KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb3_in

Found invariant X₀ ≤ X₃ for location eval_jama_ex7_bb1_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb2_in

Found invariant 1 ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb4_in

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location eval_jama_ex7_stop

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location eval_jama_ex7_bb5_in

Problem after Preprocessing

Start: eval_jama_ex7_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_jama_ex7_bb0_in, eval_jama_ex7_bb1_in, eval_jama_ex7_bb2_in, eval_jama_ex7_bb3_in, eval_jama_ex7_bb4_in, eval_jama_ex7_bb5_in, eval_jama_ex7_start, eval_jama_ex7_stop
Transitions:
t₁: eval_jama_ex7_bb0_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb1_in(X₃, X₁, X₂, X₃)
t₂: eval_jama_ex7_bb1_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb2_in(X₀, X₂, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃
t₃: eval_jama_ex7_bb1_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb5_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₄: eval_jama_ex7_bb2_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb3_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂
t₅: eval_jama_ex7_bb2_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb4_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂
t₆: eval_jama_ex7_bb3_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂
t₇: eval_jama_ex7_bb4_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₂
t₈: eval_jama_ex7_bb5_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₀: eval_jama_ex7_start(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_jama_ex7_bb1_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb2_in(X₀, X₂, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_jama_ex7_bb1_in: [X₀]
• eval_jama_ex7_bb2_in: [X₀-1]
• eval_jama_ex7_bb3_in: [X₀-1]
• eval_jama_ex7_bb4_in: [X₀-1]

MPRF for transition t₅: eval_jama_ex7_bb2_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb4_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_jama_ex7_bb1_in: [X₀]
• eval_jama_ex7_bb2_in: [X₀]
• eval_jama_ex7_bb3_in: [X₀]
• eval_jama_ex7_bb4_in: [X₀-1]

MPRF for transition t₇: eval_jama_ex7_bb4_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_jama_ex7_bb1_in: [X₀]
• eval_jama_ex7_bb2_in: [X₀]
• eval_jama_ex7_bb3_in: [X₀]
• eval_jama_ex7_bb4_in: [X₀]

MPRF for transition t₄: eval_jama_ex7_bb2_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb3_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₂⋅X₃+X₂ {O(n^2)}

MPRF:

• eval_jama_ex7_bb1_in: [X₂]
• eval_jama_ex7_bb2_in: [X₁]
• eval_jama_ex7_bb3_in: [X₁-1]
• eval_jama_ex7_bb4_in: [X₁]

MPRF for transition t₆: eval_jama_ex7_bb3_in(X₀, X₁, X₂, X₃) → eval_jama_ex7_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₂⋅X₃+X₂ {O(n^2)}

MPRF:

• eval_jama_ex7_bb1_in: [X₂]
• eval_jama_ex7_bb2_in: [X₁]
• eval_jama_ex7_bb3_in: [X₁]
• eval_jama_ex7_bb4_in: [X₁]

Found invariant X₀ ≤ X₃ for location eval_jama_ex7_bb1_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb2_in

Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb3_in_v2

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb2_in_v1

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location eval_jama_ex7_bb5_in

Found invariant 1 ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb4_in

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location eval_jama_ex7_stop

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_jama_ex7_bb3_in_v1

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₃+2⋅X₂+3⋅X₃+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃ {O(n)}
t₃: 1 {O(1)}
t₄: X₂⋅X₃+X₂ {O(n^2)}
t₅: X₃ {O(n)}
t₆: X₂⋅X₃+X₂ {O(n^2)}
t₇: X₃ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₃+2⋅X₂+3⋅X₃+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃ {O(n)}
t₃: 1 {O(1)}
t₄: X₂⋅X₃+X₂ {O(n^2)}
t₅: X₃ {O(n)}
t₆: X₂⋅X₃+X₂ {O(n^2)}
t₇: X₃ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₃ {O(n)}
t₂, X₁: 2⋅X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 2⋅X₃ {O(n)}
t₃, X₁: 4⋅X₂+X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₃ {O(n)}
t₄, X₁: 2⋅X₂ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₃ {O(n)}
t₅, X₁: 4⋅X₂ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₃ {O(n)}
t₆, X₁: 2⋅X₂ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₃ {O(n)}
t₇, X₁: 4⋅X₂ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: 2⋅X₃ {O(n)}
t₈, X₁: 4⋅X₂+X₁ {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}