KoAT2 Proof Output

Initial Problem

Start: eval_size08_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: eval_size08_bb0_in, eval_size08_bb1_in, eval_size08_bb2_in, eval_size08_bb3_in, eval_size08_bb4_in, eval_size08_bb5_in, eval_size08_bb6_in, eval_size08_start, eval_size08_stop
Transitions:
t₁: eval_size08_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅, X₆) :|: 1+X₆ ≤ 0
t₂: eval_size08_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb3_in(X₀, X₁, X₅, X₃, X₄, X₅, X₆) :|: 0 ≤ X₆
t₃: eval_size08_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀+(X₁)²
t₄: eval_size08_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb3_in(X₀, X₁, X₁, X₃, X₄, X₅, X₆) :|: X₀+(X₁)² ≤ 0
t₅: eval_size08_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb1_in(X₀+(X₁)²*X₆, X₁-2⋅(X₆)², X₂, X₃, X₄, X₅, X₆)
t₆: eval_size08_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb4_in(X₀, X₁, X₂, (X₂)², X₄, X₅, X₆)
t₇: eval_size08_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₃
t₈: eval_size08_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ 0
t₉: eval_size08_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb4_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆)
t₁₀: eval_size08_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_size08_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Found invariant 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ for location eval_size08_bb1_in

Found invariant X₂ ≤ X₅ for location eval_size08_bb3_in

Found invariant X₂ ≤ X₅ ∧ 1 ≤ X₃ for location eval_size08_bb5_in

Found invariant X₂ ≤ X₅ ∧ 0 ≤ X₃ for location eval_size08_bb4_in

Found invariant X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ for location eval_size08_stop

Found invariant 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ for location eval_size08_bb2_in

Found invariant X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ for location eval_size08_bb6_in

Problem after Preprocessing

Start: eval_size08_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: eval_size08_bb0_in, eval_size08_bb1_in, eval_size08_bb2_in, eval_size08_bb3_in, eval_size08_bb4_in, eval_size08_bb5_in, eval_size08_bb6_in, eval_size08_start, eval_size08_stop
Transitions:
t₁: eval_size08_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅, X₆) :|: 1+X₆ ≤ 0
t₂: eval_size08_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb3_in(X₀, X₁, X₅, X₃, X₄, X₅, X₆) :|: 0 ≤ X₆
t₃: eval_size08_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀+(X₁)² ∧ 1+X₆ ≤ 0 ∧ X₁ ≤ X₅
t₄: eval_size08_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb3_in(X₀, X₁, X₁, X₃, X₄, X₅, X₆) :|: X₀+(X₁)² ≤ 0 ∧ 1+X₆ ≤ 0 ∧ X₁ ≤ X₅
t₅: eval_size08_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb1_in(X₀+(X₁)²*X₆, X₁-2⋅(X₆)², X₂, X₃, X₄, X₅, X₆) :|: 1+X₆ ≤ 0 ∧ X₁ ≤ X₅
t₆: eval_size08_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb4_in(X₀, X₁, X₂, (X₂)², X₄, X₅, X₆) :|: X₂ ≤ X₅
t₇: eval_size08_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₃ ∧ X₂ ≤ X₅ ∧ 0 ≤ X₃
t₈: eval_size08_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ 0 ∧ X₂ ≤ X₅ ∧ 0 ≤ X₃
t₉: eval_size08_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb4_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆) :|: 1 ≤ X₃ ∧ X₂ ≤ X₅
t₁₀: eval_size08_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₂ ≤ X₅ ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
t₀: eval_size08_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_size08_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

TWN: t₅: eval_size08_bb2_in→eval_size08_bb1_in

cycle: [t₅: eval_size08_bb2_in→eval_size08_bb1_in; t₃: eval_size08_bb1_in→eval_size08_bb2_in]
loop: (1 ≤ X₀+(X₁)² ∧ 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ ∧ 1+X₆ ≤ 0 ∧ X₁ ≤ X₅,(X₀,X₁,X₅,X₆) -> (X₀+(X₁)²*X₆,X₁-2⋅(X₆)²,X₅,X₆))
order: [X₆; X₁; X₅; X₀]
closed-form:

X₆: X₆
X₁: X₁ + [[n != 0]]⋅-2⋅(X₆)²⋅n^1
X₅: X₅
X₀: X₀ + [[n != 0]]⋅(X₁)²*X₆⋅n^1 + [[n != 0, n != 1]]⋅4/3⋅(X₆)⁵⋅n^3 + [[n != 0, n != 1]]⋅(-2⋅X₁*(X₆)³-2⋅(X₆)⁵)⋅n^2 + [[n != 0, n != 1]]⋅(2⋅X₁*(X₆)³+2/3⋅(X₆)⁵)⋅n^1

