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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₃, X₄, X₅, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₁-X₀, X₂, 1+X₂, X₃, X₄, X₅)
t₅: eval_foo_bb3_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

Found invariant X₅ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location eval_foo_bb2_in

Found invariant X₅ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₅ ≤ X₂ ∧ X₀ ≤ X₁ for location eval_foo_stop

Found invariant X₅ ≤ X₂ ∧ X₀ ≤ X₁ for location eval_foo_bb3_in

Problem after Preprocessing

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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₃, X₄, X₅, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₀ ∧ X₅ ≤ X₂
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₁ ∧ X₅ ≤ X₂
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₁-X₀, X₂, 1+X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₀ ∧ X₅ ≤ X₂
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₁ ∧ X₅ ≤ X₂
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

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

new bound:

8⋅X₃+8⋅X₄+8⋅X₅+7 {O(n)}

MPRF:

• eval_foo_bb1_in: [-1-X₂; X₀-X₁]
• eval_foo_bb2_in: [-2-X₂; X₀-2-X₁-X₂]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₁-X₀, X₂, 1+X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₀ ∧ X₅ ≤ X₂ of depth 2:

new bound:

8⋅X₃+8⋅X₄+8⋅X₅+15 {O(n)}

MPRF:

• eval_foo_bb1_in: [-1-X₂; X₀-1-X₁]
• eval_foo_bb2_in: [-1-X₂; 0]

All Bounds

Timebounds

Overall timebound:16⋅X₃+16⋅X₄+16⋅X₅+26 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 8⋅X₃+8⋅X₄+8⋅X₅+7 {O(n)}
t₃: 1 {O(1)}
t₄: 8⋅X₃+8⋅X₄+8⋅X₅+15 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₃+16⋅X₄+16⋅X₅+26 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 8⋅X₃+8⋅X₄+8⋅X₅+7 {O(n)}
t₃: 1 {O(1)}
t₄: 8⋅X₃+8⋅X₄+8⋅X₅+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)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₅ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: 10⋅X₃+10⋅X₄+10⋅X₅+11 {O(n)}
t₂, X₁: 2⋅X₄+2⋅X₅+2 {O(n)}
t₂, X₂: 8⋅X₃+8⋅X₄+9⋅X₅+9 {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: 2⋅X₄+3⋅X₃+3⋅X₅+3 {O(n)}
t₃, X₁: 2⋅X₄+2⋅X₅+2 {O(n)}
t₃, X₂: 3⋅X₅+1 {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₄, X₀: 2⋅X₃+2⋅X₄+3⋅X₅+3 {O(n)}
t₄, X₁: 2⋅X₅+X₄+2 {O(n)}
t₄, X₂: 2⋅X₅+1 {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: 2⋅X₄+3⋅X₃+3⋅X₅+3 {O(n)}
t₅, X₁: 2⋅X₄+2⋅X₅+2 {O(n)}
t₅, X₂: 3⋅X₅+1 {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}
t₅, X₅: 2⋅X₅ {O(n)}