KoAT2 Proof Output

Initial Problem

Start: eval_twn20_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_twn20_bb0_in, eval_twn20_bb1_in, eval_twn20_bb2_in, eval_twn20_bb3_in, eval_twn20_start, eval_twn20_stop
Transitions:
t₁: eval_twn20_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb1_in(X₂, X₃, X₂, X₃, X₄) :|: 1 ≤ X₄
t₂: eval_twn20_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0
t₃: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁
t₄: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁
t₅: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ (X₀)²+(X₄)⁵
t₆: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₇: eval_twn20_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb1_in(4⋅X₀, 9⋅X₁-8⋅(X₄)³, X₂, X₃, X₄)
t₈: eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_twn20_start(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant 1 ≤ X₄ for location eval_twn20_bb2_in

Found invariant 1 ≤ X₄ for location eval_twn20_bb1_in

Problem after Preprocessing

Start: eval_twn20_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_twn20_bb0_in, eval_twn20_bb1_in, eval_twn20_bb2_in, eval_twn20_bb3_in, eval_twn20_start, eval_twn20_stop
Transitions:
t₁: eval_twn20_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb1_in(X₂, X₃, X₂, X₃, X₄) :|: 1 ≤ X₄
t₂: eval_twn20_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0
t₃: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄
t₄: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄
t₅: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ (X₀)²+(X₄)⁵ ∧ 1 ≤ X₄
t₆: eval_twn20_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 1 ≤ X₄
t₇: eval_twn20_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb1_in(4⋅X₀, 9⋅X₁-8⋅(X₄)³, X₂, X₃, X₄) :|: 1 ≤ X₄
t₈: eval_twn20_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_twn20_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_twn20_start(X₀, X₁, X₂, X₃, X₄) → eval_twn20_bb0_in(X₀, X₁, X₂, X₃, X₄)

TWN: t₄: eval_twn20_bb1_in→eval_twn20_bb2_in

cycle: [t₃: eval_twn20_bb1_in→eval_twn20_bb2_in; t₄: eval_twn20_bb1_in→eval_twn20_bb2_in; t₇: eval_twn20_bb2_in→eval_twn20_bb1_in]
original loop: (1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄ ∨ 1 ≤ X₀ ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄,(X₀,X₁,X₄) -> (4⋅X₀,9⋅X₁-8⋅(X₄)³,X₄))
transformed loop: (1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄ ∨ 1 ≤ X₀ ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄,(X₀,X₁,X₄) -> (4⋅X₀,9⋅X₁-8⋅(X₄)³,X₄))
loop: (1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄ ∨ 1 ≤ X₀ ∧ 1+(X₀)²+(X₄)⁵ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄,(X₀,X₁,X₄) -> (4⋅X₀,9⋅X₁-8⋅(X₄)³,X₄))
order: [X₄; X₁; X₀]
closed-form:

X₄: X₄
X₁: X₁⋅(9)^n + [[n != 0]]⋅-(X₄)³⋅(9)^n + [[n != 0]]⋅(X₄)³
X₀: X₀⋅(4)^n

Termination: true
Formula:

(X₄)³ ≤ 1+(X₄)⁵ ∧ 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 1+(X₄)⁵ ≤ (X₄)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ (X₄)³ ≤ 1+(X₄)⁵ ∧ 1+X₀ ≤ 0 ∧ 1 ≤ X₄ ∧ 1+(X₄)⁵ ≤ (X₄)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 1 ≤ X₀ ∧ 1+(X₀)² ≤ 0 ∧ 1 ≤ X₄
∨ 1 ≤ X₀ ∧ 1+(X₄)³ ≤ X₁ ∧ 1 ≤ X₄ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2+(X₄)⁵ ≤ (X₄)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 1+X₀ ≤ 0 ∧ 1+(X₀)² ≤ 0 ∧ 1 ≤ X₄
∨ 1+X₀ ≤ 0 ∧ 1+(X₄)³ ≤ X₁ ∧ 1 ≤ X₄ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0
∨ 1+X₀ ≤ 0 ∧ 1 ≤ X₄ ∧ 2+(X₄)⁵ ≤ (X₄)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³

Stabilization-Threshold for: 1+(X₀)²+(X₄)⁵ ≤ X₁

alphas_abs: X₁+(X₄)³+(X₄)⁵
M': 1
N: 1
Bound: 8⋅log(X₄)+log(X₁)+4 {O(log(n))}

TWN - Lifting for [3: eval_twn20_bb1_in->eval_twn20_bb2_in; 4: eval_twn20_bb1_in->eval_twn20_bb2_in; 7: eval_twn20_bb2_in->eval_twn20_bb1_in] of 8⋅log(X₄)+log(X₁)+6 {O(log(n))}

relevant size-bounds w.r.t. t₁: eval_twn20_bb0_in→eval_twn20_bb1_in:
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 8⋅log(X₄)+log(X₃)+6 {O(log(n))}

All Bounds

Timebounds

Overall timebound:24⋅log(X₄)+3⋅log(X₃)+24 {O(log(n))}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 8⋅log(X₄)+log(X₃)+6 {O(log(n))}
t₄: 8⋅log(X₄)+log(X₃)+6 {O(log(n))}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 8⋅log(X₄)+log(X₃)+6 {O(log(n))}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 24⋅log(X₄)+3⋅log(X₃)+24 {O(log(n))}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 8⋅log(X₄)+log(X₃)+6 {O(log(n))}
t₄: 8⋅log(X₄)+log(X₃)+6 {O(log(n))}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 8⋅log(X₄)+log(X₃)+6 {O(log(n))}
t₈: 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, X₀: 4096⋅4^(8⋅log(X₄))⋅X₂⋅X₃ {O(EXP)}
t₃, X₁: 150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₃+150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₄⋅X₄⋅X₄+X₄⋅X₄⋅X₄ {O(EXP)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₄, X₀: 4096⋅4^(8⋅log(X₄))⋅X₂⋅X₃ {O(EXP)}
t₄, X₁: 150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₃+150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₄⋅X₄⋅X₄+X₄⋅X₄⋅X₄ {O(EXP)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: 4096⋅4^(8⋅log(X₄))⋅X₂⋅X₃+X₂ {O(EXP)}
t₅, X₁: 150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₃+150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₄⋅X₄⋅X₄+X₄⋅X₄⋅X₄+X₃ {O(EXP)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}
t₆, X₀: 0 {O(1)}
t₆, X₁: 150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₃+150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₄⋅X₄⋅X₄+X₄⋅X₄⋅X₄+X₃ {O(EXP)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 2⋅X₃ {O(n)}
t₆, X₄: 2⋅X₄ {O(n)}
t₇, X₀: 4096⋅4^(8⋅log(X₄))⋅X₂⋅X₃ {O(EXP)}
t₇, X₁: 150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₃+150094635296999121⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₄⋅X₄⋅X₄+X₄⋅X₄⋅X₄ {O(EXP)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₈, X₀: 4096⋅4^(8⋅log(X₄))⋅X₂⋅X₃+X₀+X₂ {O(EXP)}
t₈, X₁: 300189270593998242⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₃+300189270593998242⋅9^(24⋅log(X₄))⋅9^(3⋅log(X₃))⋅X₄⋅X₄⋅X₄+2⋅X₄⋅X₄⋅X₄+2⋅X₃+X₁ {O(EXP)}
t₈, X₂: 5⋅X₂ {O(n)}
t₈, X₃: 5⋅X₃ {O(n)}
t₈, X₄: 5⋅X₄ {O(n)}