KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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, X₄) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₄, X₂, X₃, X₄)
t₅: 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

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

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location eval_foo_stop

Found invariant X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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, X₄) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ X₂ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₀ ∧ X₂ ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₄, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₀ ∧ X₂ ≤ X₀
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

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

new bound:

16⋅X₂+8⋅X₃+8⋅X₄+7 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₄-1-X₀; X₁-X₀]
• eval_foo_bb2_in: [X₄-2-X₀; X₁+X₄-2-2⋅X₀]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₄, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ of depth 2:

new bound:

16⋅X₂+8⋅X₃+8⋅X₄+15 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₄-1-X₀; X₁-1-X₀]
• eval_foo_bb2_in: [X₄-1-X₀; 0]

All Bounds

Timebounds

Overall timebound:16⋅X₃+16⋅X₄+32⋅X₂+26 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₂+8⋅X₃+8⋅X₄+7 {O(n)}
t₃: 1 {O(1)}
t₄: 16⋅X₂+8⋅X₃+8⋅X₄+15 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₃+16⋅X₄+32⋅X₂+26 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₂+8⋅X₃+8⋅X₄+7 {O(n)}
t₃: 1 {O(1)}
t₄: 16⋅X₂+8⋅X₃+8⋅X₄+15 {O(n)}
t₅: 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: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 17⋅X₂+8⋅X₃+8⋅X₄+9 {O(n)}
t₂, X₁: X₃+X₄ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 3⋅X₂+X₄+1 {O(n)}
t₃, X₁: X₃+X₄ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 2⋅X₂+X₄+1 {O(n)}
t₄, X₁: X₄ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: 3⋅X₂+X₄+1 {O(n)}
t₅, X₁: X₃+X₄ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}