KoAT2 Proof Output

Initial Problem

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

Preprocessing

Found invariant 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant 1+X₀ ≤ 0 for location eval_foo_stop

Found invariant 1+X₀ ≤ 0 for location eval_foo_bb3_in

Problem after Preprocessing

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

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

new bound:

2⋅X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+2⋅X₀+X₁]
• eval_foo_bb2_in: [1+2⋅X₀+X₁]

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ of depth 1:

new bound:

10⋅X₂⋅X₃⋅X₃+16⋅X₂⋅X₂⋅X₃+2⋅X₃⋅X₃⋅X₃+8⋅X₂⋅X₂⋅X₂+18⋅X₂⋅X₂+23⋅X₂⋅X₃+7⋅X₃⋅X₃+16⋅X₂+9⋅X₃+6 {O(n^3)}

MPRF:

• eval_foo_bb1_in: [2+X₀]
• eval_foo_bb2_in: [1+X₀]

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

new bound:

10⋅X₂⋅X₃⋅X₃+16⋅X₂⋅X₂⋅X₃+2⋅X₃⋅X₃⋅X₃+8⋅X₂⋅X₂⋅X₂+18⋅X₂⋅X₂+23⋅X₂⋅X₃+7⋅X₃⋅X₃+14⋅X₂+8⋅X₃+4 {O(n^3)}

MPRF:

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

Found invariant 1+X₃ ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁ for location eval_foo_bb1_in_v1

Found invariant 1+X₁+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v2

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_foo_bb3_in

Found invariant 1+X₁+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v2

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant 1+X₃ ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v3

Analysing control-flow refined program

MPRF for transition t₃₉: eval_foo_bb1_in_v2(X₀, X₁, X₂, X₃) → eval_foo_bb2_in_v2(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1+X₁+X₃ ≤ 0 ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₂+3⋅X₃+4 {O(n)}

MPRF:

• eval_foo_bb1_in_v1: [1+X₀]
• eval_foo_bb1_in_v2: [2+X₀+X₁]
• eval_foo_bb2_in_v2: [1+X₀+X₁]
• eval_foo_bb2_in_v3: [1+X₀]

MPRF for transition t₄₀: eval_foo_bb2_in_v2(X₀, X₁, X₂, X₃) → eval_foo_bb1_in_v1(X₀+X₁, -X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ 0 ≤ 1+X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1+X₁+X₃ ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₂+3⋅X₃+4 {O(n)}

MPRF:

• eval_foo_bb1_in_v1: [1+X₀]
• eval_foo_bb1_in_v2: [2+X₀+X₁]
• eval_foo_bb2_in_v2: [2+X₀+X₁]
• eval_foo_bb2_in_v3: [1+X₀]

MPRF for transition t₄₂: eval_foo_bb1_in_v1(X₀, X₁, X₂, X₃) → eval_foo_bb2_in_v3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₂+3⋅X₃+3 {O(n)}

MPRF:

• eval_foo_bb1_in_v1: [1+X₀]
• eval_foo_bb1_in_v2: [1+X₀+X₁]
• eval_foo_bb2_in_v2: [1+X₀+X₁]
• eval_foo_bb2_in_v3: [X₀]

MPRF for transition t₄₃: eval_foo_bb2_in_v3(X₀, X₁, X₂, X₃) → eval_foo_bb1_in_v2(X₀+X₁, -1-X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₂+3⋅X₃+3 {O(n)}

MPRF:

• eval_foo_bb1_in_v1: [1+X₀]
• eval_foo_bb1_in_v2: [1+X₀+X₁]
• eval_foo_bb2_in_v2: [1+X₀+X₁]
• eval_foo_bb2_in_v3: [1+X₀]

CFR: Improvement to new bound with the following program:

method: PartialEvaluation new bound:

O(n)

cfr-program:

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb1_in_v1, eval_foo_bb1_in_v2, eval_foo_bb2_in_v1, eval_foo_bb2_in_v2, eval_foo_bb2_in_v3, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₃₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in_v1(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃
t₃₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃
t₄₂: eval_foo_bb1_in_v1(X₀, X₁, X₂, X₃) → eval_foo_bb2_in_v3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂
t₄₁: eval_foo_bb1_in_v1(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ 0 ≤ 1+X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂
t₃₉: eval_foo_bb1_in_v2(X₀, X₁, X₂, X₃) → eval_foo_bb2_in_v2(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1+X₁+X₃ ≤ 0 ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₂
t₃₇: eval_foo_bb2_in_v1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in_v1(X₀+X₁, -X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂
t₃₈: eval_foo_bb2_in_v1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in_v2(X₀+X₁, -1-X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂
t₄₀: eval_foo_bb2_in_v2(X₀, X₁, X₂, X₃) → eval_foo_bb1_in_v1(X₀+X₁, -X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ 0 ≤ 1+X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₃ ∧ 1+X₁+X₃ ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₂
t₄₃: eval_foo_bb2_in_v3(X₀, X₁, X₂, X₃) → eval_foo_bb1_in_v2(X₀+X₁, -1-X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

