KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₀ ≤ X₃ for location eval_foo_bb5_in

Found invariant 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₃ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₃ for location eval_foo_stop

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb3_in

Found invariant 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb4_in

Cut unsatisfiable transition [t₇: eval_foo_bb2_in→eval_foo_bb4_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_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, 1, X₃, X₄) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0 ∧ X₀ ≤ X₃
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 0 ∧ X₀ ≤ X₃
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, 2⋅X₂, X₃, X₄) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃
t₁₀: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₃
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, 1, X₃, X₄) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

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

MPRF for transition t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+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: [X₀]

MPRF for transition t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+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]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃⋅X₃+3⋅X₃+1 {O(n^2)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀-X₂]
• eval_foo_bb3_in: [X₀-2⋅X₂]
• eval_foo_bb4_in: [-X₂]

MPRF for transition t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, 2⋅X₂, X₃, X₄) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ of depth 1:

new bound:

2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}

MPRF:

• eval_foo_bb1_in: [2⋅X₃]
• eval_foo_bb2_in: [2⋅X₃-2-X₂]
• eval_foo_bb3_in: [2⋅X₃-1-2⋅X₂]
• eval_foo_bb4_in: [2⋅X₃-2-X₂]

Found invariant 3 ≤ X₃ ∧ 5 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ 6 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 5 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location eval_foo_bb3_in_v2

Found invariant X₀ ≤ X₃ for location eval_foo_bb5_in

Found invariant X₀ ≤ X₃ for location eval_foo_stop

Found invariant 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₃ for location eval_foo_bb1_in

Found invariant 2 ≤ X₃ ∧ 4 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb3_in_v1

All Bounds

Timebounds

Overall timebound:3⋅X₃⋅X₃+10⋅X₃+9 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₃⋅X₃+3⋅X₃+1 {O(n^2)}
t₆: X₃+1 {O(n)}
t₈: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₉: X₃+1 {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₃⋅X₃+10⋅X₃+9 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₃⋅X₃+3⋅X₃+1 {O(n^2)}
t₆: X₃+1 {O(n)}
t₈: 2⋅X₃⋅X₃+4⋅X₃ {O(n^2)}
t₉: X₃+1 {O(n)}
t₁₀: 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: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: X₃+1 {O(n)}
t₂, X₁: 2^(X₃)⋅4+X₄+1 {O(EXP)}
t₂, X₂: 1 {O(1)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2⋅X₃+1 {O(n)}
t₃, X₁: 2^(X₃)⋅4+X₄+1 {O(EXP)}
t₃, X₂: 2^(X₃)⋅4+X₂+1 {O(EXP)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: X₃ {O(n)}
t₄, X₁: X₄ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: X₃+1 {O(n)}
t₅, X₁: 2^(X₃)⋅4+X₄+1 {O(EXP)}
t₅, X₂: 2^(X₃)⋅8 {O(EXP)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: X₃+1 {O(n)}
t₆, X₁: 2^(X₃)⋅8+2⋅X₄+2 {O(EXP)}
t₆, X₂: 2^(X₃)⋅4+1 {O(EXP)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₈, X₀: X₃+1 {O(n)}
t₈, X₁: 2^(X₃)⋅4+X₄+1 {O(EXP)}
t₈, X₂: 2^(X₃)⋅4 {O(EXP)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₄ {O(n)}
t₉, X₀: X₃+1 {O(n)}
t₉, X₁: 2^(X₃)⋅4+1 {O(EXP)}
t₉, X₂: 2^(X₃)⋅4+1 {O(EXP)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₁₀, X₀: 3⋅X₃+1 {O(n)}
t₁₀, X₁: 2^(X₃)⋅4+2⋅X₄+1 {O(EXP)}
t₁₀, X₂: 2^(X₃)⋅4+2⋅X₂+1 {O(EXP)}
t₁₀, X₃: 3⋅X₃ {O(n)}
t₁₀, X₄: 3⋅X₄ {O(n)}