KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 1 ∧ 0 ≤ X₂ for location eval_start_bb1_in

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location eval_start_stop

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ for location eval_start_bb2_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_bb3_in

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location eval_start_bb4_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_bb4_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃, X₄) → eval_start_1(X₀, X₁, X₂, X₃, X₄)
t₃: eval_start_1(X₀, X₁, X₂, X₃, X₄) → eval_start_2(X₀, X₁, X₂, X₃, X₄)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄) → eval_start_3(X₀, X₁, X₂, X₃, X₄)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄) → eval_start_4(X₀, X₁, X₂, X₃, X₄)
t₆: eval_start_4(X₀, X₁, X₂, X₃, X₄) → eval_start_5(X₀, X₁, X₂, X₃, X₄)
t₇: eval_start_5(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₄, X₁, 1, X₃, X₄)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_start_0(X₀, X₁, X₂, X₃, X₄)
t₈: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₀, X₂, 0, X₄) :|: 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂
t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂
t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₁, X₁, X₃, X₃, X₄) :|: X₁ ≤ 0 ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₁₀: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₁₂: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁-1, X₂, 1, X₄) :|: X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₄ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄
t₁₃: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₁₀: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₁]
• eval_start_bb3_in: [X₁-1]

MPRF for transition t₁₂: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁-1, X₂, 1, X₄) :|: X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₄ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₁]
• eval_start_bb3_in: [X₁]

TWN: t₁₁: eval_start_bb2_in→eval_start_bb1_in

cycle: [t₁₁: eval_start_bb2_in→eval_start_bb1_in; t₈: eval_start_bb1_in→eval_start_bb2_in]
original loop: (1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₂ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1 ∧ 0 ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ 0 ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ 0 ≤ 0,(X₀,X₂,X₄) -> (X₀,0,X₄))
transformed loop: (1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₂ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1 ∧ 0 ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ 0 ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ 0 ≤ 0,(X₀,X₂,X₄) -> (X₀,0,X₄))
loop: (1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₂ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1 ∧ 0 ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ 0 ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ 0 ≤ 0,(X₀,X₂,X₄) -> (X₀,0,X₄))
order: [X₄; X₂; X₀]
closed-form:

X₄: X₄
X₂: [[n == 0]]⋅X₂
X₀: X₀

Termination: true
Formula:

0 ≤ 0 ∧ X₂ ≤ 2 ∧ 0 ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂

original loop: (X₁ ≤ 0 ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃,(X₀,X₁,X₂,X₃,X₄) -> (X₁,X₁,X₃,0,X₄))
transformed loop: (X₁ ≤ 0 ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃,(X₀,X₁,X₂,X₃,X₄) -> (X₁,X₁,X₃,0,X₄))
loop: (X₁ ≤ 0 ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃,(X₀,X₁,X₂,X₃,X₄) -> (X₁,X₁,X₃,0,X₄))
order: [X₃; X₁; X₄; X₂; X₀]
closed-form:

X₃: [[n == 0]]⋅X₃
X₁: X₁
X₄: X₄
X₂: [[n == 0]]⋅X₂ + [[n != 0, n == 1]]⋅X₃
X₀: [[n == 0]]⋅X₀ + [[n != 0]]⋅X₁

Termination: true
Formula:

0 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃
∨ 0 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 1 ∧ 1 ≤ 0 ∧ X₁ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ X₃ ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ X₃ ≤ 1 ∧ 1 ≤ 0 ∧ X₁ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₃

TWN - Lifting for [8: eval_start_bb1_in->eval_start_bb2_in; 11: eval_start_bb2_in->eval_start_bb1_in] of 2 {O(1)}

relevant size-bounds w.r.t. t₇: eval_start_5→eval_start_bb1_in:
Runtime-bound of t₇: 1 {O(1)}
Results in: 2 {O(1)}

TWN - Lifting for [8: eval_start_bb1_in->eval_start_bb2_in; 11: eval_start_bb2_in->eval_start_bb1_in] of 7 {O(1)}

relevant size-bounds w.r.t. t₁₂: eval_start_bb3_in→eval_start_bb2_in:
Runtime-bound of t₁₂: X₄ {O(n)}
Results in: 7⋅X₄ {O(n)}

All Bounds

Timebounds

Overall timebound:16⋅X₄+14 {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: 7⋅X₄+2 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₄ {O(n)}
t₁₁: 7⋅X₄+2 {O(n)}
t₁₂: X₄ {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₄+14 {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: 7⋅X₄+2 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₄ {O(n)}
t₁₁: 7⋅X₄+2 {O(n)}
t₁₂: X₄ {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)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₇, X₀: X₄ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 1 {O(1)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₈, X₀: 2⋅X₄ {O(n)}
t₈, X₁: X₄ {O(n)}
t₈, X₂: 1 {O(1)}
t₈, X₃: 0 {O(1)}
t₈, X₄: X₄ {O(n)}
t₉, X₀: X₄ {O(n)}
t₉, X₁: 2⋅X₄ {O(n)}
t₉, X₂: 0 {O(1)}
t₉, X₃: 1 {O(1)}
t₉, X₄: X₄ {O(n)}
t₁₀, X₀: 2⋅X₄ {O(n)}
t₁₀, X₁: X₄ {O(n)}
t₁₀, X₂: 1 {O(1)}
t₁₀, X₃: 1 {O(1)}
t₁₀, X₄: X₄ {O(n)}
t₁₁, X₀: X₄ {O(n)}
t₁₁, X₁: 2⋅X₄ {O(n)}
t₁₁, X₂: 1 {O(1)}
t₁₁, X₃: 1 {O(1)}
t₁₁, X₄: X₄ {O(n)}
t₁₂, X₀: 2⋅X₄ {O(n)}
t₁₂, X₁: X₄ {O(n)}
t₁₂, X₂: 1 {O(1)}
t₁₂, X₃: 1 {O(1)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₀: X₄ {O(n)}
t₁₃, X₁: 2⋅X₄ {O(n)}
t₁₃, X₂: 0 {O(1)}
t₁₃, X₃: 1 {O(1)}
t₁₃, X₄: X₄ {O(n)}