Initial Problem

Start: evalwhile2start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalwhile2bb1in, evalwhile2bb2in, evalwhile2bb3in, evalwhile2bb4in, evalwhile2entryin, evalwhile2returnin, evalwhile2start, evalwhile2stop
Transitions:
t₆: evalwhile2bb1in(X₀, X₁, X₂) → evalwhile2bb2in(X₀, X₁, X₂-1)
t₄: evalwhile2bb2in(X₀, X₁, X₂) → evalwhile2bb1in(X₀, X₁, X₂) :|: 1 ≤ X₂
t₅: evalwhile2bb2in(X₀, X₁, X₂) → evalwhile2bb3in(X₀, X₁, X₂) :|: X₂ ≤ 0
t₇: evalwhile2bb3in(X₀, X₁, X₂) → evalwhile2bb4in(X₀-1, X₁, X₂)
t₂: evalwhile2bb4in(X₀, X₁, X₂) → evalwhile2bb2in(X₀, X₁, X₁) :|: 1 ≤ X₀
t₃: evalwhile2bb4in(X₀, X₁, X₂) → evalwhile2returnin(X₀, X₁, X₂) :|: X₀ ≤ 0
t₁: evalwhile2entryin(X₀, X₁, X₂) → evalwhile2bb4in(X₁, X₁, X₂)
t₈: evalwhile2returnin(X₀, X₁, X₂) → evalwhile2stop(X₀, X₁, X₂)
t₀: evalwhile2start(X₀, X₁, X₂) → evalwhile2entryin(X₀, X₁, X₂)

Preprocessing

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb3in

Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalwhile2stop

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb1in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb2in

Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalwhile2returnin

Found invariant X₀ ≤ X₁ for location evalwhile2bb4in

Problem after Preprocessing

Start: evalwhile2start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalwhile2bb1in, evalwhile2bb2in, evalwhile2bb3in, evalwhile2bb4in, evalwhile2entryin, evalwhile2returnin, evalwhile2start, evalwhile2stop
Transitions:
t₆: evalwhile2bb1in(X₀, X₁, X₂) → evalwhile2bb2in(X₀, X₁, X₂-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁
t₄: evalwhile2bb2in(X₀, X₁, X₂) → evalwhile2bb1in(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₅: evalwhile2bb2in(X₀, X₁, X₂) → evalwhile2bb3in(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₇: evalwhile2bb3in(X₀, X₁, X₂) → evalwhile2bb4in(X₀-1, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₂: evalwhile2bb4in(X₀, X₁, X₂) → evalwhile2bb2in(X₀, X₁, X₁) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁
t₃: evalwhile2bb4in(X₀, X₁, X₂) → evalwhile2returnin(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ X₁
t₁: evalwhile2entryin(X₀, X₁, X₂) → evalwhile2bb4in(X₁, X₁, X₂)
t₈: evalwhile2returnin(X₀, X₁, X₂) → evalwhile2stop(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ X₁
t₀: evalwhile2start(X₀, X₁, X₂) → evalwhile2entryin(X₀, X₁, X₂)

MPRF for transition t₂: evalwhile2bb4in(X₀, X₁, X₂) → evalwhile2bb2in(X₀, X₁, X₁) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

• evalwhile2bb1in: [X₀-1]
• evalwhile2bb2in: [X₀-1]
• evalwhile2bb3in: [X₀-1]
• evalwhile2bb4in: [X₀]

MPRF for transition t₅: evalwhile2bb2in(X₀, X₁, X₂) → evalwhile2bb3in(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₁ {O(n)}

• evalwhile2bb1in: [X₀]
• evalwhile2bb2in: [X₀]
• evalwhile2bb3in: [X₀-1]
• evalwhile2bb4in: [X₀]

MPRF for transition t₇: evalwhile2bb3in(X₀, X₁, X₂) → evalwhile2bb4in(X₀-1, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 of depth 1:

new bound:

X₁ {O(n)}

• evalwhile2bb1in: [X₀]
• evalwhile2bb2in: [X₀]
• evalwhile2bb3in: [X₀]
• evalwhile2bb4in: [X₀]

TWN: t₄: evalwhile2bb2in→evalwhile2bb1in

Show Graph

Termination: true
Formula:

TWN - Lifting for [4: evalwhile2bb2in->evalwhile2bb1in; 6: evalwhile2bb1in->evalwhile2bb2in] of 12⋅X₂+4⋅X₀+4⋅X₁+20 {O(n)}

Cut unsatisfiable transition [t₅: evalwhile2bb2in→evalwhile2bb3in; t₄₆: evalwhile2bb2in→evalwhile2bb3in]

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb1in_v1

Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalwhile2stop

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb2in_v1

Found invariant X₀ ≤ X₁ for location evalwhile2bb4in

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb1in_v2

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb3in

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalwhile2bb2in

Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalwhile2returnin

All Bounds

Timebounds

Overall timebound:64⋅X₁⋅X₁+43⋅X₁+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: 32⋅X₁⋅X₁+20⋅X₁ {O(n^2)}
t₅: X₁ {O(n)}
t₆: 32⋅X₁⋅X₁+20⋅X₁ {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 64⋅X₁⋅X₁+43⋅X₁+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: 32⋅X₁⋅X₁+20⋅X₁ {O(n^2)}
t₅: X₁ {O(n)}
t₆: 32⋅X₁⋅X₁+20⋅X₁ {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₁ {O(n)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 2⋅X₁ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 0 {O(1)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 2⋅X₁ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 0 {O(1)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: X₂ {O(n)}