KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_foo_.critedge_in, eval_foo_3, eval_foo_4, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₁: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₉: eval_foo_3(X₀, X₁, X₂, X₃, X₄) → eval_foo_4(X₀, X₁, nondef.0, X₃, X₄)
t₁₀: eval_foo_4(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₀, X₂, X₃, X₄)
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₄, X₃, X₂, X₃, X₄)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: 100 ≤ X₀
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 99 ∧ 1 ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 9+2⋅X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 10+2⋅X₁ ≤ X₀
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_3(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ 1 ≤ X₀ for location eval_foo_3

Found invariant X₀ ≤ 99 ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ 1 ≤ X₀ for location eval_foo_4

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_foo_.critedge_in, eval_foo_3, eval_foo_4, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₁: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₉: eval_foo_3(X₀, X₁, X₂, X₃, X₄) → eval_foo_4(X₀, X₁, nondef.0, X₃, X₄) :|: X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ 1 ≤ X₀
t₁₀: eval_foo_4(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₀, X₂, X₃, X₄) :|: X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ 1 ≤ X₀
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₄, X₃, X₂, X₃, X₄)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: 100 ≤ X₀
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 99 ∧ 1 ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 9+2⋅X₁ ∧ X₀ ≤ 99 ∧ 1 ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 10+2⋅X₁ ≤ X₀ ∧ X₀ ≤ 99 ∧ 1 ≤ X₀
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_3(X₀, X₁, X₂, X₃, X₄) :|: X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ 1 ≤ X₀
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 10+2⋅X₁ ≤ X₀ ∧ X₀ ≤ 99 ∧ 1 ≤ X₀ of depth 1:

new bound:

106⋅X₃+4718 {O(n)}

MPRF:

• eval_foo_3: [5187-47⋅X₀-12⋅X₁]
• eval_foo_4: [5234-94⋅X₀-12⋅X₁]
• eval_foo_bb1_in: [4718-106⋅X₁]
• eval_foo_bb2_in: [4718-106⋅X₁]
• eval_foo_bb3_in: [5187-47⋅X₀-12⋅X₁]

MPRF for transition t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_3(X₀, X₁, X₂, X₃, X₄) :|: X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ 1 ≤ X₀ of depth 1:

new bound:

160⋅X₃+7118 {O(n)}

MPRF:

• eval_foo_3: [7029-71⋅X₀]
• eval_foo_4: [7029-71⋅X₀]
• eval_foo_bb1_in: [7118-160⋅X₁]
• eval_foo_bb2_in: [7118-160⋅X₁]
• eval_foo_bb3_in: [7888-77⋅X₀-6⋅X₁]

MPRF for transition t₉: eval_foo_3(X₀, X₁, X₂, X₃, X₄) → eval_foo_4(X₀, X₁, nondef.0, X₃, X₄) :|: X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ 1 ≤ X₀ of depth 1:

new bound:

18⋅X₃+802 {O(n)}

MPRF:

• eval_foo_3: [892-9⋅X₀]
• eval_foo_4: [792-8⋅X₀]
• eval_foo_bb1_in: [802-18⋅X₁]
• eval_foo_bb2_in: [802-18⋅X₁]
• eval_foo_bb3_in: [892-9⋅X₀]

MPRF for transition t₁₀: eval_foo_4(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₀, X₂, X₃, X₄) :|: X₀+X₁ ≤ 143 ∧ X₀ ≤ 99 ∧ X₁ ≤ 44 ∧ X₁ ≤ 43+X₀ ∧ 1 ≤ X₀ of depth 1:

new bound:

160⋅X₃+7118 {O(n)}

MPRF:

• eval_foo_3: [7030-71⋅X₀]
• eval_foo_4: [7030-71⋅X₀]
• eval_foo_bb1_in: [7118-160⋅X₁]
• eval_foo_bb2_in: [7118-160⋅X₁]
• eval_foo_bb3_in: [7888-77⋅X₀-6⋅X₁]

knowledge_propagation leads to new time bound 160⋅X₃+7119 {O(n)} for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 99 ∧ 1 ≤ X₀

All Bounds

Timebounds

Overall timebound:604⋅X₃+26881 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 160⋅X₃+7119 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 106⋅X₃+4718 {O(n)}
t₆: 1 {O(1)}
t₇: 160⋅X₃+7118 {O(n)}
t₉: 18⋅X₃+802 {O(n)}
t₁₀: 160⋅X₃+7118 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 604⋅X₃+26881 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 160⋅X₃+7119 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 106⋅X₃+4718 {O(n)}
t₆: 1 {O(1)}
t₇: 160⋅X₃+7118 {O(n)}
t₉: 18⋅X₃+802 {O(n)}
t₁₀: 160⋅X₃+7118 {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: X₄ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 99 {O(1)}
t₂, X₁: X₃+99 {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₁: X₃+99 {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₁: X₃+99 {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₀: 99 {O(1)}
t₅, X₁: X₃+99 {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 99 {O(1)}
t₆, X₁: X₃+99 {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₇, X₀: 99 {O(1)}
t₇, X₁: X₃+99 {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₉, X₀: 99 {O(1)}
t₉, X₁: X₃+99 {O(n)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₁₀, X₁: 99 {O(1)}
t₁₀, X₃: X₃ {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₁, X₁: 3⋅X₃+297 {O(n)}
t₁₁, X₃: 5⋅X₃ {O(n)}
t₁₁, X₄: 5⋅X₄ {O(n)}