KoAT2 Proof Output

Initial Problem

Start: evalEx3start
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: evalEx3bb1in, evalEx3bb2in, evalEx3bb3in, evalEx3bb4in, evalEx3bbin, evalEx3entryin, evalEx3returnin, evalEx3start, evalEx3stop
Transitions:
t₁₀: evalEx3bb1in(X₀, X₁, X₂) → evalEx3bb2in(X₀, X₁, X₂-1)
t₆: evalEx3bb2in(X₀, X₁, X₂) → evalEx3bb3in(X₀, X₁, X₂) :|: 1 ≤ X₂
t₅: evalEx3bb2in(X₀, X₁, X₂) → evalEx3bb4in(X₂, X₁, X₂) :|: X₂ ≤ 0
t₇: evalEx3bb3in(X₀, X₁, X₂) → evalEx3bb1in(X₀, X₁, X₂)
t₈: evalEx3bb3in(X₀, X₁, X₂) → evalEx3bb4in(X₂, X₁, X₂) :|: 1+D ≤ X₁
t₉: evalEx3bb3in(X₀, X₁, X₂) → evalEx3bb4in(X₂, X₁, X₂) :|: 1+X₁ ≤ D
t₂: evalEx3bb4in(X₀, X₁, X₂) → evalEx3bbin(X₀, X₁, X₂) :|: 1 ≤ X₀
t₃: evalEx3bb4in(X₀, X₁, X₂) → evalEx3returnin(X₀, X₁, X₂) :|: X₀ ≤ 0
t₄: evalEx3bbin(X₀, X₁, X₂) → evalEx3bb2in(X₀, D, X₀)
t₁: evalEx3entryin(X₀, X₁, X₂) → evalEx3bb4in(X₀, X₁, X₂)
t₁₁: evalEx3returnin(X₀, X₁, X₂) → evalEx3stop(X₀, X₁, X₂)
t₀: evalEx3start(X₀, X₁, X₂) → evalEx3entryin(X₀, X₁, X₂)

Preprocessing

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location evalEx3bb1in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location evalEx3bb3in

Found invariant X₀ ≤ 0 for location evalEx3returnin

Found invariant X₀ ≤ 0 for location evalEx3stop

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location evalEx3bb2in

Found invariant 1 ≤ X₀ for location evalEx3bbin

Problem after Preprocessing

Start: evalEx3start
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: evalEx3bb1in, evalEx3bb2in, evalEx3bb3in, evalEx3bb4in, evalEx3bbin, evalEx3entryin, evalEx3returnin, evalEx3start, evalEx3stop
Transitions:
t₁₀: evalEx3bb1in(X₀, X₁, X₂) → evalEx3bb2in(X₀, X₁, X₂-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀
t₆: evalEx3bb2in(X₀, X₁, X₂) → evalEx3bb3in(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂
t₅: evalEx3bb2in(X₀, X₁, X₂) → evalEx3bb4in(X₂, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂
t₇: evalEx3bb3in(X₀, X₁, X₂) → evalEx3bb1in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀
t₈: evalEx3bb3in(X₀, X₁, X₂) → evalEx3bb4in(X₂, X₁, X₂) :|: 1+D ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀
t₉: evalEx3bb3in(X₀, X₁, X₂) → evalEx3bb4in(X₂, X₁, X₂) :|: 1+X₁ ≤ D ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀
t₂: evalEx3bb4in(X₀, X₁, X₂) → evalEx3bbin(X₀, X₁, X₂) :|: 1 ≤ X₀
t₃: evalEx3bb4in(X₀, X₁, X₂) → evalEx3returnin(X₀, X₁, X₂) :|: X₀ ≤ 0
t₄: evalEx3bbin(X₀, X₁, X₂) → evalEx3bb2in(X₀, D, X₀) :|: 1 ≤ X₀
t₁: evalEx3entryin(X₀, X₁, X₂) → evalEx3bb4in(X₀, X₁, X₂)
t₁₁: evalEx3returnin(X₀, X₁, X₂) → evalEx3stop(X₀, X₁, X₂) :|: X₀ ≤ 0
t₀: evalEx3start(X₀, X₁, X₂) → evalEx3entryin(X₀, X₁, X₂)

MPRF for transition t₅: evalEx3bb2in(X₀, X₁, X₂) → evalEx3bb4in(X₂, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₀ {O(n)}

MPRF:

• evalEx3bb1in: [2⋅X₀]
• evalEx3bb2in: [2⋅X₀]
• evalEx3bb3in: [2⋅X₀]
• evalEx3bb4in: [2⋅X₀]
• evalEx3bbin: [2⋅X₀]

MPRF for transition t₇: evalEx3bb3in(X₀, X₁, X₂) → evalEx3bb1in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalEx3bb1in: [X₂-1]
• evalEx3bb2in: [X₂]
• evalEx3bb3in: [X₂]
• evalEx3bb4in: [X₀]
• evalEx3bbin: [X₀]

MPRF for transition t₁₀: evalEx3bb1in(X₀, X₁, X₂) → evalEx3bb2in(X₀, X₁, X₂-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalEx3bb1in: [X₂]
• evalEx3bb2in: [X₂]
• evalEx3bb3in: [X₂]
• evalEx3bb4in: [X₀]
• evalEx3bbin: [X₀]

Cut unreachable locations [evalEx3bb3in] from the program graph

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location evalEx3bb1in

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location evalEx3bb4in_v1

Found invariant X₀ ≤ 0 for location evalEx3returnin

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location evalEx3bbin_v1

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location evalEx3bb3in_v2

Found invariant X₀ ≤ 0 for location evalEx3stop

Found invariant 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location evalEx3bb2in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location evalEx3bb2in_v1

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location evalEx3bb4in_v2

Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 2 ≤ X₀ for location evalEx3bb3in_v1

Found invariant 1 ≤ X₀ for location evalEx3bbin_v2

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: 2⋅X₀ {O(n)}
t₆: inf {Infinity}
t₇: X₀ {O(n)}
t₈: inf {Infinity}
t₉: inf {Infinity}
t₁₀: X₀ {O(n)}
t₁₁: 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: 2⋅X₀ {O(n)}
t₆: inf {Infinity}
t₇: X₀ {O(n)}
t₈: inf {Infinity}
t₉: inf {Infinity}
t₁₀: X₀ {O(n)}
t₁₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₂: 2⋅X₀+X₂ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₂: X₀ {O(n)}
t₅, X₀: 0 {O(1)}
t₅, X₂: 0 {O(1)}
t₆, X₀: X₀ {O(n)}
t₆, X₂: X₀ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₂: X₀ {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₂: X₀ {O(n)}
t₉, X₀: X₀ {O(n)}
t₉, X₂: X₀ {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₂: X₀ {O(n)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₂: X₂ {O(n)}