KoAT2 Proof Output

Initial Problem

Start: eval_non_linear07_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
Temp_Vars:
Locations: eval_non_linear07_.critedge_in, eval_non_linear07_bb0_in, eval_non_linear07_bb1_in, eval_non_linear07_bb2_in, eval_non_linear07_start, eval_non_linear07_stop
Transitions:
t₇: eval_non_linear07_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₁: eval_non_linear07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb1_in(X₆, X₇, X₄, X₅, X₄, X₅, X₆, X₇)
t₄: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₀+2⋅X₁ ≤ X₃ ∧ X₂ ≤ X₃
t₅: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃ ≤ 0
t₂: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1+X₃ ≤ X₀+2⋅X₁ ∧ 1 ≤ X₃
t₃: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1+X₃ ≤ X₂ ∧ 1 ≤ X₃
t₆: eval_non_linear07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb1_in(2⋅X₀+2⋅(X₀)², 2⋅X₁-(X₀)², 3⋅X₂, 4⋅X₃, X₄, X₅, X₆, X₇)
t₀: eval_non_linear07_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)

Preprocessing

Found invariant X₅ ≤ X₃ for location eval_non_linear07_stop

Found invariant X₅ ≤ X₃ for location eval_non_linear07_.critedge_in

Found invariant X₅ ≤ X₃ for location eval_non_linear07_bb1_in

Found invariant X₅ ≤ X₃ ∧ 1 ≤ X₃ for location eval_non_linear07_bb2_in

Problem after Preprocessing

Start: eval_non_linear07_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
Temp_Vars:
Locations: eval_non_linear07_.critedge_in, eval_non_linear07_bb0_in, eval_non_linear07_bb1_in, eval_non_linear07_bb2_in, eval_non_linear07_start, eval_non_linear07_stop
Transitions:
t₇: eval_non_linear07_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₅ ≤ X₃
t₁: eval_non_linear07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb1_in(X₆, X₇, X₄, X₅, X₄, X₅, X₆, X₇)
t₄: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₀+2⋅X₁ ≤ X₃ ∧ X₂ ≤ X₃ ∧ X₅ ≤ X₃
t₅: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃ ≤ 0 ∧ X₅ ≤ X₃
t₂: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1+X₃ ≤ X₀+2⋅X₁ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃
t₃: eval_non_linear07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1+X₃ ≤ X₂ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃
t₆: eval_non_linear07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb1_in(2⋅X₀+2⋅(X₀)², 2⋅X₁-(X₀)², 3⋅X₂, 4⋅X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₃ ∧ X₅ ≤ X₃
t₀: eval_non_linear07_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_non_linear07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)

TWN: t₂: eval_non_linear07_bb1_in→eval_non_linear07_bb2_in

cycle: [t₂: eval_non_linear07_bb1_in→eval_non_linear07_bb2_in; t₃: eval_non_linear07_bb1_in→eval_non_linear07_bb2_in; t₆: eval_non_linear07_bb2_in→eval_non_linear07_bb1_in]
original loop: (1+Temp_Int₃₄₂₇+X₃ ≤ 0 ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∨ 1+X₃ ≤ X₂ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃,(Temp_Int₃₄₂₇,X₂,X₃,X₅) -> (2⋅Temp_Int₃₄₂₇,3⋅X₂,4⋅X₃,X₅))
transformed loop: (1+Temp_Int₃₄₂₇+X₃ ≤ 0 ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∨ 1+X₃ ≤ X₂ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃,(Temp_Int₃₄₂₇,X₂,X₃,X₅) -> (2⋅Temp_Int₃₄₂₇,3⋅X₂,4⋅X₃,X₅))
loop: (1+Temp_Int₃₄₂₇+X₃ ≤ 0 ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∨ 1+X₃ ≤ X₂ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₅ ≤ X₃,(Temp_Int₃₄₂₇,X₂,X₃,X₅) -> (2⋅Temp_Int₃₄₂₇,3⋅X₂,4⋅X₃,X₅))
order: [X₅; X₃; X₂; Temp_Int₃₄₂₇]
closed-form:

X₅: X₅
X₃: X₃⋅(4)^n
X₂: X₂⋅(3)^n
Temp_Int₃₄₂₇: Temp_Int₃₄₂₇⋅(2)^n

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 1+X₅ ≤ 0 ∧ 0 ≤ Temp_Int₃₄₂₇ ∧ Temp_Int₃₄₂₇ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 1+X₅ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ Temp_Int₃₄₂₇ ∧ Temp_Int₃₄₂₇ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ Temp_Int₃₄₂₇ ∧ Temp_Int₃₄₂₇ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 1+X₅ ≤ 0 ∧ 0 ≤ Temp_Int₃₄₂₇ ∧ Temp_Int₃₄₂₇ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 1+X₅ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ Temp_Int₃₄₂₇ ∧ Temp_Int₃₄₂₇ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ Temp_Int₃₄₂₇ ∧ Temp_Int₃₄₂₇ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
∨ 1+Temp_Int₃₄₂₇ ≤ 0 ∧ 1 ≤ X₃ ∧ 1+X₅ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1+Temp_Int₃₄₂₇ ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1+Temp_Int₃₄₂₇ ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
∨ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 1+X₅ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
∨ 1 ≤ X₃ ∧ 1+X₃ ≤ 0
∨ 1 ≤ X₃ ∧ 1+X₃ ≤ 0 ∧ 1+X₅ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ X₃ ∧ 1+X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 0 ≤ X₅ ∧ X₅ ≤ 0

