KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₈: eval_foo_0(X₀, X₁, X₂, X₃, X₄) → eval_foo_1(X₀, X₁, nondef.0, X₃, X₄)
t₉: eval_foo_1(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, 0, X₂, X₃, X₄)
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 0
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_0(X₀, X₁, X₂, X₃, X₄)
t₁₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃, X₄)
t₁₁: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ 0 for location eval_foo_bb5_in

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_0

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ X₄ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_1

Found invariant X₁ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₄ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₈: eval_foo_0(X₀, X₁, X₂, X₃, X₄) → eval_foo_1(X₀, X₁, nondef.0, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₉: eval_foo_1(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, 0, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₁ ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₁ ≤ X₄
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₄
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₁ ≤ X₄
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_0(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₁₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄
t₁₁: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₁ ≤ X₄
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_0: [X₁-1]
• eval_foo_1: [X₁-1]
• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁-1]
• eval_foo_bb4_in: [X₁]

MPRF for transition t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_0(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_foo_0: [1]
• eval_foo_1: [1]
• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [1+X₁]
• eval_foo_bb3_in: [2]
• eval_foo_bb4_in: [X₁]

MPRF for transition t₈: eval_foo_0(X₀, X₁, X₂, X₃, X₄) → eval_foo_1(X₀, X₁, nondef.0, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_0: [1]
• eval_foo_1: [0]
• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁]
• eval_foo_bb4_in: [X₁]

MPRF for transition t₉: eval_foo_1(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, 0, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_0: [X₁]
• eval_foo_1: [1]
• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁]
• eval_foo_bb4_in: [X₁]

Cut unreachable locations [eval_foo_bb2_in] from the program graph

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ 0 for location eval_foo_bb5_in

Found invariant 1+X₁ ≤ X₄ ∧ 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_foo_bb4_in_v2

Found invariant 1+X₁ ≤ X₄ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v1

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_0

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant 1+X₁ ≤ X₄ ∧ 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v2

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_1

Found invariant X₁ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₄ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant X₁ ≤ X₄ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_foo_bb4_in_v1

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: inf {Infinity}
t₆: X₄+1 {O(n)}
t₈: X₄ {O(n)}
t₉: X₄ {O(n)}
t₁₀: inf {Infinity}
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: X₄ {O(n)}
t₅: inf {Infinity}
t₆: X₄+1 {O(n)}
t₈: X₄ {O(n)}
t₉: X₄ {O(n)}
t₁₀: inf {Infinity}
t₁₁: 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: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₃: 3⋅X₃ {O(n)}
t₃, X₄: 3⋅X₄ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₄ {O(n)}
t₉, X₁: 0 {O(1)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₁, X₃: 3⋅X₃ {O(n)}
t₁₁, X₄: 3⋅X₄ {O(n)}