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_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) → eval_foo_bb1_in(X₂, X₁, X₂, X₃, X₄) :|: 0 ≤ X₂
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₂ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, 1, X₂, X₃, X₄) :|: 2 ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 1
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, 2⋅X₁, X₂, X₃, X₄)
t₁₀: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃, X₄)
t₁₁: eval_foo_bb6_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

Eliminate variables [X₃; X₄] that do not contribute to the problem

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb5_in

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb3_in

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant 0 ≤ X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ 1+X₀ for location eval_foo_bb1_in

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ 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:
t₂₃: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂) :|: 0 ≤ X₂
t₂₄: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb6_in(X₀, X₁, X₂) :|: 1+X₂ ≤ 0
t₂₅: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₂₆: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb6_in(X₀, X₁, X₂) :|: 1+X₀ ≤ 0 ∧ 0 ≤ 1+X₀ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₂₇: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, 1, X₂) :|: 2 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₂₈: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: X₀ ≤ 1 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₂₉: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₃₀: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₃₁: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, 2⋅X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₃₂: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-1, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₃₃: eval_foo_bb6_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂)
t₃₄: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂₅: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ 1+X₀ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

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

MPRF for transition t₂₇: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, 1, X₂) :|: 2 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

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

MPRF for transition t₂₈: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: X₀ ≤ 1 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀+X₂]
• eval_foo_bb2_in: [1+X₀+X₂]
• eval_foo_bb3_in: [X₀+X₂]
• eval_foo_bb4_in: [X₀+X₂]
• eval_foo_bb5_in: [X₀+X₂]

MPRF for transition t₃₀: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+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: [X₀-1]
• eval_foo_bb4_in: [X₀-1]
• eval_foo_bb5_in: [X₀-2]

MPRF for transition t₃₂: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-1, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

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

MPRF for transition t₂₉: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂⋅X₂+2⋅X₂ {O(n^2)}

MPRF:

• eval_foo_bb1_in: [X₂]
• eval_foo_bb2_in: [X₂]
• eval_foo_bb3_in: [X₂-X₁]
• eval_foo_bb4_in: [X₂-2⋅X₁]
• eval_foo_bb5_in: [X₂]

MPRF for transition t₃₁: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, 2⋅X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

2⋅X₂⋅X₂+4⋅X₂ {O(n^2)}

MPRF:

• eval_foo_bb1_in: [2⋅X₂]
• eval_foo_bb2_in: [2⋅X₂]
• eval_foo_bb3_in: [X₂-X₁]
• eval_foo_bb4_in: [X₂-X₁]
• eval_foo_bb5_in: [2⋅X₂]

Cut unsatisfiable transition [t₃₀: eval_foo_bb3_in→eval_foo_bb5_in; t₆₁: eval_foo_bb3_in→eval_foo_bb5_in]

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb5_in

Found invariant 3 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 6 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location eval_foo_bb4_in_v2

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb3_in

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant 0 ≤ X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ 1+X₀ for location eval_foo_bb1_in

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb4_in_v1

Found invariant 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb3_in_v1

All Bounds

Timebounds

Overall timebound:3⋅X₂⋅X₂+12⋅X₂+11 {O(n^2)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: X₂+2 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₂+1 {O(n)}
t₂₈: 2⋅X₂+1 {O(n)}
t₂₉: X₂⋅X₂+2⋅X₂ {O(n^2)}
t₃₀: X₂+1 {O(n)}
t₃₁: 2⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
t₃₂: X₂+1 {O(n)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂⋅X₂+12⋅X₂+11 {O(n^2)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: X₂+2 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₂+1 {O(n)}
t₂₈: 2⋅X₂+1 {O(n)}
t₂₉: X₂⋅X₂+2⋅X₂ {O(n^2)}
t₃₀: X₂+1 {O(n)}
t₃₁: 2⋅X₂⋅X₂+4⋅X₂ {O(n^2)}
t₃₂: X₂+1 {O(n)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}

Sizebounds

t₂₃, X₀: X₂ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: X₀ {O(n)}
t₂₄, X₁: X₁ {O(n)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: X₂+2 {O(n)}
t₂₅, X₁: 16⋅2^(X₂)+X₁ {O(EXP)}
t₂₅, X₂: X₂ {O(n)}
t₂₆, X₀: 1 {O(1)}
t₂₆, X₁: 16⋅2^(X₂)+X₁ {O(EXP)}
t₂₆, X₂: X₂ {O(n)}
t₂₇, X₀: X₂+2 {O(n)}
t₂₇, X₁: 1 {O(1)}
t₂₇, X₂: X₂ {O(n)}
t₂₈, X₀: 1 {O(1)}
t₂₈, X₁: 16⋅2^(X₂)+X₁ {O(EXP)}
t₂₈, X₂: X₂ {O(n)}
t₂₉, X₀: X₂+2 {O(n)}
t₂₉, X₁: 16⋅2^(X₂) {O(EXP)}
t₂₉, X₂: X₂ {O(n)}
t₃₀, X₀: X₂+2 {O(n)}
t₃₀, X₁: 16⋅2^(X₂) {O(EXP)}
t₃₀, X₂: X₂ {O(n)}
t₃₁, X₀: X₂+2 {O(n)}
t₃₁, X₁: 16⋅2^(X₂) {O(EXP)}
t₃₁, X₂: X₂ {O(n)}
t₃₂, X₀: X₂+2 {O(n)}
t₃₂, X₁: 16⋅2^(X₂)+X₁ {O(EXP)}
t₃₂, X₂: X₂ {O(n)}
t₃₃, X₀: X₀+1 {O(n)}
t₃₃, X₁: 16⋅2^(X₂)+2⋅X₁ {O(EXP)}
t₃₃, X₂: 2⋅X₂ {O(n)}
t₃₄, X₀: X₀ {O(n)}
t₃₄, X₁: X₁ {O(n)}
t₃₄, X₂: X₂ {O(n)}