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)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀+X₁ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₀+X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄, X₅)
t₅: 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
Eliminate variables [X₂] that do not contribute to the problem
Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb2_in
Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb1_in
Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_stop
Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb3_in
Problem after Preprocessing
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:
t₁₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀+X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₆: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)
MPRF for transition t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ of depth 1:
new bound:
X₂+X₃+X₄ {O(n)}
MPRF:
• eval_foo_bb1_in: [X₂-X₁-X₃]
• eval_foo_bb2_in: [X₂-1-X₁-X₃]
TWN: t₁₂: eval_foo_bb1_in→eval_foo_bb2_in
cycle: [t₁₂: eval_foo_bb1_in→eval_foo_bb2_in; t₁₄: eval_foo_bb2_in→eval_foo_bb1_in]
original loop: (1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁,(X₀,X₁,X₂,X₃,X₄) -> (1+X₀,1+X₁,X₂,X₃,X₄))
transformed loop: (1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁,(X₀,X₁,X₂,X₃,X₄) -> (1+X₀,1+X₁,X₂,X₃,X₄))
loop: (1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁,(X₀,X₁,X₂,X₃,X₄) -> (1+X₀,1+X₁,X₂,X₃,X₄))
order: [X₄; X₃; X₂; X₁; X₀]
closed-form:X₄: X₄
X₃: X₃
X₂: X₂
X₁: X₁ + [[n != 0]]⋅n^1
X₀: X₀ + [[n != 0]]⋅n^1
Termination: true
Formula:
0 ≤ 1 ∧ X₂ ≤ 1+X₀+X₁ ∧ 1 ≤ 0 ∧ 1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
∨ 1 ≤ 0 ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
Stabilization-Threshold for: 1+X₀+X₁ ≤ X₂
alphas_abs: X₀+X₁+X₂
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2⋅X₂+2 {O(n)}
TWN - Lifting for [12: eval_foo_bb1_in->eval_foo_bb2_in; 14: eval_foo_bb2_in->eval_foo_bb1_in] of 2⋅X₀+2⋅X₁+2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₁₁: eval_foo_bb0_in→eval_foo_bb1_in:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₁₁: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
All Bounds
Timebounds
Overall timebound:3⋅X₂+3⋅X₃+3⋅X₄+8 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+X₃+X₄ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
Costbounds
Overall costbound: 3⋅X₂+3⋅X₃+3⋅X₄+8 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+X₃+X₄ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
Sizebounds
t₁₁, X₀: X₃ {O(n)}
t₁₁, X₁: X₄ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₂, X₀: 2⋅X₂+2⋅X₄+3⋅X₃+4 {O(n)}
t₁₂, X₁: 2⋅X₂+2⋅X₃+3⋅X₄+4 {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₀: 2⋅X₂+2⋅X₄+4⋅X₃+4 {O(n)}
t₁₃, X₁: 2⋅X₂+2⋅X₃+4⋅X₄+4 {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}
t₁₃, X₃: 2⋅X₃ {O(n)}
t₁₃, X₄: 2⋅X₄ {O(n)}
t₁₄, X₀: 2⋅X₂+2⋅X₄+3⋅X₃+4 {O(n)}
t₁₄, X₁: 2⋅X₂+2⋅X₃+3⋅X₄+4 {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₅, X₀: 2⋅X₂+2⋅X₄+4⋅X₃+4 {O(n)}
t₁₅, X₁: 2⋅X₂+2⋅X₃+4⋅X₄+4 {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₅, X₄: 2⋅X₄ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₆, X₄: X₄ {O(n)}