Termination: true
Formula:

X₀+(X₁)² ≤ 1 ∧ 1 ≤ X₀+(X₁)² ∧ 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ ∧ 6⋅X₁*(X₆)³+3⋅(X₁)²*X₆+2⋅(X₆)⁵ ≤ 12⋅X₁*(X₆)² ∧ 12⋅X₁*(X₆)² ≤ 6⋅X₁*(X₆)³+3⋅(X₁)²*X₆+2⋅(X₆)⁵ ∧ 2⋅(X₆)⁴ ≤ X₁*(X₆)³+(X₆)⁵ ∧ X₁*(X₆)³+(X₆)⁵ ≤ 2⋅(X₆)⁴ ∧ 0 ≤ (X₆)⁵ ∧ (X₆)⁵ ≤ 0
∨ 1+12⋅X₁*(X₆)² ≤ 6⋅X₁*(X₆)³+3⋅(X₁)²*X₆+2⋅(X₆)⁵ ∧ 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ ∧ 2⋅(X₆)⁴ ≤ X₁*(X₆)³+(X₆)⁵ ∧ X₁*(X₆)³+(X₆)⁵ ≤ 2⋅(X₆)⁴ ∧ 0 ≤ (X₆)⁵ ∧ (X₆)⁵ ≤ 0
∨ 1+6⋅X₁*(X₆)³+6⋅(X₆)⁵ ≤ 12⋅(X₆)⁴ ∧ 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ ∧ 0 ≤ (X₆)⁵ ∧ (X₆)⁵ ≤ 0
∨ 1+X₆ ≤ 0 ∧ 1 ≤ 4⋅(X₆)⁵ ∧ X₁ ≤ X₅
∨ 1+X₆ ≤ 0 ∧ 4 ≤ 3⋅X₀+3⋅(X₁)² ∧ X₁ ≤ X₅ ∧ 6⋅X₁*(X₆)³+3⋅(X₁)²*X₆+2⋅(X₆)⁵ ≤ 12⋅X₁*(X₆)² ∧ 12⋅X₁*(X₆)² ≤ 6⋅X₁*(X₆)³+3⋅(X₁)²*X₆+2⋅(X₆)⁵ ∧ 2⋅(X₆)⁴ ≤ X₁*(X₆)³+(X₆)⁵ ∧ X₁*(X₆)³+(X₆)⁵ ≤ 2⋅(X₆)⁴ ∧ 0 ≤ (X₆)⁵ ∧ (X₆)⁵ ≤ 0

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

alphas_abs: 3⋅X₀+12⋅X₁*(X₆)²+6⋅X₁*(X₆)³+3⋅(X₁)²+3⋅(X₁)²*X₆+12⋅(X₆)⁴+6⋅(X₆)⁵
M: 0
N: 3
Bound: 12⋅X₆⋅X₆⋅X₆⋅X₆⋅X₆+12⋅X₁⋅X₆⋅X₆⋅X₆+24⋅X₆⋅X₆⋅X₆⋅X₆+24⋅X₁⋅X₆⋅X₆+6⋅X₁⋅X₁⋅X₆+6⋅X₁⋅X₁+6⋅X₀+4 {O(n^5)}

TWN - Lifting for [3: eval_size08_bb1_in->eval_size08_bb2_in; 5: eval_size08_bb2_in->eval_size08_bb1_in] of 12⋅X₆⋅X₆⋅X₆⋅X₆⋅X₆+12⋅X₁⋅X₆⋅X₆⋅X₆+24⋅X₆⋅X₆⋅X₆⋅X₆+24⋅X₁⋅X₆⋅X₆+6⋅X₁⋅X₁⋅X₆+6⋅X₁⋅X₁+6⋅X₀+6 {O(n^5)}

relevant size-bounds w.r.t. t₁: eval_size08_bb0_in→eval_size08_bb1_in:
X₀: X₄ {O(n)}
X₁: X₅ {O(n)}
X₆: X₆ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 12⋅X₆⋅X₆⋅X₆⋅X₆⋅X₆+12⋅X₅⋅X₆⋅X₆⋅X₆+24⋅X₆⋅X₆⋅X₆⋅X₆+24⋅X₅⋅X₆⋅X₆+6⋅X₅⋅X₅⋅X₆+6⋅X₅⋅X₅+6⋅X₄+6 {O(n^5)}

Found invariant 1+X₆ ≤ 0 ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₄ for location eval_size08_bb1_in

