Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: U, V
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(U, X₁, X₂, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 3 ∧ 1 ≤ X₀ ∧ 0 ≤ V ∧ V ≤ 0
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁
t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁-1, X₂, X₃, X₄, X₅)
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅)
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1+X₃ ≤ 0
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1 ≤ X₃

Preprocessing

Found invariant 1 ≤ X₁ for location l2

Found invariant 1 ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: U, V
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(U, X₁, X₂, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 3 ∧ 1 ≤ X₀ ∧ 0 ≤ V ∧ V ≤ 0
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁
t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁-1, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1+X₃ ≤ 0 ∧ 1 ≤ X₁
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁

MPRF for transition t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁-1, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

• l1: [X₁]
• l2: [X₁]

MPRF for transition t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

• l1: [X₁]
• l2: [X₁-1]

Found invariant 1 ≤ X₁ for location l2

Found invariant X₀ ≤ 4 ∧ 2 ≤ X₀ for location l1_v1

Found invariant 1 ≤ X₁ for location l3

Analysing control-flow refined program

knowledge_propagation leads to new time bound X₁+1 {O(n)} for transition t₃₆: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁

knowledge_propagation leads to new time bound X₁+1 {O(n)} for transition t₃₇: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1_v1(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 3 ∧ 1 ≤ X₀ ∧ 0 ≤ V ∧ V ≤ 0

MPRF for transition t₃₈: l1_v1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁ ∧ X₀ ≤ 4 ∧ 1 ≤ X₀ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

• l1: [X₁]
• l1_v1: [X₁]
• l2: [X₁-1]

MPRF for transition t₃₉: l1_v1(X₀, X₁, X₂, X₃, X₄, X₅) → l1_v1(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 3 ∧ 1 ≤ X₀ ∧ 0 ≤ V ∧ V ≤ 0 ∧ X₀ ≤ 4 ∧ 2 ≤ X₀ of depth 1:

new bound:

17⋅X₁+26 {O(n)}

• l1: [9]
• l1_v1: [13-X₀]
• l2: [9]

CFR: Improvement to new bound with the following program:

method: PartialEvaluation new bound:

O(n)

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: U, V
Locations: l0, l1, l1_v1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(U, X₁, X₂, X₃, X₄, X₅)
t₃₇: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1_v1(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 3 ∧ 1 ≤ X₀ ∧ 0 ≤ V ∧ V ≤ 0
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁
t₃₆: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁
t₃₉: l1_v1(X₀, X₁, X₂, X₃, X₄, X₅) → l1_v1(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 3 ∧ 1 ≤ X₀ ∧ 0 ≤ V ∧ V ≤ 0 ∧ X₀ ≤ 4 ∧ 2 ≤ X₀
t₃₈: l1_v1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: V ≤ 1 ∧ 1 ≤ V ∧ 1 ≤ X₁ ∧ X₀ ≤ 4 ∧ 1 ≤ X₀ ∧ 2 ≤ X₀
t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁-1, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1+X₃ ≤ 0 ∧ 1 ≤ X₁
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1 ≤ X₃ ∧ 1 ≤ X₁

TWN: t₅: l3→l3

Show Graph

Termination: true
Formula:

TWN - Lifting for [5: l3->l3; 6: l3->l3] of 28⋅log(X₂)+4⋅log(X₄)+4⋅log(X₅)+41 {O(log(n))}

All Bounds

Timebounds

Overall timebound:22⋅X₁+56⋅log(X₂)+8⋅log(X₄)+8⋅log(X₅)+112 {O(n)}
t₀: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: X₁ {O(n)}
t₄: 1 {O(1)}
t₅: 28⋅log(X₂)+4⋅log(X₄)+4⋅log(X₅)+41 {O(log(n))}
t₆: 28⋅log(X₂)+4⋅log(X₄)+4⋅log(X₅)+41 {O(log(n))}
t₃₆: X₁+1 {O(n)}
t₃₇: X₁+1 {O(n)}
t₃₈: X₁ {O(n)}
t₃₉: 17⋅X₁+26 {O(n)}

Costbounds

Overall costbound: 22⋅X₁+56⋅log(X₂)+8⋅log(X₄)+8⋅log(X₅)+112 {O(n)}
t₀: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: X₁ {O(n)}
t₄: 1 {O(1)}
t₅: 28⋅log(X₂)+4⋅log(X₄)+4⋅log(X₅)+41 {O(log(n))}
t₆: 28⋅log(X₂)+4⋅log(X₄)+4⋅log(X₅)+41 {O(log(n))}
t₃₆: X₁+1 {O(n)}
t₃₇: X₁+1 {O(n)}
t₃₈: X₁ {O(n)}
t₃₉: 17⋅X₁+26 {O(n)}

Sizebounds

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)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: 2^(28⋅log(X₂))⋅2^(28⋅log(X₂))⋅2^(4⋅log(X₄))⋅2^(4⋅log(X₄))⋅2^(4⋅log(X₅))⋅2^(4⋅log(X₅))⋅4835703278458516698824704⋅X₃ {O(EXP)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: 2^(28⋅log(X₂))⋅2^(28⋅log(X₂))⋅2^(4⋅log(X₄))⋅2^(4⋅log(X₄))⋅2^(4⋅log(X₅))⋅2^(4⋅log(X₅))⋅4835703278458516698824704⋅X₃ {O(EXP)}
t₃₆, X₁: X₁ {O(n)}
t₃₆, X₂: X₂ {O(n)}
t₃₆, X₃: X₃ {O(n)}
t₃₆, X₄: X₄ {O(n)}
t₃₆, X₅: X₅ {O(n)}
t₃₇, X₀: 4 {O(1)}
t₃₇, X₁: X₁ {O(n)}
t₃₇, X₂: X₂ {O(n)}
t₃₇, X₃: X₃ {O(n)}
t₃₇, X₄: X₄ {O(n)}
t₃₇, X₅: X₅ {O(n)}
t₃₈, X₀: 4 {O(1)}
t₃₈, X₁: X₁ {O(n)}
t₃₈, X₂: X₂ {O(n)}
t₃₈, X₃: X₃ {O(n)}
t₃₈, X₄: X₄ {O(n)}
t₃₈, X₅: X₅ {O(n)}
t₃₉, X₀: 4 {O(1)}
t₃₉, X₁: X₁ {O(n)}
t₃₉, X₂: X₂ {O(n)}
t₃₉, X₃: X₃ {O(n)}
t₃₉, X₄: X₄ {O(n)}
t₃₉, X₅: X₅ {O(n)}