Stabilization-Threshold for: X₅ ≤ X₃

alphas_abs: 1+X₅
M': 1
N: 1
Bound: log(X₅)+3 {O(log(n))}

Stabilization-Threshold for: 1+X₃ ≤ X₂

alphas_abs: X₂
M': 1
N: 1
Bound: log(X₂)+2 {O(log(n))}

Stabilization-Threshold for: 1+Temp_Int₃₄₂₇+X₃ ≤ 0

alphas_abs: Temp_Int₃₄₂₇
M': 1
N: 1
Bound: log(Temp_Int₃₄₂₇)+2 {O(log(n))}

TWN - Lifting for [2: eval_non_linear07_bb1_in->eval_non_linear07_bb2_in; 3: eval_non_linear07_bb1_in->eval_non_linear07_bb2_in; 6: eval_non_linear07_bb2_in->eval_non_linear07_bb1_in] of log(X₀)+log(X₁)+log(X₂)+log(X₅)+10 {O(log(n))}

relevant size-bounds w.r.t. t₁: eval_non_linear07_bb0_in→eval_non_linear07_bb1_in:
X₀: X₆ {O(n)}
X₁: X₇ {O(n)}
X₂: X₄ {O(n)}
X₅: X₅ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: log(X₄)+log(X₅)+log(X₆)+log(X₇)+10 {O(log(n))}

All Bounds

Timebounds

Overall timebound:3⋅log(X₄)+3⋅log(X₅)+3⋅log(X₆)+3⋅log(X₇)+35 {O(log(n))}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: log(X₄)+log(X₅)+log(X₆)+log(X₇)+10 {O(log(n))}
t₃: log(X₄)+log(X₅)+log(X₆)+log(X₇)+10 {O(log(n))}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: log(X₄)+log(X₅)+log(X₆)+log(X₇)+10 {O(log(n))}
t₇: 1 {O(1)}

Costbounds

Overall costbound: 3⋅log(X₄)+3⋅log(X₅)+3⋅log(X₆)+3⋅log(X₇)+35 {O(log(n))}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: log(X₄)+log(X₅)+log(X₆)+log(X₇)+10 {O(log(n))}
t₃: log(X₄)+log(X₅)+log(X₆)+log(X₇)+10 {O(log(n))}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: log(X₄)+log(X₅)+log(X₆)+log(X₇)+10 {O(log(n))}
t₇: 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)}
t₀, X₇: X₇ {O(n)}
t₁, X₀: X₆ {O(n)}
t₁, X₁: X₇ {O(n)}
t₁, X₂: X₄ {O(n)}
t₁, X₃: X₅ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₁, X₆: X₆ {O(n)}
t₁, X₇: X₇ {O(n)}
t₂, X₂: 59049⋅X₄⋅X₄⋅X₅⋅X₆⋅X₇ {O(n^5)}
t₂, X₃: 1048576⋅X₄⋅X₅⋅X₅⋅X₆⋅X₇ {O(n^5)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₂, X₆: X₆ {O(n)}
t₂, X₇: X₇ {O(n)}
t₃, X₂: 59049⋅X₄⋅X₄⋅X₅⋅X₆⋅X₇ {O(n^5)}
t₃, X₃: 1048576⋅X₄⋅X₅⋅X₅⋅X₆⋅X₇ {O(n^5)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₃, X₆: X₆ {O(n)}
t₃, X₇: X₇ {O(n)}
t₄, X₂: 59049⋅X₄⋅X₄⋅X₅⋅X₆⋅X₇+X₄ {O(n^5)}
t₄, X₃: 1048576⋅X₄⋅X₅⋅X₅⋅X₆⋅X₇+X₅ {O(n^5)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₄, X₆: 2⋅X₆ {O(n)}
t₄, X₇: 2⋅X₇ {O(n)}
t₅, X₀: X₆ {O(n)}
t₅, X₁: X₇ {O(n)}
t₅, X₂: X₄ {O(n)}
t₅, X₃: X₅ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₅, X₆: X₆ {O(n)}
t₅, X₇: X₇ {O(n)}
t₆, X₂: 59049⋅X₄⋅X₄⋅X₅⋅X₆⋅X₇ {O(n^5)}
t₆, X₃: 1048576⋅X₄⋅X₅⋅X₅⋅X₆⋅X₇ {O(n^5)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₆, X₆: X₆ {O(n)}
t₆, X₇: X₇ {O(n)}
t₇, X₂: 59049⋅X₄⋅X₄⋅X₅⋅X₆⋅X₇+2⋅X₄ {O(n^5)}
t₇, X₃: 1048576⋅X₄⋅X₅⋅X₅⋅X₆⋅X₇+2⋅X₅ {O(n^5)}
t₇, X₄: 3⋅X₄ {O(n)}
t₇, X₅: 3⋅X₅ {O(n)}
t₇, X₆: 3⋅X₆ {O(n)}
t₇, X₇: 3⋅X₇ {O(n)}