KoAT2 Proof Output

Initial Problem

Start: eval_t10_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_t10_0, eval_t10_1, eval_t10_bb0_in, eval_t10_bb1_in, eval_t10_bb2_in, eval_t10_bb3_in, eval_t10_start, eval_t10_stop
Transitions:
t₆: eval_t10_0(X₀, X₁, X₂, X₃, X₄) → eval_t10_1(X₀, X₁, nondef.0, X₃, X₄)
t₇: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀, 1+X₁, X₂, X₃, X₄) :|: 1 ≤ X₂
t₈: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀-1, 1+X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₂ ≤ 0
t₉: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₂ ≤ 0
t₁₀: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0
t₁: eval_t10_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_t10_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀
t₃: eval_t10_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₄: eval_t10_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_0(X₀, X₁, X₂, X₃, X₄)
t₁₁: eval_t10_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_t10_start(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Cut unsatisfiable transition [t₈: eval_t10_1→eval_t10_bb1_in; t₉: eval_t10_1→eval_t10_bb1_in]

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ X₃ for location eval_t10_bb1_in

Found invariant 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ for location eval_t10_bb2_in

Found invariant 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ for location eval_t10_0

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location eval_t10_bb3_in

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location eval_t10_stop

Found invariant 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ for location eval_t10_1

Problem after Preprocessing

Start: eval_t10_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_t10_0, eval_t10_1, eval_t10_bb0_in, eval_t10_bb1_in, eval_t10_bb2_in, eval_t10_bb3_in, eval_t10_start, eval_t10_stop
Transitions:
t₆: eval_t10_0(X₀, X₁, X₂, X₃, X₄) → eval_t10_1(X₀, X₁, nondef.0, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₇: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀, 1+X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₁₀: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₁: eval_t10_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_t10_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₃: eval_t10_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₄: eval_t10_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_0(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₁₁: eval_t10_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₀: eval_t10_start(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂: eval_t10_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_t10_0: [X₀-1-X₁]
• eval_t10_1: [X₀-1-X₁]
• eval_t10_bb1_in: [X₀-X₁]
• eval_t10_bb2_in: [X₀-1-X₁]

MPRF for transition t₄: eval_t10_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_t10_0(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_t10_0: [X₀-1-X₁]
• eval_t10_1: [X₀-1-X₁]
• eval_t10_bb1_in: [X₀-X₁]
• eval_t10_bb2_in: [X₀-X₁]

MPRF for transition t₆: eval_t10_0(X₀, X₁, X₂, X₃, X₄) → eval_t10_1(X₀, X₁, nondef.0, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

2⋅X₃+2⋅X₄+1 {O(n)}

MPRF:

• eval_t10_0: [X₀+X₃-1-2⋅X₁]
• eval_t10_1: [X₀+X₃-2-2⋅X₁]
• eval_t10_bb1_in: [X₀+X₃-1-2⋅X₁]
• eval_t10_bb2_in: [X₀+X₃-1-2⋅X₁]

MPRF for transition t₇: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀, 1+X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_t10_0: [X₃-X₁]
• eval_t10_1: [X₃-X₁]
• eval_t10_bb1_in: [X₃-X₁]
• eval_t10_bb2_in: [X₃-X₁]

MPRF for transition t₁₀: eval_t10_1(X₀, X₁, X₂, X₃, X₄) → eval_t10_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_t10_0: [X₀-X₄]
• eval_t10_1: [X₀-X₄]
• eval_t10_bb1_in: [X₀-X₄]
• eval_t10_bb2_in: [X₀-X₄]

All Bounds

Timebounds

Overall timebound:6⋅X₃+6⋅X₄+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+X₄ {O(n)}
t₃: 1 {O(1)}
t₄: X₃+X₄ {O(n)}
t₆: 2⋅X₃+2⋅X₄+1 {O(n)}
t₇: X₃+X₄ {O(n)}
t₁₀: X₃+X₄ {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₃+6⋅X₄+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+X₄ {O(n)}
t₃: 1 {O(1)}
t₄: X₃+X₄ {O(n)}
t₆: 2⋅X₃+2⋅X₄+1 {O(n)}
t₇: X₃+X₄ {O(n)}
t₁₀: X₃+X₄ {O(n)}
t₁₁: 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)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₃+X₄ {O(n)}
t₂, X₁: 2⋅X₄+X₃ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2⋅X₄+5⋅X₃ {O(n)}
t₃, X₁: 2⋅X₃+5⋅X₄ {O(n)}
t₃, X₃: 3⋅X₃ {O(n)}
t₃, X₄: 3⋅X₄ {O(n)}
t₄, X₀: 2⋅X₃+X₄ {O(n)}
t₄, X₁: 2⋅X₄+X₃ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₆, X₀: 2⋅X₃+X₄ {O(n)}
t₆, X₁: 2⋅X₄+X₃ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₇, X₀: 2⋅X₃+X₄ {O(n)}
t₇, X₁: 2⋅X₄+X₃ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₁₀, X₀: 2⋅X₃+X₄ {O(n)}
t₁₀, X₁: 2⋅X₄+X₃ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₁, X₀: 2⋅X₄+5⋅X₃ {O(n)}
t₁₁, X₁: 2⋅X₃+5⋅X₄ {O(n)}
t₁₁, X₃: 3⋅X₃ {O(n)}
t₁₁, X₄: 3⋅X₄ {O(n)}