KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars: nondef.0
Locations: eval_foo_.critedge_in, eval_foo_2, eval_foo_3, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_start, eval_foo_stop
Transitions:
t₉: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₇: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_3(X₀, X₁, X₂, nondef.0, X₄, X₅, X₆)
t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₁, X₃, X₁, X₃, X₄, X₅, X₆)
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₅, X₆, X₄, X₃, X₄, X₅, X₆)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₁+X₂
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Found invariant X₁+X₂ ≤ 0 ∧ 1 ≤ X₀ for location eval_foo_2

Found invariant X₁+X₂ ≤ 0 ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁+X₂ ≤ 0 ∧ 1 ≤ X₀ for location eval_foo_3

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars: nondef.0
Locations: eval_foo_.critedge_in, eval_foo_2, eval_foo_3, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_start, eval_foo_stop
Transitions:
t₉: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₇: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_3(X₀, X₁, X₂, nondef.0, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0
t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₁, X₃, X₁, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₅, X₆, X₄, X₃, X₄, X₅, X₆)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₁+X₂
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0 of depth 2:

new bound:

24⋅X₅+8⋅X₄+15 {O(n)}

MPRF:

• eval_foo_2: [2⋅X₀-4-X₂; X₁]
• eval_foo_3: [2⋅X₀-4-X₂; X₁]
• eval_foo_bb1_in: [2⋅X₀-2-X₂; X₀]
• eval_foo_bb2_in: [2⋅X₀-4-X₂; X₁]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0 of depth 2:

new bound:

24⋅X₅+8⋅X₄+7 {O(n)}

MPRF:

• eval_foo_2: [X₁-1; X₀+X₁-1]
• eval_foo_3: [X₁-1; X₀+X₁-1]
• eval_foo_bb1_in: [2⋅X₀-1-X₂; X₀]
• eval_foo_bb2_in: [X₁; X₀]

MPRF for transition t₇: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_3(X₀, X₁, X₂, nondef.0, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0 of depth 2:

new bound:

24⋅X₅+8⋅X₄+7 {O(n)}

MPRF:

• eval_foo_2: [-X₂; X₀]
• eval_foo_3: [X₁-1; X₀+X₁-1]
• eval_foo_bb1_in: [2⋅X₀-1-X₂; X₀]
• eval_foo_bb2_in: [-X₂; X₀]

MPRF for transition t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₁, X₃, X₁, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁+X₂ ≤ 0 of depth 2:

new bound:

24⋅X₅+8⋅X₄+23 {O(n)}

MPRF:

• eval_foo_2: [2⋅X₀-2-X₂; 0]
• eval_foo_3: [-X₂; 0]
• eval_foo_bb1_in: [2⋅X₀-2-X₂; X₀-1]
• eval_foo_bb2_in: [2⋅X₀-2-X₂; 0]

All Bounds

Timebounds

Overall timebound:32⋅X₄+96⋅X₅+57 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 24⋅X₅+8⋅X₄+15 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 24⋅X₅+8⋅X₄+7 {O(n)}
t₇: 24⋅X₅+8⋅X₄+7 {O(n)}
t₈: 24⋅X₅+8⋅X₄+23 {O(n)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 32⋅X₄+96⋅X₅+57 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 24⋅X₅+8⋅X₄+15 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 24⋅X₅+8⋅X₄+7 {O(n)}
t₇: 24⋅X₅+8⋅X₄+7 {O(n)}
t₈: 24⋅X₅+8⋅X₄+23 {O(n)}
t₉: 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)}
t₀, X₅: X₅ {O(n)}
t₀, X₆: X₆ {O(n)}
t₁, X₀: X₅ {O(n)}
t₁, X₁: X₆ {O(n)}
t₁, X₂: X₄ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₁, X₆: X₆ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₂, X₆: X₆ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₃, X₆: 2⋅X₆ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₄, X₆: 2⋅X₆ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₅, X₆: X₆ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₇, X₆: X₆ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₅ {O(n)}
t₈, X₆: X₆ {O(n)}
t₉, X₄: 4⋅X₄ {O(n)}
t₉, X₅: 4⋅X₅ {O(n)}
t₉, X₆: 4⋅X₆ {O(n)}