KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: 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:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₀, X₂, X₃) :|: X₀ ≤ 9
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: 10 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, 1+X₁, X₂, X₃)
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃)
t₈: eval_foo_bb5_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Eliminate variables [X₃] that do not contribute to the problem

Found invariant X₂ ≤ X₀ ∧ 10 ≤ X₀ for location eval_foo_bb5_in

Found invariant X₂ ≤ 9 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 18 ∧ X₀ ≤ X₁ ∧ X₀ ≤ 9 for location eval_foo_bb2_in

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₂ ≤ X₀ ∧ 10 ≤ X₀ for location eval_foo_stop

Found invariant X₂ ≤ 9 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 18 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 9 for location eval_foo_bb3_in

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 0 ∧ X₁ ≤ 0 ∧ X₀+X₁ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: 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:
t₁₇: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₀, X₂) :|: X₀ ≤ 9 ∧ X₂ ≤ X₀
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: 10 ≤ X₀ ∧ X₂ ≤ X₀
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₀+X₂ ≤ 18 ∧ X₀ ≤ 9 ∧ X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁
t₂₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₀+X₂ ≤ 18 ∧ X₀ ≤ 9 ∧ X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, 1+X₁, X₂) :|: X₀+X₂ ≤ 18 ∧ X₀ ≤ 9 ∧ X₂ ≤ 9 ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁
t₂₃: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀+X₁ ≤ 0 ∧ X₀+X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁+X₂ ≤ 0 ∧ X₂ ≤ 0
t₂₄: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: 10 ≤ X₀ ∧ X₂ ≤ X₀
t₂₅: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₁₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₀, X₂) :|: X₀ ≤ 9 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+10 {O(n)}

MPRF:

• eval_foo_bb1_in: [10-X₀]
• eval_foo_bb2_in: [9-X₀]
• eval_foo_bb3_in: [9-X₀]
• eval_foo_bb4_in: [9-X₀]

MPRF for transition t₂₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₀+X₂ ≤ 18 ∧ X₀ ≤ 9 ∧ X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₂+10 {O(n)}

MPRF:

• eval_foo_bb1_in: [10-X₀]
• eval_foo_bb2_in: [10-X₀]
• eval_foo_bb3_in: [10-X₀]
• eval_foo_bb4_in: [9-X₀]

MPRF for transition t₂₃: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀+X₁ ≤ 0 ∧ X₀+X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁+X₂ ≤ 0 ∧ X₂ ≤ 0 of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1-X₀]
• eval_foo_bb2_in: [1-X₀]
• eval_foo_bb3_in: [1-X₀]
• eval_foo_bb4_in: [1-X₀]

Found invariant X₂ ≤ 9 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 18 ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 9 ∧ 1 ≤ X₀ for location eval_foo_bb3_in_v2

Found invariant X₂ ≤ X₀ ∧ 10 ≤ X₀ for location eval_foo_bb5_in

Found invariant X₂ ≤ X₀ ∧ 10 ≤ X₀ for location eval_foo_stop

Found invariant X₂ ≤ 9 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 18 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 18 ∧ X₁ ≤ 9 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ X₁ ∧ X₀ ≤ 9 for location eval_foo_bb2_in

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₂ ≤ 9 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 18 ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 9 ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant X₂ ≤ 9 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 18 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 18 ∧ X₁ ≤ 9 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 18 ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 9 ∧ 1 ≤ X₀ for location eval_foo_bb3_in_v1

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 0 ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_foo_bb4_in

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₁₇: 1 {O(1)}
t₁₈: X₂+10 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: inf {Infinity}
t₂₁: X₂+10 {O(n)}
t₂₂: inf {Infinity}
t₂₃: X₂+1 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₁₇: 1 {O(1)}
t₁₈: X₂+10 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: inf {Infinity}
t₂₁: X₂+10 {O(n)}
t₂₂: inf {Infinity}
t₂₃: X₂+1 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₂ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₈, X₀: X₂+1 {O(n)}
t₁₈, X₁: 2⋅X₂+1 {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₉, X₀: X₂ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₂₀, X₀: X₂+1 {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: X₂+1 {O(n)}
t₂₁, X₁: 2⋅X₂+1 {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₂, X₀: X₂+1 {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: X₂+1 {O(n)}
t₂₃, X₁: 2⋅X₂+1 {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: X₂ {O(n)}
t₂₄, X₁: X₁ {O(n)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}