KoAT2 Proof Output

Initial Problem

Start: eval_twn13_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_twn13_bb0_in, eval_twn13_bb1_in, eval_twn13_bb2_in, eval_twn13_bb3_in, eval_twn13_start, eval_twn13_stop
Transitions:
t₁: eval_twn13_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb1_in(X₂, X₃, X₂, X₃, X₄) :|: 1+X₄ ≤ 0
t₂: eval_twn13_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₄
t₃: eval_twn13_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀+(X₁)²
t₄: eval_twn13_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀+(X₁)² ≤ 0
t₅: eval_twn13_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb1_in(X₀+(X₁)²*X₄, X₁-2⋅(X₄)², X₂, X₃, X₄)
t₆: eval_twn13_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_twn13_start(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ for location eval_twn13_bb1_in

Found invariant 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ for location eval_twn13_bb2_in

Problem after Preprocessing

Start: eval_twn13_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_twn13_bb0_in, eval_twn13_bb1_in, eval_twn13_bb2_in, eval_twn13_bb3_in, eval_twn13_start, eval_twn13_stop
Transitions:
t₁: eval_twn13_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb1_in(X₂, X₃, X₂, X₃, X₄) :|: 1+X₄ ≤ 0
t₂: eval_twn13_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₄
t₃: eval_twn13_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀+(X₁)² ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃
t₄: eval_twn13_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀+(X₁)² ≤ 0 ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃
t₅: eval_twn13_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb1_in(X₀+(X₁)²*X₄, X₁-2⋅(X₄)², X₂, X₃, X₄) :|: 1+X₄ ≤ 0 ∧ X₁ ≤ X₃
t₆: eval_twn13_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_twn13_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_twn13_start(X₀, X₁, X₂, X₃, X₄) → eval_twn13_bb0_in(X₀, X₁, X₂, X₃, X₄)

TWN: t₅: eval_twn13_bb2_in→eval_twn13_bb1_in

cycle: [t₅: eval_twn13_bb2_in→eval_twn13_bb1_in; t₃: eval_twn13_bb1_in→eval_twn13_bb2_in]
loop: (1 ≤ X₀+(X₁)² ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃,(X₀,X₁,X₃,X₄) -> (X₀+(X₁)²*X₄,X₁-2⋅(X₄)²,X₃,X₄))
order: [X₄; X₁; X₃; X₀]
closed-form:

X₄: X₄
X₁: X₁ + [[n != 0]]⋅-2⋅(X₄)²⋅n^1
X₃: X₃
X₀: X₀ + [[n != 0]]⋅(X₁)²*X₄⋅n^1 + [[n != 0, n != 1]]⋅4/3⋅(X₄)⁵⋅n^3 + [[n != 0, n != 1]]⋅(-2⋅X₁*(X₄)³-2⋅(X₄)⁵)⋅n^2 + [[n != 0, n != 1]]⋅(2⋅X₁*(X₄)³+2/3⋅(X₄)⁵)⋅n^1

Termination: true
Formula:

X₀+(X₁)² ≤ 1 ∧ 1 ≤ X₀+(X₁)² ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ ∧ 6⋅X₁*(X₄)³+3⋅(X₁)²*X₄+2⋅(X₄)⁵ ≤ 12⋅X₁*(X₄)² ∧ 12⋅X₁*(X₄)² ≤ 6⋅X₁*(X₄)³+3⋅(X₁)²*X₄+2⋅(X₄)⁵ ∧ 2⋅(X₄)⁴ ≤ X₁*(X₄)³+(X₄)⁵ ∧ X₁*(X₄)³+(X₄)⁵ ≤ 2⋅(X₄)⁴ ∧ 0 ≤ (X₄)⁵ ∧ (X₄)⁵ ≤ 0
∨ 1+12⋅X₁*(X₄)² ≤ 6⋅X₁*(X₄)³+3⋅(X₁)²*X₄+2⋅(X₄)⁵ ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ ∧ 2⋅(X₄)⁴ ≤ X₁*(X₄)³+(X₄)⁵ ∧ X₁*(X₄)³+(X₄)⁵ ≤ 2⋅(X₄)⁴ ∧ 0 ≤ (X₄)⁵ ∧ (X₄)⁵ ≤ 0
∨ 1+6⋅X₁*(X₄)³+6⋅(X₄)⁵ ≤ 12⋅(X₄)⁴ ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ ∧ 0 ≤ (X₄)⁵ ∧ (X₄)⁵ ≤ 0
∨ 1+X₄ ≤ 0 ∧ 1 ≤ 4⋅(X₄)⁵ ∧ X₁ ≤ X₃
∨ 1+X₄ ≤ 0 ∧ 4 ≤ 3⋅X₀+3⋅(X₁)² ∧ X₁ ≤ X₃ ∧ 6⋅X₁*(X₄)³+3⋅(X₁)²*X₄+2⋅(X₄)⁵ ≤ 12⋅X₁*(X₄)² ∧ 12⋅X₁*(X₄)² ≤ 6⋅X₁*(X₄)³+3⋅(X₁)²*X₄+2⋅(X₄)⁵ ∧ 2⋅(X₄)⁴ ≤ X₁*(X₄)³+(X₄)⁵ ∧ X₁*(X₄)³+(X₄)⁵ ≤ 2⋅(X₄)⁴ ∧ 0 ≤ (X₄)⁵ ∧ (X₄)⁵ ≤ 0

Stabilization-Threshold for: 1 ≤ X₀+(X₁)²

alphas_abs: 3⋅X₀+12⋅X₁*(X₄)²+6⋅X₁*(X₄)³+3⋅(X₁)²+3⋅(X₁)²*X₄+12⋅(X₄)⁴+6⋅(X₄)⁵
M: 0
N: 3
Bound: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₁⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₁⋅X₄⋅X₄+6⋅X₁⋅X₁⋅X₄+6⋅X₁⋅X₁+6⋅X₀+4 {O(n^5)}

TWN - Lifting for [3: eval_twn13_bb1_in->eval_twn13_bb2_in; 5: eval_twn13_bb2_in->eval_twn13_bb1_in] of 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₁⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₁⋅X₄⋅X₄+6⋅X₁⋅X₁⋅X₄+6⋅X₁⋅X₁+6⋅X₀+6 {O(n^5)}

relevant size-bounds w.r.t. t₁: eval_twn13_bb0_in→eval_twn13_bb1_in:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

Found invariant 1+X₄ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location eval_twn13_bb1_in

Found invariant 1+X₄ ≤ 0 ∧ 2+X₁ ≤ X₃ for location eval_twn13_bb1_in_v1

Found invariant 1+X₄ ≤ 0 ∧ 2+X₁ ≤ X₃ for location eval_twn13_bb2_in_v2

Found invariant 1+X₄ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location eval_twn13_bb2_in_v1

All Bounds

Timebounds

Overall timebound:24⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄⋅X₄+48⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₃⋅X₄+48⋅X₃⋅X₄⋅X₄+12⋅X₃⋅X₃+12⋅X₂+17 {O(n^5)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}
t₄: 1 {O(1)}
t₅: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 24⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄⋅X₄+48⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₃⋅X₄+48⋅X₃⋅X₄⋅X₄+12⋅X₃⋅X₃+12⋅X₂+17 {O(n^5)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}
t₄: 1 {O(1)}
t₅: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}
t₆: 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: X₂ {O(n)}
t₁, X₁: X₃ {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, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}
t₆, X₄: 3⋅X₄ {O(n)}