KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₇: eval_foo_0(X₀, X₁, X₂, X₃, X₄) → eval_foo_1(X₀, nondef.0, X₂, X₃, X₄)
t₈: eval_foo_1(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ 0
t₉: eval_foo_1(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁
t₁₀: eval_foo_1(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₁ ∧ X₁ ≤ 0
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1, X₁, X₂, X₃, X₄) :|: 1 ≤ X₃
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₃
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₃ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_0(X₀, X₁, X₂, X₃, X₄)
t₁₁: 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

Eliminate variables [X₂; X₄] 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 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_0

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_1

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ for location eval_foo_bb1_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
Temp_Vars: nondef.0
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₂₃: eval_foo_0(X₀, X₁, X₂) → eval_foo_1(X₀, nondef.0, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₄: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1+X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₅: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₆: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₇: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(1, X₁, X₂) :|: 1 ≤ X₂
t₂₈: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ 0
t₂₉: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂
t₃₀: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂
t₃₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_0(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₃₂: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂)
t₃₃: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂₃: eval_foo_0(X₀, X₁, X₂) → eval_foo_1(X₀, nondef.0, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

• eval_foo_0: [1+X₂-X₀]
• eval_foo_1: [X₂-X₀]
• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [1+X₂-X₀]

MPRF for transition t₂₄: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1+X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

• eval_foo_0: [1+X₂-X₀]
• eval_foo_1: [1+X₂-X₀]
• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [1+X₂-X₀]

MPRF for transition t₂₅: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

• eval_foo_0: [1+X₂-X₀]
• eval_foo_1: [1+X₂-X₀]
• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [1+X₂-X₀]

MPRF for transition t₂₆: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

• eval_foo_0: [1+X₂-X₀]
• eval_foo_1: [1+X₂-X₀]
• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [1+X₂-X₀]

MPRF for transition t₂₉: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ of depth 1:

new bound:

2⋅X₂+2 {O(n)}

MPRF:

• eval_foo_0: [2⋅X₂-X₀]
• eval_foo_1: [2⋅X₂-X₀]
• eval_foo_bb1_in: [1+2⋅X₂-X₀]
• eval_foo_bb2_in: [2⋅X₂-X₀]

MPRF for transition t₃₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_0(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

• eval_foo_0: [X₂-X₀]
• eval_foo_1: [X₂-X₀]
• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [1+X₂-X₀]

All Bounds

Timebounds

Overall timebound:7⋅X₂+17 {O(n)}
t₂₃: X₂+2 {O(n)}
t₂₄: X₂+2 {O(n)}
t₂₅: X₂+2 {O(n)}
t₂₆: X₂+2 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}
t₂₉: 2⋅X₂+2 {O(n)}
t₃₀: 1 {O(1)}
t₃₁: X₂+2 {O(n)}
t₃₂: 1 {O(1)}
t₃₃: 1 {O(1)}

Costbounds

Overall costbound: 7⋅X₂+17 {O(n)}
t₂₃: X₂+2 {O(n)}
t₂₄: X₂+2 {O(n)}
t₂₅: X₂+2 {O(n)}
t₂₆: X₂+2 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}
t₂₉: 2⋅X₂+2 {O(n)}
t₃₀: 1 {O(1)}
t₃₁: X₂+2 {O(n)}
t₃₂: 1 {O(1)}
t₃₃: 1 {O(1)}

Sizebounds

t₂₃, X₀: 3⋅X₂+7 {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: 3⋅X₂+7 {O(n)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: 3⋅X₂+7 {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₆, X₀: 3⋅X₂+7 {O(n)}
t₂₆, X₁: 0 {O(1)}
t₂₆, X₂: X₂ {O(n)}
t₂₇, X₀: 1 {O(1)}
t₂₇, X₁: X₁ {O(n)}
t₂₇, X₂: X₂ {O(n)}
t₂₈, X₀: X₀ {O(n)}
t₂₈, X₁: X₁ {O(n)}
t₂₈, X₂: X₂ {O(n)}
t₂₉, X₀: 3⋅X₂+7 {O(n)}
t₂₉, X₂: X₂ {O(n)}
t₃₀, X₀: 9⋅X₂+21 {O(n)}
t₃₀, X₂: 3⋅X₂ {O(n)}
t₃₁, X₀: 3⋅X₂+7 {O(n)}
t₃₁, X₂: X₂ {O(n)}
t₃₂, X₀: 9⋅X₂+X₀+21 {O(n)}
t₃₂, X₂: 4⋅X₂ {O(n)}
t₃₃, X₀: X₀ {O(n)}
t₃₃, X₁: X₁ {O(n)}
t₃₃, X₂: X₂ {O(n)}