Found invariant X₂ ≤ X₅ for location eval_size08_bb3_in

Found invariant 1+X₆ ≤ 0 ∧ 2+X₁ ≤ X₅ for location eval_size08_bb1_in_v1

Found invariant X₂ ≤ X₅ ∧ 1 ≤ X₃ for location eval_size08_bb5_in

Found invariant 1+X₆ ≤ 0 ∧ X₅ ≤ X₁ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₄ for location eval_size08_bb2_in_v1

Found invariant X₂ ≤ X₅ ∧ 0 ≤ X₃ for location eval_size08_bb4_in

Found invariant X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ for location eval_size08_stop

Found invariant 1+X₆ ≤ 0 ∧ 2+X₁ ≤ X₅ for location eval_size08_bb2_in_v2

Found invariant X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ for location eval_size08_bb6_in

Found invariant 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ for location eval_size08_bb1_in

Found invariant X₂ ≤ X₅ ∧ 0 ≤ X₃ for location eval_size08_bb4_in_v1

Found invariant X₂ ≤ X₅ for location eval_size08_bb3_in

Found invariant X₂ ≤ X₅ ∧ 0 ≤ X₃ for location eval_size08_bb4_in

Found invariant X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ for location eval_size08_stop

Found invariant X₂ ≤ X₅ ∧ 1 ≤ X₃ for location eval_size08_bb5_in_v1

Found invariant 1+X₆ ≤ 0 ∧ X₁ ≤ X₅ for location eval_size08_bb2_in

Found invariant X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ for location eval_size08_bb6_in

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 12⋅X₆⋅X₆⋅X₆⋅X₆⋅X₆+12⋅X₅⋅X₆⋅X₆⋅X₆+24⋅X₆⋅X₆⋅X₆⋅X₆+24⋅X₅⋅X₆⋅X₆+6⋅X₅⋅X₅⋅X₆+6⋅X₅⋅X₅+6⋅X₄+6 {O(n^5)}
t₄: 1 {O(1)}
t₅: 12⋅X₆⋅X₆⋅X₆⋅X₆⋅X₆+12⋅X₅⋅X₆⋅X₆⋅X₆+24⋅X₆⋅X₆⋅X₆⋅X₆+24⋅X₅⋅X₆⋅X₆+6⋅X₅⋅X₅⋅X₆+6⋅X₅⋅X₅+6⋅X₄+6 {O(n^5)}
t₆: 1 {O(1)}
t₇: inf {Infinity}
t₈: 1 {O(1)}
t₉: inf {Infinity}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 12⋅X₆⋅X₆⋅X₆⋅X₆⋅X₆+12⋅X₅⋅X₆⋅X₆⋅X₆+24⋅X₆⋅X₆⋅X₆⋅X₆+24⋅X₅⋅X₆⋅X₆+6⋅X₅⋅X₅⋅X₆+6⋅X₅⋅X₅+6⋅X₄+6 {O(n^5)}
t₄: 1 {O(1)}
t₅: 12⋅X₆⋅X₆⋅X₆⋅X₆⋅X₆+12⋅X₅⋅X₆⋅X₆⋅X₆+24⋅X₆⋅X₆⋅X₆⋅X₆+24⋅X₅⋅X₆⋅X₆+6⋅X₅⋅X₅⋅X₆+6⋅X₅⋅X₅+6⋅X₄+6 {O(n^5)}
t₆: 1 {O(1)}
t₇: inf {Infinity}
t₈: 1 {O(1)}
t₉: inf {Infinity}
t₁₀: 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)}
t₀, X₅: X₅ {O(n)}
t₀, X₆: X₆ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₁, X₆: X₆ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₅ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₂, X₆: X₆ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₃, X₆: X₆ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₄, X₆: 2⋅X₆ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₅, X₆: X₆ {O(n)}
t₆, X₄: 3⋅X₄ {O(n)}
t₆, X₅: 3⋅X₅ {O(n)}
t₆, X₆: 3⋅X₆ {O(n)}
t₇, X₄: 3⋅X₄ {O(n)}
t₇, X₅: 3⋅X₅ {O(n)}
t₇, X₆: 3⋅X₆ {O(n)}
t₈, X₃: 0 {O(1)}
t₈, X₄: 6⋅X₄ {O(n)}
t₈, X₅: 6⋅X₅ {O(n)}
t₈, X₆: 6⋅X₆ {O(n)}
t₉, X₄: 3⋅X₄ {O(n)}
t₉, X₅: 3⋅X₅ {O(n)}
t₉, X₆: 3⋅X₆ {O(n)}
t₁₀, X₃: 0 {O(1)}
t₁₀, X₄: 6⋅X₄ {O(n)}
t₁₀, X₅: 6⋅X₅ {O(n)}
t₁₀, X₆: 6⋅X₆ {O(n)}