KoAT2 Proof Output

Initial Problem

Preprocessing

Cut unsatisfiable transition [t₉: evalsipma91bb5in→evalsipma91bb8in; t₁₃: evalsipma91bb8in→evalsipma91bb11in; t₁₄: evalsipma91bb8in→evalsipma91bb11in]

Found invariant X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location evalsipma91bb5in

Found invariant 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 92 ≤ X₂+X₃ ∧ X₂ ≤ 99+X₃ ∧ 102 ≤ X₁+X₃ ∧ X₁ ≤ 109+X₃ ∧ 3 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ X₂ ≤ 101 ∧ 10+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 212 ∧ X₂ ≤ 98+X₀ ∧ 91 ≤ X₂ ∧ 192 ≤ X₁+X₂ ∧ X₁ ≤ 10+X₂ ∧ 93 ≤ X₀+X₂ ∧ X₁ ≤ 111 ∧ X₁ ≤ 108+X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location evalsipma91bb8in

Found invariant 1 ≤ X₀ for location evalsipma91returnin

Found invariant 1 ≤ X₀ for location evalsipma91stop

Found invariant X₁ ≤ 100 ∧ X₁ ≤ 99+X₀ ∧ 1 ≤ X₀ for location evalsipma91bb2in

Found invariant X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 1 ≤ X₀ for location evalsipma91bb3in

Found invariant X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 101 ≤ X₁ ∧ 102 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalsipma91bb11in

Problem after Preprocessing

Start: evalsipma91start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: evalsipma91bb11in, evalsipma91bb2in, evalsipma91bb3in, evalsipma91bb5in, evalsipma91bb8in, evalsipma91entryin, evalsipma91returnin, evalsipma91start, evalsipma91stop
Transitions:
t₆: evalsipma91bb11in(X₀, X₁, X₂, X₃) → evalsipma91bb5in(X₀, X₁, X₂, X₃) :|: 2 ≤ X₀ ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 1 ≤ X₀ ∧ 101 ≤ X₁ ∧ 102 ≤ X₀+X₁
t₇: evalsipma91bb11in(X₀, X₁, X₂, X₃) → evalsipma91returnin(X₀, X₁, X₂, X₃) :|: X₀ ≤ 1 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 1 ≤ X₀ ∧ 101 ≤ X₁ ∧ 102 ≤ X₀+X₁
t₅: evalsipma91bb2in(X₀, X₁, X₂, X₃) → evalsipma91bb3in(1+X₀, 11+X₁, X₂, X₃) :|: X₁ ≤ 100 ∧ X₁ ≤ 99+X₀ ∧ 1 ≤ X₀
t₄: evalsipma91bb3in(X₀, X₁, X₂, X₃) → evalsipma91bb11in(X₀, X₁, X₂, X₃) :|: 101 ≤ X₁ ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 1 ≤ X₀
t₃: evalsipma91bb3in(X₀, X₁, X₂, X₃) → evalsipma91bb2in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 100 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 1 ≤ X₀
t₁₁: evalsipma91bb5in(X₀, X₁, X₂, X₃) → evalsipma91bb11in(X₀-1, X₁-10, X₂, X₃) :|: X₀ ≤ 2 ∧ 2 ≤ X₀ ∧ 111 ≤ X₁ ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁
t₈: evalsipma91bb5in(X₀, X₁, X₂, X₃) → evalsipma91bb8in(X₀, X₁, X₁-10, X₀-1) :|: X₁ ≤ 110 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 2 ≤ X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁
t₁₀: evalsipma91bb5in(X₀, X₁, X₂, X₃) → evalsipma91bb8in(X₀, X₁, X₁-10, X₀-1) :|: 3 ≤ X₀ ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 2 ≤ X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁
t₁₂: evalsipma91bb8in(X₀, X₁, X₂, X₃) → evalsipma91bb11in(X₃, 1+X₂, X₂, X₃) :|: 101 ≤ X₂ ∧ X₁+X₂ ≤ 212 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₃ ∧ X₁ ≤ 108+X₀ ∧ X₂ ≤ 101 ∧ X₂ ≤ 99+X₃ ∧ X₂ ≤ 98+X₀ ∧ X₁ ≤ 10+X₂ ∧ X₀ ≤ 1+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₃ ∧ 10+X₂ ≤ X₁ ∧ 91 ≤ X₂ ∧ 92 ≤ X₂+X₃ ∧ 93 ≤ X₀+X₂ ∧ 101 ≤ X₁ ∧ 102 ≤ X₁+X₃ ∧ 103 ≤ X₀+X₁ ∧ 192 ≤ X₁+X₂
t₁₅: evalsipma91bb8in(X₀, X₁, X₂, X₃) → evalsipma91bb11in(1+X₃, 11+X₂, X₂, X₃) :|: X₂ ≤ 100 ∧ X₁+X₂ ≤ 212 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₃ ∧ X₁ ≤ 108+X₀ ∧ X₂ ≤ 101 ∧ X₂ ≤ 99+X₃ ∧ X₂ ≤ 98+X₀ ∧ X₁ ≤ 10+X₂ ∧ X₀ ≤ 1+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₃ ∧ 10+X₂ ≤ X₁ ∧ 91 ≤ X₂ ∧ 92 ≤ X₂+X₃ ∧ 93 ≤ X₀+X₂ ∧ 101 ≤ X₁ ∧ 102 ≤ X₁+X₃ ∧ 103 ≤ X₀+X₁ ∧ 192 ≤ X₁+X₂
t₂: evalsipma91entryin(X₀, X₁, X₂, X₃) → evalsipma91bb3in(1, X₀, X₂, X₃) :|: X₀ ≤ 100
t₁: evalsipma91entryin(X₀, X₁, X₂, X₃) → evalsipma91returnin(X₀, X₁, X₂, X₃) :|: 101 ≤ X₀
t₁₆: evalsipma91returnin(X₀, X₁, X₂, X₃) → evalsipma91stop(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀
t₀: evalsipma91start(X₀, X₁, X₂, X₃) → evalsipma91entryin(X₀, X₁, X₂, X₃)

