KoAT2 Proof Output

Initial Problem

Start: evalcomplexstart
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: evalcomplexbb10in, evalcomplexbb1in, evalcomplexbb6in, evalcomplexbb7in, evalcomplexbb8in, evalcomplexbb9in, evalcomplexentryin, evalcomplexreturnin, evalcomplexstart, evalcomplexstop
Transitions:
t₂: evalcomplexbb10in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb8in(X₀, X₁, X₀, X₁, X₄) :|: X₁ ≤ 29
t₃: evalcomplexbb10in(X₀, X₁, X₂, X₃, X₄) → evalcomplexreturnin(X₀, X₁, X₂, X₃, X₄) :|: 30 ≤ X₁
t₈: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb6in(X₀, X₁, X₂, X₃, 7+X₂) :|: X₂ ≤ 5 ∧ 3 ≤ X₂ ∧ 6 ≤ X₂
t₁₁: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb6in(X₀, X₁, X₂, X₃, 2+X₂) :|: X₂ ≤ 10 ∧ X₂ ≤ 5 ∧ 8 ≤ X₂
t₆: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 7+X₂) :|: X₂ ≤ 2 ∧ 6 ≤ X₂
t₇: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 7+X₂) :|: 6 ≤ X₂
t₉: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 2+X₂) :|: X₂ ≤ 7 ∧ X₂ ≤ 5
t₁₀: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 2+X₂) :|: X₂ ≤ 5 ∧ 11 ≤ X₂
t₁₃: evalcomplexbb6in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb8in(X₀, X₁, X₄, 10+X₃, X₄)
t₁₂: evalcomplexbb7in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb8in(X₀, X₁, X₄, 1+X₃, X₄)
t₄: evalcomplexbb8in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₂ ≤ X₃
t₅: evalcomplexbb8in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb9in(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₂
t₁₄: evalcomplexbb9in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb10in(X₂-10, 2+X₃, X₂, X₃, X₄)
t₁: evalcomplexentryin(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb10in(X₁, X₀, X₂, X₃, X₄)
t₁₅: evalcomplexreturnin(X₀, X₁, X₂, X₃, X₄) → evalcomplexstop(X₀, X₁, X₂, X₃, X₄)
t₀: evalcomplexstart(X₀, X₁, X₂, X₃, X₄) → evalcomplexentryin(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Cut unsatisfiable transition [t₆: evalcomplexbb1in→evalcomplexbb7in; t₈: evalcomplexbb1in→evalcomplexbb6in; t₁₀: evalcomplexbb1in→evalcomplexbb7in; t₁₁: evalcomplexbb1in→evalcomplexbb6in; t₁₃: evalcomplexbb6in→evalcomplexbb8in]

Cut unreachable locations [evalcomplexbb6in] from the program graph

Found invariant 30 ≤ X₁ for location evalcomplexstop

Found invariant 30 ≤ X₁ for location evalcomplexreturnin

Found invariant X₄ ≤ 6+X₃ ∧ X₄ ≤ 7+X₂ ∧ 2+X₂ ≤ X₄ ∧ 2+X₀ ≤ X₄ ∧ 1+X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 29 for location evalcomplexbb7in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 29 for location evalcomplexbb8in

Found invariant 1+X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 29 for location evalcomplexbb1in

Found invariant X₃ ≤ X₂ ∧ X₁ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 29 for location evalcomplexbb9in

Problem after Preprocessing

Start: evalcomplexstart
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: evalcomplexbb10in, evalcomplexbb1in, evalcomplexbb7in, evalcomplexbb8in, evalcomplexbb9in, evalcomplexentryin, evalcomplexreturnin, evalcomplexstart, evalcomplexstop
Transitions:
t₂: evalcomplexbb10in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb8in(X₀, X₁, X₀, X₁, X₄) :|: X₁ ≤ 29
t₃: evalcomplexbb10in(X₀, X₁, X₂, X₃, X₄) → evalcomplexreturnin(X₀, X₁, X₂, X₃, X₄) :|: 30 ≤ X₁
t₇: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 7+X₂) :|: 6 ≤ X₂ ∧ X₁ ≤ 29 ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₉: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 2+X₂) :|: X₂ ≤ 7 ∧ X₂ ≤ 5 ∧ X₁ ≤ 29 ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₂: evalcomplexbb7in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb8in(X₀, X₁, X₄, 1+X₃, X₄) :|: X₁ ≤ 29 ∧ X₄ ≤ 7+X₂ ∧ X₄ ≤ 6+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2+X₀ ≤ X₄ ∧ 2+X₂ ≤ X₄ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₄: evalcomplexbb8in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₂ ≤ X₃ ∧ X₁ ≤ 29 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₅: evalcomplexbb8in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb9in(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₂ ∧ X₁ ≤ 29 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₄: evalcomplexbb9in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb10in(X₂-10, 2+X₃, X₂, X₃, X₄) :|: X₁ ≤ 29 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₂
t₁: evalcomplexentryin(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb10in(X₁, X₀, X₂, X₃, X₄)
t₁₅: evalcomplexreturnin(X₀, X₁, X₂, X₃, X₄) → evalcomplexstop(X₀, X₁, X₂, X₃, X₄) :|: 30 ≤ X₁
t₀: evalcomplexstart(X₀, X₁, X₂, X₃, X₄) → evalcomplexentryin(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂: evalcomplexbb10in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb8in(X₀, X₁, X₀, X₁, X₄) :|: X₁ ≤ 29 of depth 1:

new bound:

X₀+30 {O(n)}

MPRF:

• evalcomplexbb10in: [30-X₁]
• evalcomplexbb1in: [27-X₃]
• evalcomplexbb7in: [27-X₃]
• evalcomplexbb8in: [28-X₃]
• evalcomplexbb9in: [28-X₃]

MPRF for transition t₄: evalcomplexbb8in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₂ ≤ X₃ ∧ X₁ ≤ 29 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

5⋅X₀+X₁+174 {O(n)}

MPRF:

• evalcomplexbb10in: [174-X₀-5⋅X₁]
• evalcomplexbb1in: [173+X₃-6⋅X₁-X₂]
• evalcomplexbb7in: [173+X₃-6⋅X₁-X₂]
• evalcomplexbb8in: [174+X₃-6⋅X₁-X₂]
• evalcomplexbb9in: [174+X₃-6⋅X₁-X₂]

MPRF for transition t₅: evalcomplexbb8in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb9in(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₂ ∧ X₁ ≤ 29 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₀+31 {O(n)}

MPRF:

• evalcomplexbb10in: [31-X₁]
• evalcomplexbb1in: [30-X₁]
• evalcomplexbb7in: [30-X₁]
• evalcomplexbb8in: [30-X₁]
• evalcomplexbb9in: [29-X₁]

MPRF for transition t₇: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 7+X₂) :|: 6 ≤ X₂ ∧ X₁ ≤ 29 ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

5⋅X₀+X₁+174 {O(n)}

MPRF:

• evalcomplexbb10in: [174-X₀-5⋅X₁]
• evalcomplexbb1in: [174+X₃-6⋅X₁-X₂]
• evalcomplexbb7in: [175+X₃-6⋅X₁-X₄]
• evalcomplexbb8in: [174+X₃-6⋅X₁-X₂]
• evalcomplexbb9in: [174+X₃-6⋅X₁-X₂]

MPRF for transition t₉: evalcomplexbb1in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb7in(X₀, X₁, X₂, X₃, 2+X₂) :|: X₂ ≤ 7 ∧ X₂ ≤ 5 ∧ X₁ ≤ 29 ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

5⋅X₀+X₁+151 {O(n)}

MPRF:

• evalcomplexbb10in: [151-X₀-5⋅X₁]
• evalcomplexbb1in: [151+X₃-6⋅X₁-X₂]
• evalcomplexbb7in: [152+X₃-6⋅X₁-X₄]
• evalcomplexbb8in: [151+X₃-6⋅X₁-X₂]
• evalcomplexbb9in: [151-X₂-5⋅X₃]

MPRF for transition t₁₂: evalcomplexbb7in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb8in(X₀, X₁, X₄, 1+X₃, X₄) :|: X₁ ≤ 29 ∧ X₄ ≤ 7+X₂ ∧ X₄ ≤ 6+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2+X₀ ≤ X₄ ∧ 2+X₂ ≤ X₄ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

5⋅X₀+X₁+174 {O(n)}

MPRF:

• evalcomplexbb10in: [174-X₀-5⋅X₁]
• evalcomplexbb1in: [174+X₃-6⋅X₁-X₂]
• evalcomplexbb7in: [174+X₃-6⋅X₁-X₂]
• evalcomplexbb8in: [174+X₃-6⋅X₁-X₂]
• evalcomplexbb9in: [174+X₃-6⋅X₁-X₂]

MPRF for transition t₁₄: evalcomplexbb9in(X₀, X₁, X₂, X₃, X₄) → evalcomplexbb10in(X₂-10, 2+X₃, X₂, X₃, X₄) :|: X₁ ≤ 29 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₂ of depth 1:

new bound:

X₀+31 {O(n)}

MPRF:

• evalcomplexbb10in: [31-X₁]
• evalcomplexbb1in: [31-X₁]
• evalcomplexbb7in: [31-X₁]
• evalcomplexbb8in: [31-X₁]
• evalcomplexbb9in: [30-X₁]

All Bounds

Timebounds

Overall timebound:23⋅X₀+4⋅X₁+769 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+30 {O(n)}
t₃: 1 {O(1)}
t₄: 5⋅X₀+X₁+174 {O(n)}
t₅: X₀+31 {O(n)}
t₇: 5⋅X₀+X₁+174 {O(n)}
t₉: 5⋅X₀+X₁+151 {O(n)}
t₁₂: 5⋅X₀+X₁+174 {O(n)}
t₁₄: X₀+31 {O(n)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 23⋅X₀+4⋅X₁+769 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+30 {O(n)}
t₃: 1 {O(1)}
t₄: 5⋅X₀+X₁+174 {O(n)}
t₅: X₀+31 {O(n)}
t₇: 5⋅X₀+X₁+174 {O(n)}
t₉: 5⋅X₀+X₁+151 {O(n)}
t₁₂: 5⋅X₀+X₁+174 {O(n)}
t₁₄: X₀+31 {O(n)}
t₁₅: 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: X₁ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 45⋅X₀+9⋅X₁+1528 {O(n)}
t₂, X₁: 9⋅X₀+X₁+236 {O(n)}
t₂, X₂: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₂, X₃: 8⋅X₀+X₁+236 {O(n)}
t₂, X₄: 16⋅X₁+90⋅X₀+X₄+3065 {O(n)}
t₃, X₀: 45⋅X₀+9⋅X₁+1528 {O(n)}
t₃, X₁: 9⋅X₀+X₁+236 {O(n)}
t₃, X₂: 45⋅X₀+8⋅X₁+X₂+1528 {O(n)}
t₃, X₃: 8⋅X₀+X₁+X₃+236 {O(n)}
t₃, X₄: 16⋅X₁+2⋅X₄+90⋅X₀+3065 {O(n)}
t₄, X₀: 45⋅X₀+9⋅X₁+1528 {O(n)}
t₄, X₁: 9⋅X₀+X₁+236 {O(n)}
t₄, X₂: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₄, X₃: 8⋅X₀+X₁+236 {O(n)}
t₄, X₄: 180⋅X₀+32⋅X₁+X₄+6130 {O(n)}
t₅, X₀: 18⋅X₁+90⋅X₀+3056 {O(n)}
t₅, X₁: 18⋅X₀+2⋅X₁+472 {O(n)}
t₅, X₂: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₅, X₃: 8⋅X₀+X₁+236 {O(n)}
t₅, X₄: 16⋅X₁+90⋅X₀+X₄+3065 {O(n)}
t₇, X₀: 45⋅X₀+9⋅X₁+1528 {O(n)}
t₇, X₁: 9⋅X₀+X₁+236 {O(n)}
t₇, X₂: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₇, X₃: 8⋅X₀+X₁+236 {O(n)}
t₇, X₄: 45⋅X₀+8⋅X₁+1535 {O(n)}
t₉, X₀: 45⋅X₀+9⋅X₁+1528 {O(n)}
t₉, X₁: 9⋅X₀+X₁+236 {O(n)}
t₉, X₂: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₉, X₃: 8⋅X₀+X₁+236 {O(n)}
t₉, X₄: 45⋅X₀+8⋅X₁+1530 {O(n)}
t₁₂, X₀: 45⋅X₀+9⋅X₁+1528 {O(n)}
t₁₂, X₁: 9⋅X₀+X₁+236 {O(n)}
t₁₂, X₂: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₁₂, X₃: 8⋅X₀+X₁+236 {O(n)}
t₁₂, X₄: 16⋅X₁+90⋅X₀+3065 {O(n)}
t₁₄, X₀: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₁₄, X₁: 8⋅X₀+X₁+236 {O(n)}
t₁₄, X₂: 45⋅X₀+8⋅X₁+1528 {O(n)}
t₁₄, X₃: 8⋅X₀+X₁+236 {O(n)}
t₁₄, X₄: 16⋅X₁+90⋅X₀+X₄+3065 {O(n)}
t₁₅, X₀: 45⋅X₀+9⋅X₁+1528 {O(n)}
t₁₅, X₁: 9⋅X₀+X₁+236 {O(n)}
t₁₅, X₂: 45⋅X₀+8⋅X₁+X₂+1528 {O(n)}
t₁₅, X₃: 8⋅X₀+X₁+X₃+236 {O(n)}
t₁₅, X₄: 16⋅X₁+2⋅X₄+90⋅X₀+3065 {O(n)}