KoAT2 Proof Output

Initial Problem

Start: eval_non_linear02_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: eval_non_linear02_bb0_in, eval_non_linear02_bb1_in, eval_non_linear02_bb2_in, eval_non_linear02_bb3_in, eval_non_linear02_start, eval_non_linear02_stop
Transitions:
t₁: eval_non_linear02_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_bb1_in(X₅, X₆, X₂, X₃, X₄, X₅, X₆)
t₂: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₀ ≤ 0 ∧ 1+(X₀)² ≤ X₁+(X₄)⁵
t₃: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ 1+(X₀)² ≤ X₁+(X₄)⁵
t₄: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₁+(X₄)⁵ ≤ (X₀)²
t₅: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₆: eval_non_linear02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_bb1_in(4⋅X₀, 9⋅X₁+(X₄)², X₂, X₃, X₄, X₅, X₆)
t₇: eval_non_linear02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_non_linear02_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear02_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Eliminate variables [X₂; X₃] that do not contribute to the problem

Problem after Preprocessing

Start: eval_non_linear02_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_non_linear02_bb0_in, eval_non_linear02_bb1_in, eval_non_linear02_bb2_in, eval_non_linear02_bb3_in, eval_non_linear02_start, eval_non_linear02_stop
Transitions:
t₁₅: eval_non_linear02_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₁₆: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0 ∧ 1+(X₀)² ≤ X₁+(X₂)⁵
t₁₇: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+(X₀)² ≤ X₁+(X₂)⁵
t₁₈: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁+(X₂)⁵ ≤ (X₀)²
t₁₉: eval_non_linear02_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₂₀: eval_non_linear02_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_bb1_in(4⋅X₀, 9⋅X₁+(X₂)², X₂, X₃, X₄)
t₂₁: eval_non_linear02_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_stop(X₀, X₁, X₂, X₃, X₄)
t₂₂: eval_non_linear02_start(X₀, X₁, X₂, X₃, X₄) → eval_non_linear02_bb0_in(X₀, X₁, X₂, X₃, X₄)

TWN: t₁₆: eval_non_linear02_bb1_in→eval_non_linear02_bb2_in

cycle: [t₁₆: eval_non_linear02_bb1_in→eval_non_linear02_bb2_in; t₁₇: eval_non_linear02_bb1_in→eval_non_linear02_bb2_in; t₂₀: eval_non_linear02_bb2_in→eval_non_linear02_bb1_in]
original loop: (1+X₀ ≤ 0 ∧ 1+(X₀)² ≤ X₁+(X₂)⁵ ∨ 1 ≤ X₀ ∧ 1+(X₀)² ≤ X₁+(X₂)⁵,(X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁+(X₂)²,X₂))
transformed loop: (1+X₀ ≤ 0 ∧ 1+(X₀)² ≤ X₁+(X₂)⁵ ∨ 1 ≤ X₀ ∧ 1+(X₀)² ≤ X₁+(X₂)⁵,(X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁+(X₂)²,X₂))
loop: (1+X₀ ≤ 0 ∧ 1+(X₀)² ≤ X₁+(X₂)⁵ ∨ 1 ≤ X₀ ∧ 1+(X₀)² ≤ X₁+(X₂)⁵,(X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁+(X₂)²,X₂))
order: [X₂; X₁; X₀]
closed-form:

X₂: X₂
X₁: X₁⋅(9)^n + [[n != 0]]⋅1/8⋅(X₂)²⋅(9)^n + [[n != 0]]⋅-1/8⋅(X₂)²
X₀: X₀⋅(4)^n

Termination: true
Formula:

8⋅(X₂)⁵ ≤ 8+(X₂)² ∧ 1 ≤ X₀ ∧ 8+(X₂)² ≤ 8⋅(X₂)⁵ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ 0 ≤ 8⋅X₁+(X₂)² ∧ 8⋅X₁+(X₂)² ≤ 0
∨ 8⋅(X₂)⁵ ≤ 8+(X₂)² ∧ 1+X₀ ≤ 0 ∧ 8+(X₂)² ≤ 8⋅(X₂)⁵ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ 0 ≤ 8⋅X₁+(X₂)² ∧ 8⋅X₁+(X₂)² ≤ 0
∨ 1 ≤ X₀ ∧ 1+8⋅(X₀)² ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 8⋅X₁+(X₂)² ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0
∨ 1 ≤ X₀ ∧ 9+(X₂)² ≤ 8⋅(X₂)⁵ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ 0 ≤ 8⋅X₁+(X₂)² ∧ 8⋅X₁+(X₂)² ≤ 0
∨ 1+X₀ ≤ 0 ∧ 1+8⋅(X₀)² ≤ 0
∨ 1+X₀ ≤ 0 ∧ 1 ≤ 8⋅X₁+(X₂)² ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0
∨ 1+X₀ ≤ 0 ∧ 9+(X₂)² ≤ 8⋅(X₂)⁵ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ 0 ≤ 8⋅X₁+(X₂)² ∧ 8⋅X₁+(X₂)² ≤ 0

