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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, 20, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0
t₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁
t₉: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₁₀: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(0, X₁-1, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₁: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁-1, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁-1, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₃: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(0, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₄: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(0, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₁₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁
t₁₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₁₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₁₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₁₉: eval_foo_bb4_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

Cut unsatisfiable transition [t₇: eval_foo_bb2_in→eval_foo_bb3_in; t₉: eval_foo_bb2_in→eval_foo_bb3_in; t₁₁: eval_foo_bb3_in→eval_foo_bb1_in; t₁₂: eval_foo_bb3_in→eval_foo_bb1_in; t₁₃: eval_foo_bb3_in→eval_foo_bb1_in; t₁₄: eval_foo_bb3_in→eval_foo_bb1_in; t₁₆: eval_foo_bb3_in→eval_foo_bb1_in; t₁₇: eval_foo_bb3_in→eval_foo_bb1_in]

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

Found invariant X₁ ≤ 20 ∧ X₁ ≤ 20+X₀ ∧ X₀+X₁ ≤ 40 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 20 ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ 20 ∧ 0 ≤ X₁ for location eval_foo_bb1_in

Found invariant X₁ ≤ 20 ∧ 0 ≤ X₁ for location eval_foo_stop

Found invariant X₁ ≤ 20 ∧ X₁ ≤ 20+X₀ ∧ X₀+X₁ ≤ 40 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 20 ∧ 0 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ 20 ∧ 0 ≤ X₁ for location eval_foo_bb4_in

Cut unsatisfiable transition [t₇₂: eval_foo_bb3_in→eval_foo_bb1_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_start, eval_foo_stop
Transitions:
t₆₃: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, 20, X₂)
t₆₄: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ 20 ∧ 0 ≤ X₁
t₆₅: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 1+X₀ ≤ 0 ∧ X₁ ≤ 20 ∧ 0 ≤ X₁
t₆₆: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ 20 ∧ 0 ≤ X₁
t₆₇: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₆₈: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁
t₆₉: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁
t₇₀: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb1_in(0, X₁-1, X₂) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₇₁: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁
t₇₃: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: X₁ ≤ 20 ∧ 0 ≤ X₁
t₇₄: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₆₈: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+461 {O(n)}

MPRF:

• eval_foo_bb1_in: [41+21⋅X₁-X₀]
• eval_foo_bb2_in: [41+21⋅X₁-X₀]
• eval_foo_bb3_in: [40+21⋅X₁-X₀]

MPRF for transition t₆₉: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+461 {O(n)}

MPRF:

• eval_foo_bb1_in: [41+21⋅X₁-X₀]
• eval_foo_bb2_in: [41+21⋅X₁-X₀]
• eval_foo_bb3_in: [40+21⋅X₁-X₀]

MPRF for transition t₇₀: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb1_in(0, X₁-1, X₂) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

19⋅X₂+8000 {O(n)}

MPRF:

• eval_foo_bb1_in: [400⋅X₁-19⋅X₀]
• eval_foo_bb2_in: [400⋅X₁-19⋅X₀]
• eval_foo_bb3_in: [1+399⋅X₁-19⋅X₀]

MPRF for transition t₇₁: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ X₀+X₁ ≤ 40 ∧ X₁ ≤ 20+X₀ ∧ X₀ ≤ 20 ∧ X₁ ≤ 20 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₂+401 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+20⋅X₁-X₀]
• eval_foo_bb2_in: [1+20⋅X₁-X₀]
• eval_foo_bb3_in: [1+20⋅X₁-X₀]

Cut unreachable locations [eval_foo_bb2_in] from the program graph

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

Found invariant X₁ ≤ 20 ∧ 0 ≤ X₁ for location eval_foo_stop

Found invariant X₁ ≤ 20 ∧ X₁ ≤ 20+X₀ ∧ X₀+X₁ ≤ 40 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 20 ∧ 0 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v2

Found invariant X₁ ≤ 20 ∧ 0 ≤ X₁ for location eval_foo_bb1_in

Found invariant X₁ ≤ 20 ∧ X₁ ≤ 20+X₀ ∧ X₀+X₁ ≤ 40 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 20 ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant X₁ ≤ 20 ∧ 0 ≤ X₁ for location eval_foo_bb4_in

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₆₃: 1 {O(1)}
t₆₄: inf {Infinity}
t₆₅: 1 {O(1)}
t₆₆: 1 {O(1)}
t₆₇: inf {Infinity}
t₆₈: X₂+461 {O(n)}
t₆₉: X₂+461 {O(n)}
t₇₀: 19⋅X₂+8000 {O(n)}
t₇₁: X₂+401 {O(n)}
t₇₃: 1 {O(1)}
t₇₄: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₆₃: 1 {O(1)}
t₆₄: inf {Infinity}
t₆₅: 1 {O(1)}
t₆₆: 1 {O(1)}
t₆₇: inf {Infinity}
t₆₈: X₂+461 {O(n)}
t₆₉: X₂+461 {O(n)}
t₇₀: 19⋅X₂+8000 {O(n)}
t₇₁: X₂+401 {O(n)}
t₇₃: 1 {O(1)}
t₇₄: 1 {O(1)}

Sizebounds

t₆₃, X₀: X₂ {O(n)}
t₆₃, X₁: 20 {O(1)}
t₆₃, X₂: X₂ {O(n)}
t₆₄, X₀: 20 {O(1)}
t₆₄, X₁: 20 {O(1)}
t₆₄, X₂: X₂ {O(n)}
t₆₅, X₀: X₂ {O(n)}
t₆₅, X₁: 20 {O(1)}
t₆₅, X₂: X₂ {O(n)}
t₆₆, X₀: X₂ {O(n)}
t₆₆, X₁: 20 {O(1)}
t₆₆, X₂: X₂ {O(n)}
t₆₇, X₀: 0 {O(1)}
t₆₇, X₁: 0 {O(1)}
t₆₇, X₂: X₂ {O(n)}
t₆₈, X₀: 20 {O(1)}
t₆₈, X₁: 20 {O(1)}
t₆₈, X₂: X₂ {O(n)}
t₆₉, X₀: 19 {O(1)}
t₆₉, X₁: 20 {O(1)}
t₆₉, X₂: X₂ {O(n)}
t₇₀, X₀: 0 {O(1)}
t₇₀, X₁: 19 {O(1)}
t₇₀, X₂: X₂ {O(n)}
t₇₁, X₀: 20 {O(1)}
t₇₁, X₁: 20 {O(1)}
t₇₁, X₂: X₂ {O(n)}
t₇₃, X₀: 2⋅X₂ {O(n)}
t₇₃, X₁: 20 {O(1)}
t₇₃, X₂: 2⋅X₂ {O(n)}
t₇₄, X₀: X₀ {O(n)}
t₇₄, X₁: X₁ {O(n)}
t₇₄, X₂: X₂ {O(n)}