KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalcousot9bb2in

Found invariant X₁ ≤ X₂ ∧ X₁ ≤ 0 for location evalcousot9returnin

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₁ for location evalcousot9bbin

Found invariant X₁ ≤ X₂ ∧ X₁ ≤ 0 for location evalcousot9stop

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalcousot9bb1in

Found invariant X₁ ≤ X₂ for location evalcousot9bb3in

Problem after Preprocessing

Start: evalcousot9start
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: evalcousot9bb1in, evalcousot9bb2in, evalcousot9bb3in, evalcousot9bbin, evalcousot9entryin, evalcousot9returnin, evalcousot9start, evalcousot9stop
Transitions:
t₆: evalcousot9bb1in(X₀, X₁, X₂) → evalcousot9bb3in(X₀-1, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂
t₇: evalcousot9bb2in(X₀, X₁, X₂) → evalcousot9bb3in(X₂, X₁-1, X₂) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₂
t₂: evalcousot9bb3in(X₀, X₁, X₂) → evalcousot9bbin(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₁ ≤ X₂
t₃: evalcousot9bb3in(X₀, X₁, X₂) → evalcousot9returnin(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₁ ≤ X₂
t₄: evalcousot9bbin(X₀, X₁, X₂) → evalcousot9bb1in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂
t₅: evalcousot9bbin(X₀, X₁, X₂) → evalcousot9bb2in(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂
t₁: evalcousot9entryin(X₀, X₁, X₂) → evalcousot9bb3in(D, X₂, X₂)
t₈: evalcousot9returnin(X₀, X₁, X₂) → evalcousot9stop(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₁ ≤ X₂
t₀: evalcousot9start(X₀, X₁, X₂) → evalcousot9entryin(X₀, X₁, X₂)

MPRF for transition t₅: evalcousot9bbin(X₀, X₁, X₂) → evalcousot9bb2in(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• evalcousot9bb1in: [X₁]
• evalcousot9bb2in: [X₁-1]
• evalcousot9bb3in: [X₁]
• evalcousot9bbin: [X₁]

MPRF for transition t₇: evalcousot9bb2in(X₀, X₁, X₂) → evalcousot9bb3in(X₂, X₁-1, X₂) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₂ of depth 1:

new bound:

2⋅X₂ {O(n)}

MPRF:

• evalcousot9bb1in: [X₁+X₂]
• evalcousot9bb2in: [X₁+X₂]
• evalcousot9bb3in: [X₁+X₂]
• evalcousot9bbin: [X₁+X₂]

Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalcousot9bb3in_v3

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalcousot9bbin_v5

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalcousot9bb3in_v1

Found invariant X₁ ≤ X₂ ∧ X₁ ≤ 0 for location evalcousot9stop

Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location evalcousot9bb1in_v2

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalcousot9bb3in_v2

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₁ for location evalcousot9bbin_v1

Found invariant X₁ ≤ X₂ ∧ X₁ ≤ 0 for location evalcousot9returnin

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalcousot9bb1in_v3

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalcousot9bb1in_v1

Found invariant X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location evalcousot9bb3in

Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalcousot9bbin_v4

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location evalcousot9bb2in_v2

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalcousot9bb2in_v1

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalcousot9bbin_v2

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location evalcousot9bb2in_v3

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalcousot9bb3in_v4

Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location evalcousot9bbin_v3

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: inf {Infinity}
t₅: X₂ {O(n)}
t₆: inf {Infinity}
t₇: 2⋅X₂ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: inf {Infinity}
t₅: X₂ {O(n)}
t₆: inf {Infinity}
t₇: 2⋅X₂ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₁: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₁: X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₁: 2⋅X₂ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₄, X₁: X₂ {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₁: X₂ {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₁: X₂ {O(n)}
t₆, X₂: X₂ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: X₂ {O(n)}
t₇, X₂: X₂ {O(n)}
t₈, X₁: 2⋅X₂ {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}