Stabilization-Threshold for: 1+(X₀)² ≤ X₁+(X₂)⁵

alphas_abs: 8⋅X₁+(X₂)²+8⋅(X₂)⁵
M': 1
N: 1
Bound: 7⋅log(X₂)+log(X₁)+10 {O(log(n))}

TWN - Lifting for [16: eval_non_linear02_bb1_in->eval_non_linear02_bb2_in; 17: eval_non_linear02_bb1_in->eval_non_linear02_bb2_in; 20: eval_non_linear02_bb2_in->eval_non_linear02_bb1_in] of 7⋅log(X₂)+log(X₁)+12 {O(log(n))}

relevant size-bounds w.r.t. t₁₅: eval_non_linear02_bb0_in→eval_non_linear02_bb1_in:
X₁: X₄ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 7⋅log(X₂)+log(X₄)+12 {O(log(n))}

All Bounds

Timebounds

Overall timebound:21⋅log(X₂)+3⋅log(X₄)+41 {O(log(n))}
t₁₅: 1 {O(1)}
t₁₆: 7⋅log(X₂)+log(X₄)+12 {O(log(n))}
t₁₇: 7⋅log(X₂)+log(X₄)+12 {O(log(n))}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 7⋅log(X₂)+log(X₄)+12 {O(log(n))}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}

Costbounds

Overall costbound: 21⋅log(X₂)+3⋅log(X₄)+41 {O(log(n))}
t₁₅: 1 {O(1)}
t₁₆: 7⋅log(X₂)+log(X₄)+12 {O(log(n))}
t₁₇: 7⋅log(X₂)+log(X₄)+12 {O(log(n))}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 7⋅log(X₂)+log(X₄)+12 {O(log(n))}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}

Sizebounds

t₁₅, X₀: X₃ {O(n)}
t₁₅, X₁: X₄ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₆, X₀: 16777216⋅4^(7⋅log(X₂))⋅X₃⋅X₄ {O(EXP)}
t₁₆, X₁: 22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₂⋅X₂+22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₄+X₂⋅X₂ {O(EXP)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₆, X₄: X₄ {O(n)}
t₁₇, X₀: 16777216⋅4^(7⋅log(X₂))⋅X₃⋅X₄ {O(EXP)}
t₁₇, X₁: 22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₂⋅X₂+22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₄+X₂⋅X₂ {O(EXP)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₈, X₀: 16777216⋅4^(7⋅log(X₂))⋅X₃⋅X₄+X₃ {O(EXP)}
t₁₈, X₁: 22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₂⋅X₂+22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₄+X₂⋅X₂+X₄ {O(EXP)}
t₁₈, X₂: 2⋅X₂ {O(n)}
t₁₈, X₃: 2⋅X₃ {O(n)}
t₁₈, X₄: 2⋅X₄ {O(n)}
t₁₉, X₀: 0 {O(1)}
t₁₉, X₁: 22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₂⋅X₂+22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₄+X₂⋅X₂+X₄ {O(EXP)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₁₉, X₄: 2⋅X₄ {O(n)}
t₂₀, X₀: 16777216⋅4^(7⋅log(X₂))⋅X₃⋅X₄ {O(EXP)}
t₂₀, X₁: 22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₂⋅X₂+22528399544939174411840147874772641⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₄+X₂⋅X₂ {O(EXP)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₁, X₀: 16777216⋅4^(7⋅log(X₂))⋅X₃⋅X₄+X₃ {O(EXP)}
t₂₁, X₁: 45056799089878348823680295749545282⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₂⋅X₂+45056799089878348823680295749545282⋅9^(21⋅log(X₂))⋅9^(3⋅log(X₄))⋅X₄+2⋅X₂⋅X₂+2⋅X₄ {O(EXP)}
t₂₁, X₂: 4⋅X₂ {O(n)}
t₂₁, X₃: 4⋅X₃ {O(n)}
t₂₁, X₄: 4⋅X₄ {O(n)}
t₂₂, X₀: X₀ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}