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, 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) :|: 1 ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₁-1, X₀, 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

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

Found invariant 1 ≤ X₁ for location eval_foo_bb2_in

Found invariant X₁ ≤ 0 for location eval_foo_stop

Found invariant X₁ ≤ 0 for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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:
t₁₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁
t₁₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0
t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₁-1, X₀, X₂, X₃) :|: 1 ≤ X₁
t₁₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0
t₁₆: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ of depth 2:

new bound:

16⋅X₃+8⋅X₂+7 {O(n)}

MPRF:

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

MPRF for transition t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₁-1, X₀, X₂, X₃) :|: 1 ≤ X₁ of depth 2:

new bound:

16⋅X₃+8⋅X₂+15 {O(n)}

MPRF:

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

All Bounds

Timebounds

Overall timebound:16⋅X₂+32⋅X₃+26 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 16⋅X₃+8⋅X₂+7 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 16⋅X₃+8⋅X₂+15 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₂+32⋅X₃+26 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 16⋅X₃+8⋅X₂+7 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 16⋅X₃+8⋅X₂+15 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₂ {O(n)}
t₁₁, X₁: X₃ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₂, X₀: X₂+X₃ {O(n)}
t₁₂, X₁: X₂+X₃ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: 2⋅X₂+X₃ {O(n)}
t₁₃, X₁: 2⋅X₃+X₂ {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}
t₁₃, X₃: 2⋅X₃ {O(n)}
t₁₄, X₀: X₂+X₃ {O(n)}
t₁₄, X₁: X₂+X₃ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₅, X₀: 2⋅X₂+X₃ {O(n)}
t₁₅, X₁: 2⋅X₃+X₂ {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}