MPRF for transition t₃: evalsipma91bb3in(X₀, X₁, X₂, X₃) → evalsipma91bb2in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 100 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀+101 {O(n)}

MPRF:

• evalsipma91bb2in: [90-X₁]
• evalsipma91bb3in: [101-X₁]

MPRF for transition t₅: evalsipma91bb2in(X₀, X₁, X₂, X₃) → evalsipma91bb3in(1+X₀, 11+X₁, X₂, X₃) :|: X₁ ≤ 100 ∧ X₁ ≤ 99+X₀ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀+101 {O(n)}

MPRF:

• evalsipma91bb2in: [91+10⋅X₀-X₁]
• evalsipma91bb3in: [91+10⋅X₀-X₁]

MPRF for transition t₆: evalsipma91bb11in(X₀, X₁, X₂, X₃) → evalsipma91bb5in(X₀, X₁, X₂, X₃) :|: 2 ≤ X₀ ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 1 ≤ X₀ ∧ 101 ≤ X₁ ∧ 102 ≤ X₀+X₁ of depth 1:

new bound:

11⋅X₀+1334 {O(n)}

MPRF:

• evalsipma91bb11in: [101+11⋅X₀-X₁]
• evalsipma91bb5in: [100+11⋅X₀-X₁]
• evalsipma91bb8in: [101⋅X₀-X₂-90⋅X₃]

MPRF for transition t₈: evalsipma91bb5in(X₀, X₁, X₂, X₃) → evalsipma91bb8in(X₀, X₁, X₁-10, X₀-1) :|: X₁ ≤ 110 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 2 ≤ X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁ of depth 1:

new bound:

10⋅X₀+1313 {O(n)}

MPRF:

• evalsipma91bb11in: [182+10⋅X₀-X₁]
• evalsipma91bb5in: [182+10⋅X₀-X₁]
• evalsipma91bb8in: [191+10⋅X₃-X₁]

MPRF for transition t₁₀: evalsipma91bb5in(X₀, X₁, X₂, X₃) → evalsipma91bb8in(X₀, X₁, X₁-10, X₀-1) :|: 3 ≤ X₀ ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 2 ≤ X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁ of depth 1:

new bound:

10⋅X₀+1323 {O(n)}

MPRF:

• evalsipma91bb11in: [192+10⋅X₀-X₁]
• evalsipma91bb5in: [192+10⋅X₀-X₁]
• evalsipma91bb8in: [181+10⋅X₀-X₂]

MPRF for transition t₁₁: evalsipma91bb5in(X₀, X₁, X₂, X₃) → evalsipma91bb11in(X₀-1, X₁-10, X₂, X₃) :|: X₀ ≤ 2 ∧ 2 ≤ X₀ ∧ 111 ≤ X₁ ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₀ ∧ 101 ≤ X₁ ∧ 103 ≤ X₀+X₁ of depth 1:

new bound:

10⋅X₀+1101 {O(n)}

MPRF:

• evalsipma91bb11in: [81+10⋅X₀]
• evalsipma91bb5in: [81+10⋅X₀]
• evalsipma91bb8in: [81+10⋅X₀]

MPRF for transition t₁₂: evalsipma91bb8in(X₀, X₁, X₂, X₃) → evalsipma91bb11in(X₃, 1+X₂, X₂, X₃) :|: 101 ≤ X₂ ∧ X₁+X₂ ≤ 212 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₃ ∧ X₁ ≤ 108+X₀ ∧ X₂ ≤ 101 ∧ X₂ ≤ 99+X₃ ∧ X₂ ≤ 98+X₀ ∧ X₁ ≤ 10+X₂ ∧ X₀ ≤ 1+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₃ ∧ 10+X₂ ≤ X₁ ∧ 91 ≤ X₂ ∧ 92 ≤ X₂+X₃ ∧ 93 ≤ X₀+X₂ ∧ 101 ≤ X₁ ∧ 102 ≤ X₁+X₃ ∧ 103 ≤ X₀+X₁ ∧ 192 ≤ X₁+X₂ of depth 1:

new bound:

