Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, 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_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₁ ≤ 0
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₂, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₁ ≤ 0
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄) :|: X₁ ≤ 0
t₈: eval_foo_bb3_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

Cut unsatisfiable transition [t₅: eval_foo_bb2_in→eval_foo_bb1_in; t₆: eval_foo_bb2_in→eval_foo_bb1_in]

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

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location eval_foo_bb3_in

Problem after Preprocessing

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_start, eval_foo_stop
Transitions:
t₁₇: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂
t₂₃: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]

TWN: t₂₀: eval_foo_bb2_in→eval_foo_bb1_in

cycle: [t₂₀: eval_foo_bb2_in→eval_foo_bb1_in; t₁₈: eval_foo_bb1_in→eval_foo_bb2_in]
loop: (1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂,(X₀,X₁,X₂) -> (X₀,X₁-1,X₂))
order: [X₂; X₁; X₀]
closed-form:
X₂: X₂
X₁: X₁ + [[n != 0]]⋅-1⋅n^1
X₀: X₀

Termination: true
Formula:

0 ≤ 1 ∧ X₁ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ X₀ ≤ X₂
∨ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂

Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
loop: (1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂,(X₀,X₁,X₂) -> (X₀,X₁-1,X₂))
order: [X₂; X₁; X₀]
closed-form:
X₂: X₂
X₁: X₁ + [[n != 0]]⋅-1⋅n^1
X₀: X₀

Termination: true
Formula:

0 ≤ 1 ∧ X₁ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ X₀ ≤ X₂
∨ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂

Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}

TWN - Lifting for [18: eval_foo_bb1_in->eval_foo_bb2_in; 20: eval_foo_bb2_in->eval_foo_bb1_in] of 2⋅X₁+4 {O(n)}

relevant size-bounds w.r.t. t₁₇: eval_foo_bb0_in→eval_foo_bb1_in:
X₁: X₃ {O(n)}
Runtime-bound of t₁₇: 1 {O(1)}
Results in: 2⋅X₃+4 {O(n)}

TWN - Lifting for [18: eval_foo_bb1_in->eval_foo_bb2_in; 20: eval_foo_bb2_in->eval_foo_bb1_in] of 2⋅X₁+4 {O(n)}

relevant size-bounds w.r.t. t₂₁: eval_foo_bb2_in→eval_foo_bb1_in:
X₁: X₂ {O(n)}
Runtime-bound of t₂₁: X₂ {O(n)}
Results in: 2⋅X₂⋅X₂+4⋅X₂ {O(n^2)}

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 ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v1

Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb1_in_v3

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v5

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb1_in_v2

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location eval_foo_bb3_in

Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v4

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v2

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb1_in_v4

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v3

All Bounds

Timebounds

Overall timebound:4⋅X₂⋅X₂+4⋅X₃+9⋅X₂+12 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₂⋅X₂+2⋅X₃+4⋅X₂+4 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: 2⋅X₂⋅X₂+2⋅X₃+4⋅X₂+4 {O(n^2)}
t₂₁: X₂ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂⋅X₂+4⋅X₃+9⋅X₂+12 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₂⋅X₂+2⋅X₃+4⋅X₂+4 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: 2⋅X₂⋅X₂+2⋅X₃+4⋅X₂+4 {O(n^2)}
t₂₁: X₂ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₂ {O(n)}
t₁₇, X₁: X₃ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: X₂ {O(n)}
t₁₈, X₁: X₂+X₃ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: 2⋅X₂ {O(n)}
t₁₉, X₁: X₂+X₃ {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₂₀, X₀: X₂ {O(n)}
t₂₀, X₁: X₂+X₃ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₂ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: 2⋅X₂ {O(n)}
t₂₂, X₁: X₂+X₃ {O(n)}
t₂₂, X₂: 2⋅X₂ {O(n)}
t₂₂, X₃: 2⋅X₃ {O(n)}
t₂₃, X₀: X₀ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}