All Bounds

Timebounds

Overall timebound:12⋅X₃+8⋅X₂+23 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₃: 1 {O(1)}
t₆: 1 {O(1)}
t₃₅: 1 {O(1)}
t₃₆: 1 {O(1)}
t₃₇: 1 {O(1)}
t₃₈: 1 {O(1)}
t₃₉: 2⋅X₂+3⋅X₃+4 {O(n)}
t₄₀: 2⋅X₂+3⋅X₃+4 {O(n)}
t₄₁: 1 {O(1)}
t₄₂: 2⋅X₂+3⋅X₃+3 {O(n)}
t₄₃: 2⋅X₂+3⋅X₃+3 {O(n)}

Costbounds

Overall costbound: 12⋅X₃+8⋅X₂+23 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₃: 1 {O(1)}
t₆: 1 {O(1)}
t₃₅: 1 {O(1)}
t₃₆: 1 {O(1)}
t₃₇: 1 {O(1)}
t₃₈: 1 {O(1)}
t₃₉: 2⋅X₂+3⋅X₃+4 {O(n)}
t₄₀: 2⋅X₂+3⋅X₃+4 {O(n)}
t₄₁: 1 {O(1)}
t₄₂: 2⋅X₂+3⋅X₃+3 {O(n)}
t₄₃: 2⋅X₂+3⋅X₃+3 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₆, X₀: 3⋅X₃+5⋅X₂ {O(n)}
t₆, X₁: 2⋅X₂+9⋅X₃+6 {O(n)}
t₆, X₂: 5⋅X₂ {O(n)}
t₆, X₃: 5⋅X₃ {O(n)}
t₃₅, X₀: X₂ {O(n)}
t₃₅, X₁: X₃ {O(n)}
t₃₅, X₂: X₂ {O(n)}
t₃₅, X₃: X₃ {O(n)}
t₃₆, X₀: X₂ {O(n)}
t₃₆, X₁: X₃ {O(n)}
t₃₆, X₂: X₂ {O(n)}
t₃₆, X₃: X₃ {O(n)}
t₃₇, X₀: X₂+X₃ {O(n)}
t₃₇, X₁: X₃ {O(n)}
t₃₇, X₂: X₂ {O(n)}
t₃₇, X₃: X₃ {O(n)}
t₃₈, X₀: X₂+X₃ {O(n)}
t₃₈, X₁: X₃+1 {O(n)}
t₃₈, X₂: X₂ {O(n)}
t₃₈, X₃: X₃ {O(n)}
t₃₉, X₀: 2⋅X₂+2⋅X₃ {O(n)}
t₃₉, X₁: 2⋅X₂+6⋅X₃+6 {O(n)}
t₃₉, X₂: 2⋅X₂ {O(n)}
t₃₉, X₃: 2⋅X₃ {O(n)}
t₄₀, X₀: 2⋅X₂+2⋅X₃ {O(n)}
t₄₀, X₁: 2⋅X₂+6⋅X₃+6 {O(n)}
t₄₀, X₂: 2⋅X₂ {O(n)}
t₄₀, X₃: 2⋅X₃ {O(n)}
t₄₁, X₀: 3⋅X₂+3⋅X₃ {O(n)}
t₄₁, X₁: 2⋅X₂+7⋅X₃+6 {O(n)}
t₄₁, X₂: 3⋅X₂ {O(n)}
t₄₁, X₃: 3⋅X₃ {O(n)}
t₄₂, X₀: 4⋅X₂+4⋅X₃+4 {O(n)}
t₄₂, X₁: 2⋅X₂+4⋅X₃+5 {O(n)}
t₄₂, X₂: 2⋅X₂ {O(n)}
t₄₂, X₃: 2⋅X₃ {O(n)}
t₄₃, X₀: 4⋅X₂+4⋅X₃+4 {O(n)}
t₄₃, X₁: 2⋅X₂+4⋅X₃+5 {O(n)}
t₄₃, X₂: 2⋅X₂ {O(n)}
t₄₃, X₃: 2⋅X₃ {O(n)}