X₀+103 {O(n)}

MPRF:

• evalsipma91bb11in: [X₀-1]
• evalsipma91bb5in: [X₀-1]
• evalsipma91bb8in: [X₃]

MPRF for transition t₁₅: evalsipma91bb8in(X₀, X₁, X₂, X₃) → evalsipma91bb11in(1+X₃, 11+X₂, X₂, X₃) :|: X₂ ≤ 100 ∧ X₁+X₂ ≤ 212 ∧ X₁ ≤ 111 ∧ X₁ ≤ 109+X₃ ∧ X₁ ≤ 108+X₀ ∧ X₂ ≤ 101 ∧ X₂ ≤ 99+X₃ ∧ X₂ ≤ 98+X₀ ∧ X₁ ≤ 10+X₂ ∧ X₀ ≤ 1+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 3 ≤ X₀+X₃ ∧ 10+X₂ ≤ X₁ ∧ 91 ≤ X₂ ∧ 92 ≤ X₂+X₃ ∧ 93 ≤ X₀+X₂ ∧ 101 ≤ X₁ ∧ 102 ≤ X₁+X₃ ∧ 103 ≤ X₀+X₁ ∧ 192 ≤ X₁+X₂ of depth 1:

new bound:

10⋅X₀+1313 {O(n)}

MPRF:

• evalsipma91bb11in: [182+10⋅X₀-X₁]
• evalsipma91bb5in: [182+10⋅X₀-X₁]
• evalsipma91bb8in: [192+10⋅X₃-X₁]

All Bounds

Timebounds

Overall timebound:54⋅X₀+6695 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+101 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+101 {O(n)}
t₆: 11⋅X₀+1334 {O(n)}
t₇: 1 {O(1)}
t₈: 10⋅X₀+1313 {O(n)}
t₁₀: 10⋅X₀+1323 {O(n)}
t₁₁: 10⋅X₀+1101 {O(n)}
t₁₂: X₀+103 {O(n)}
t₁₅: 10⋅X₀+1313 {O(n)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 54⋅X₀+6695 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+101 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+101 {O(n)}
t₆: 11⋅X₀+1334 {O(n)}
t₇: 1 {O(1)}
t₈: 10⋅X₀+1313 {O(n)}
t₁₀: 10⋅X₀+1323 {O(n)}
t₁₁: 10⋅X₀+1101 {O(n)}
t₁₂: X₀+103 {O(n)}
t₁₅: 10⋅X₀+1313 {O(n)}
t₁₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 1 {O(1)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀+102 {O(n)}
t₃, X₁: 12⋅X₀+1111 {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₀+102 {O(n)}
t₄, X₁: 111 {O(1)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₀+102 {O(n)}
t₅, X₁: 12⋅X₀+1111 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₀+102 {O(n)}
t₆, X₁: 111 {O(1)}
t₆, X₂: X₂+201 {O(n)}
t₆, X₃: 3⋅X₀+X₃+306 {O(n)}
t₇, X₀: 1 {O(1)}
t₇, X₁: 110 {O(1)}
t₇, X₂: 2⋅X₂+201 {O(n)}
t₇, X₃: 2⋅X₃+3⋅X₀+306 {O(n)}
t₈, X₀: X₀+102 {O(n)}
t₈, X₁: 110 {O(1)}
t₈, X₂: 100 {O(1)}
t₈, X₃: X₀+102 {O(n)}
t₁₀, X₀: X₀+102 {O(n)}
t₁₀, X₁: 111 {O(1)}
t₁₀, X₂: 101 {O(1)}
t₁₀, X₃: X₀+102 {O(n)}
t₁₁, X₀: 1 {O(1)}
t₁₁, X₁: 101 {O(1)}
t₁₁, X₂: X₂+201 {O(n)}
t₁₁, X₃: 3⋅X₀+X₃+306 {O(n)}
t₁₂, X₀: X₀+102 {O(n)}
t₁₂, X₁: 102 {O(1)}
t₁₂, X₂: 101 {O(1)}
t₁₂, X₃: X₀+102 {O(n)}
t₁₅, X₀: X₀+102 {O(n)}
t₁₅, X₁: 111 {O(1)}
t₁₅, X₂: 100 {O(1)}
t₁₅, X₃: 2⋅X₀+204 {O(n)}
t₁₆, X₀: X₀+1 {O(n)}
t₁₆, X₁: X₁+110 {O(n)}
t₁₆, X₂: 3⋅X₂+201 {O(n)}
t₁₆, X₃: 3⋅X₀+3⋅X₃+306 {O(n)}