KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
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, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ X₃
t₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₃ ≤ X₂
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₁
t₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀+X₂-1-X₃, X₁+X₃-1-X₂, X₂, X₃, X₄, X₅)
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Found invariant X₃ ≤ X₂ ∧ X₂ ≤ X₃ for location eval_foo_bb2_in

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

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
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, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ X₃
t₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₃ ≤ X₂
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀+X₂-1-X₃, X₁+X₃-1-X₂, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [X₁]

TWN: t₇: eval_foo_bb2_in→eval_foo_bb1_in

cycle: [t₇: eval_foo_bb2_in→eval_foo_bb1_in; t₄: eval_foo_bb1_in→eval_foo_bb2_in; t₅: eval_foo_bb1_in→eval_foo_bb2_in]
original loop: (0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∨ 0 ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃,(X₀,X₁,X₂,X₃) -> (X₀+X₂-1-X₃,X₁+X₃-1-X₂,X₂,X₃))
transformed loop: (0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∨ 0 ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃,(X₀,X₁,X₂,X₃) -> (X₀+X₂-1-X₃,X₁+X₃-1-X₂,X₂,X₃))
loop: (0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∨ 0 ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃,(X₀,X₁,X₂,X₃) -> (X₀+X₂-1-X₃,X₁+X₃-1-X₂,X₂,X₃))
order: [X₂; X₃; X₁; X₀]
closed-form:

X₂: X₂
X₃: X₃
X₁: X₁ + [[n != 0]]⋅(X₃-1-X₂)⋅n^1
X₀: X₀ + [[n != 0]]⋅(X₂-1-X₃)⋅n^1

Termination: true
Formula:

X₃ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 1+X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
∨ X₃ ≤ 1+X₂ ∧ 1+X₂ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
∨ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
∨ X₂ ≤ 1+X₃ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
∨ 2+X₃ ≤ X₂ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
∨ 2+X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃

Stabilization-Threshold for: 0 ≤ X₁

alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}

Stabilization-Threshold for: 0 ≤ X₀

alphas_abs: 1+X₀
M: 0
N: 1
Bound: 2⋅X₀+4 {O(n)}

TWN - Lifting for [4: eval_foo_bb1_in->eval_foo_bb2_in; 5: eval_foo_bb1_in->eval_foo_bb2_in; 7: eval_foo_bb2_in->eval_foo_bb1_in] of 2⋅X₀+2⋅X₁+10 {O(n)}

relevant size-bounds w.r.t. t₁: eval_foo_bb0_in→eval_foo_bb1_in:
X₀: X₄ {O(n)}
X₁: X₅ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₄+2⋅X₅+10 {O(n)}

All Bounds

Timebounds

Overall timebound:3⋅X₄+3⋅X₅+18 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: X₄+1 {O(n)}
t₅: X₅+1 {O(n)}
t₆: 1 {O(1)}
t₇: 2⋅X₄+2⋅X₅+10 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₄+3⋅X₅+18 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: X₄+1 {O(n)}
t₅: X₅+1 {O(n)}
t₆: 1 {O(1)}
t₇: 2⋅X₄+2⋅X₅+10 {O(n)}
t₈: 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)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₀: 2⋅X₅+3⋅X₄+10 {O(n)}
t₄, X₁: 2⋅X₄+3⋅X₅+10 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: 2⋅X₅+3⋅X₄+10 {O(n)}
t₅, X₁: 2⋅X₄+3⋅X₅+10 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₆, X₀: 2⋅X₅+4⋅X₄+10 {O(n)}
t₆, X₁: 2⋅X₄+4⋅X₅+10 {O(n)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 2⋅X₃ {O(n)}
t₆, X₄: 2⋅X₄ {O(n)}
t₆, X₅: 2⋅X₅ {O(n)}
t₇, X₀: 2⋅X₅+3⋅X₄+10 {O(n)}
t₇, X₁: 2⋅X₄+3⋅X₅+10 {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₀: 2⋅X₀+2⋅X₅+4⋅X₄+10 {O(n)}
t₈, X₁: 2⋅X₁+2⋅X₄+4⋅X₅+10 {O(n)}
t₈, X₂: 4⋅X₂ {O(n)}
t₈, X₃: 4⋅X₃ {O(n)}
t₈, X₄: 4⋅X₄ {O(n)}
t₈, X₅: 4⋅X₅ {O(n)}