KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_bb6_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₁, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 2 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₀-2, X₂, X₃, X₄, X₅)
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 2 ≤ X₁
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁-2, X₂, X₃, X₄, X₅)
t₈: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₁, X₁, X₂, X₃, X₄, X₅)
t₉: eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Eliminate variables [X₂; X₃; X₅] that do not contribute to the problem

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb5_in

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 1 for location eval_foo_stop

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb3_in

Found invariant 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 1 for location eval_foo_bb6_in

Found invariant 4 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 8 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 4 ≤ X₀ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_bb6_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₉: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂)
t₂₀: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 2 ≤ X₀ ∧ X₀ ≤ X₂
t₂₁: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb6_in(X₀, X₁, X₂) :|: X₀ ≤ 1 ∧ X₀ ≤ X₂
t₂₂: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₀-2, X₂) :|: 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₃: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 2 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₂₄: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₂₅: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁-2, X₂) :|: 2+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 2+X₁ ≤ X₂ ∧ 4 ≤ X₀ ∧ 4 ≤ X₂ ∧ 6 ≤ X₀+X₁ ∧ 6 ≤ X₁+X₂ ∧ 8 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₆: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₁, X₁, X₂) :|: X₁ ≤ 1 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₂₇: eval_foo_bb6_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: X₀ ≤ 1 ∧ X₀ ≤ X₂
t₂₈: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂₀: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 2 ≤ X₀ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-1]
• eval_foo_bb2_in: [X₀-2]
• eval_foo_bb3_in: [X₀-2]
• eval_foo_bb4_in: [X₀-2]
• eval_foo_bb5_in: [X₀-2]

MPRF for transition t₂₂: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₀-2, X₂) :|: 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-1]
• eval_foo_bb2_in: [X₀-1]
• eval_foo_bb3_in: [X₁]
• eval_foo_bb4_in: [X₁]
• eval_foo_bb5_in: [X₁]

MPRF for transition t₂₃: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 2 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [1+X₁]
• eval_foo_bb4_in: [X₁]
• eval_foo_bb5_in: [1+X₁]

MPRF for transition t₂₄: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-1]
• eval_foo_bb2_in: [X₀-1]
• eval_foo_bb3_in: [X₀-1]
• eval_foo_bb4_in: [X₀-1]
• eval_foo_bb5_in: [X₀-2]

MPRF for transition t₂₅: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁-2, X₂) :|: 2+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 2+X₁ ≤ X₂ ∧ 4 ≤ X₀ ∧ 4 ≤ X₂ ∧ 6 ≤ X₀+X₁ ∧ 6 ≤ X₁+X₂ ∧ 8 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-2]
• eval_foo_bb2_in: [X₀-2]
• eval_foo_bb3_in: [X₁-1]
• eval_foo_bb4_in: [X₁-1]
• eval_foo_bb5_in: [X₁-1]

MPRF for transition t₂₆: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₁, X₁, X₂) :|: X₁ ≤ 1 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-1]
• eval_foo_bb2_in: [X₀-1]
• eval_foo_bb3_in: [X₀-1]
• eval_foo_bb4_in: [X₀-1]
• eval_foo_bb5_in: [1+X₁]

All Bounds

Timebounds

Overall timebound:6⋅X₂+10 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+1 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: X₂+1 {O(n)}
t₂₃: X₂ {O(n)}
t₂₄: X₂+1 {O(n)}
t₂₅: X₂+2 {O(n)}
t₂₆: X₂+1 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₂+10 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+1 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: X₂+1 {O(n)}
t₂₃: X₂ {O(n)}
t₂₄: X₂+1 {O(n)}
t₂₅: X₂+2 {O(n)}
t₂₆: X₂+1 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Sizebounds

t₁₉, X₀: X₂ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₂₀, X₀: X₂+2 {O(n)}
t₂₀, X₁: X₁+1 {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: X₂+2 {O(n)}
t₂₁, X₁: X₁+1 {O(n)}
t₂₁, X₂: 2⋅X₂ {O(n)}
t₂₂, X₀: X₂+2 {O(n)}
t₂₂, X₁: X₂+2 {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: X₂+2 {O(n)}
t₂₃, X₁: X₂+2 {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: 2⋅X₂+4 {O(n)}
t₂₄, X₁: 1 {O(1)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: X₂+2 {O(n)}
t₂₅, X₁: X₂+2 {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₆, X₀: 2 {O(1)}
t₂₆, X₁: 1 {O(1)}
t₂₆, X₂: X₂ {O(n)}
t₂₇, X₀: X₂+2 {O(n)}
t₂₇, X₁: X₁+1 {O(n)}
t₂₇, X₂: 2⋅X₂ {O(n)}
t₂₈, X₀: X₀ {O(n)}
t₂₈, X₁: X₁ {O(n)}
t₂₈, X₂: X₂ {O(n)}