3(X₀, X₁, X₂, X₃)
t₅: eval_start_3(X₀, X₁, X₂, X₃) → eval_start_4(X₀, X₁, X₂, X₃)
t₆: eval_start_4(X₀, X₁, X₂, X₃) → eval_start_5(X₀, X₁, X₂, X₃)
t₇: eval_start_5(X₀, X₁, X₂, X₃) → eval_start_6(X₀, X₁, X₂, X₃)
t₈: eval_start_6(X₀, X₁, X₂, X₃) → eval_start_7(X₀, X₁, X₂, X₃)
t₉: eval_start_7(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₂, X₃, X₂, X₃)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃) → eval_start_0(X₀, X₁, X₂, X₃)
t₁₀: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀
t₁₁: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁₂: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₁, X₀-1, X₂, X₃)
t₁₃: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃)
t₀: eval_start_start(X₀, X₁, X₂, X₃) → eval_start_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₀ ≤ 0 for location eval_start_stop

Found invariant 1 ≤ X₀ for location eval_start_bb2_in

Found invariant X₀ ≤ 0 for location eval_start_bb3_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_6, eval_start_7, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃) → eval_start_1(X₀, X₁, X₂, X₃)
t₃: eval_start_1(X₀, X₁, X₂, X₃) → eval_start_2(X₀, X₁, X₂, X₃)
t₄: eval_start_2(X₀, X₁, X₂, X₃) → eval_start_3(X₀, X₁, X₂, X₃)
t₅: eval_start_3(X₀, X₁, X₂, X₃) → eval_start_4(X₀, X₁, X₂, X₃)
t₆: eval_start_4(X₀, X₁, X₂, X₃) → eval_start_5(X₀, X₁, X₂, X₃)
t₇: eval_start_5(X₀, X₁, X₂, X₃) → eval_start_6(X₀, X₁, X₂, X₃)
t₈: eval_start_6(X₀, X₁, X₂, X₃) → eval_start_7(X₀, X₁, X₂, X₃)
t₉: eval_start_7(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₂, X₃, X₂, X₃)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃) → eval_start_0(X₀, X₁, X₂, X₃)
t₁₀: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀
t₁₁: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁₂: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₁, X₀-1, X₂, X₃) :|: 1 ≤ X₀
t₁₃: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₀: eval_start_start(X₀, X₁, X₂, X₃) → eval_start_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₁₀: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ of depth 2:

new bound:

16⋅X₂+8⋅X₃+7 {O(n)}

MPRF:

• eval_start_bb1_in: [X₀+X₁-1; X₀]
• eval_start_bb2_in: [X₀+X₁-2; X₁]

MPRF for transition t₁₂: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₁, X₀-1, X₂, X₃) :|: 1 ≤ X₀ of depth 2:

new bound:

16⋅X₂+8⋅X₃+15 {O(n)}

MPRF:

• eval_start_bb1_in: [X₀+X₁-1; X₀-1]
• eval_start_bb2_in: [X₀+X₁-1; 0]

All Bounds

Timebounds

Overall timebound:16⋅X₃+32⋅X₂+34 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: 1 {O(1)}
t₁₀: 16⋅X₂+8⋅X₃+7 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 16⋅X₂+8⋅X₃+15 {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₃+32⋅X₂+34 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: 1 {O(1)}
t₁₀: 16⋅X₂+8⋅X₃+7 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 16⋅X₂+8⋅X₃+15 {O(n)}
t₁₃: 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: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₀: X₂ {O(n)}
t₉, X₁: X₃ {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₃ {O(n)}
t₁₀, X₀: X₂+X₃ {O(n)}
t₁₀, X₁: X₂+X₃ {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₁, X₀: 2⋅X₂+X₃ {O(n)}
t₁₁, X₁: 2⋅X₃+X₂ {O(n)}
t₁₁, X₂: 2⋅X₂ {O(n)}
t₁₁, X₃: 2⋅X₃ {O(n)}
t₁₂, X₀: X₂+X₃ {O(n)}
t₁₂, X₁: X₂+X₃ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: 2⋅X₂+X₃ {O(n)}
t₁₃, X₁: 2⋅X₃+X₂ {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}
t₁₃, X₃: 2⋅X₃ {O(n)}