KoAT2 Proof Output

Initial Problem

Start: eval_size02_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈
Temp_Vars:
Locations: eval_size02_bb0_in, eval_size02_bb1_in, eval_size02_bb2_in, eval_size02_bb3_in, eval_size02_bb4_in, eval_size02_bb5_in, eval_size02_start, eval_size02_stop
Transitions:
t₁: eval_size02_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb1_in(X₅, X₆, X₇, X₃, X₄, X₅, X₆, X₇, X₈)
t₂: eval_size02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 1 ≤ X₀
t₃: eval_size02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb3_in(X₀, X₁, X₂, X₁, X₂, X₅, X₆, X₇, X₈) :|: X₀ ≤ 0
t₄: eval_size02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb1_in(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₅: eval_size02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 1 ≤ X₃+X₄
t₆: eval_size02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: X₃+X₄ ≤ 0
t₇: eval_size02_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb3_in(X₀, X₁, X₂, X₃-1, X₄-1, X₅, X₆, X₇, X₈)
t₈: eval_size02_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₀: eval_size02_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_size02_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)

Preprocessing

Eliminate variables [X₈] that do not contribute to the problem

Found invariant X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size02_bb3_in

Found invariant X₀ ≤ X₅ ∧ X₃+X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size02_bb5_in

Found invariant X₀ ≤ X₅ for location eval_size02_bb1_in

Found invariant X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₁+X₄ ∧ X₃ ≤ X₁ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ 0 for location eval_size02_bb4_in

Found invariant X₀ ≤ X₅ ∧ X₃+X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size02_stop

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₀ for location eval_size02_bb2_in

Problem after Preprocessing

Start: eval_size02_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
Temp_Vars:
Locations: eval_size02_bb0_in, eval_size02_bb1_in, eval_size02_bb2_in, eval_size02_bb3_in, eval_size02_bb4_in, eval_size02_bb5_in, eval_size02_start, eval_size02_stop
Transitions:
t₁₇: eval_size02_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb1_in(X₅, X₆, X₇, X₃, X₄, X₅, X₆, X₇)
t₁₈: eval_size02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₀ ∧ X₀ ≤ X₅
t₁₉: eval_size02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb3_in(X₀, X₁, X₂, X₁, X₂, X₅, X₆, X₇) :|: X₀ ≤ 0 ∧ X₀ ≤ X₅
t₂₀: eval_size02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb1_in(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₀ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅
t₂₁: eval_size02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₃+X₄ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₂
t₂₂: eval_size02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃+X₄ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₂
t₂₃: eval_size02_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb3_in(X₀, X₁, X₂, X₃-1, X₄-1, X₅, X₆, X₇) :|: 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃+X₄ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₂
t₂₄: eval_size02_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₀ ≤ 0 ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₂ ∧ X₃+X₄ ≤ 0
t₂₅: eval_size02_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)

MPRF for transition t₁₈: eval_size02_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₀ ∧ X₀ ≤ X₅ of depth 1:

new bound:

X₅ {O(n)}

MPRF:

• eval_size02_bb1_in: [X₀]
• eval_size02_bb2_in: [X₀-1]

MPRF for transition t₂₀: eval_size02_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb1_in(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₀ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ of depth 1:

new bound:

X₅ {O(n)}

MPRF:

• eval_size02_bb1_in: [X₀]
• eval_size02_bb2_in: [X₀]

MPRF for transition t₂₁: eval_size02_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₃+X₄ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₂ of depth 1:

new bound:

10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+X₆+X₇ {O(EXP)}

MPRF:

• eval_size02_bb3_in: [X₂+X₃]
• eval_size02_bb4_in: [X₂+X₃-1]

MPRF for transition t₂₃: eval_size02_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_size02_bb3_in(X₀, X₁, X₂, X₃-1, X₄-1, X₅, X₆, X₇) :|: 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃+X₄ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₂ of depth 1:

new bound:

10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+X₆+X₇ {O(EXP)}

MPRF:

• eval_size02_bb3_in: [X₂+X₃]
• eval_size02_bb4_in: [X₂+X₃]

Found invariant X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₂ ≤ X₄ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₀ ≤ 0 for location eval_size02_bb3_in

Found invariant X₀ ≤ X₅ ∧ X₃+X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size02_bb5_in

Found invariant X₀ ≤ X₅ for location eval_size02_bb1_in

Found invariant X₀ ≤ X₅ ∧ X₃+X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size02_stop

Found invariant X₀ ≤ X₅ ∧ 1+X₄ ≤ X₂ ∧ 0 ≤ 1+X₃+X₄ ∧ 0 ≤ X₁+X₄ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ 0 for location eval_size02_bb3_in_v1

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₀ for location eval_size02_bb2_in

Found invariant X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ 1 ≤ X₃+X₄ ∧ X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ X₃ ≤ X₁ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ 0 for location eval_size02_bb4_in_v1

Found invariant X₀ ≤ X₅ ∧ 1+X₄ ≤ X₂ ∧ 1 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ 1+X₃ ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ X₀ ≤ 0 for location eval_size02_bb4_in_v2

All Bounds

Timebounds

Overall timebound:10^(X₅)⋅4⋅X₆+10^(X₅)⋅4⋅X₇+2⋅X₅+2⋅X₆+2⋅X₇+5 {O(EXP)}
t₁₇: 1 {O(1)}
t₁₈: X₅ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₅ {O(n)}
t₂₁: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+X₆+X₇ {O(EXP)}
t₂₂: 1 {O(1)}
t₂₃: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+X₆+X₇ {O(EXP)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 10^(X₅)⋅4⋅X₆+10^(X₅)⋅4⋅X₇+2⋅X₅+2⋅X₆+2⋅X₇+5 {O(EXP)}
t₁₇: 1 {O(1)}
t₁₈: X₅ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₅ {O(n)}
t₂₁: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+X₆+X₇ {O(EXP)}
t₂₂: 1 {O(1)}
t₂₃: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+X₆+X₇ {O(EXP)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₅ {O(n)}
t₁₇, X₁: X₆ {O(n)}
t₁₇, X₂: X₇ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₇, X₅: X₅ {O(n)}
t₁₇, X₆: X₆ {O(n)}
t₁₇, X₇: X₇ {O(n)}
t₁₈, X₀: X₅ {O(n)}
t₁₈, X₁: 10^(X₅)⋅X₆+10^(X₅)⋅X₇ {O(EXP)}
t₁₈, X₂: 10^(X₅)⋅X₆+10^(X₅)⋅X₇ {O(EXP)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₈, X₅: X₅ {O(n)}
t₁₈, X₆: X₆ {O(n)}
t₁₈, X₇: X₇ {O(n)}
t₁₉, X₀: 2⋅X₅ {O(n)}
t₁₉, X₁: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₆ {O(EXP)}
t₁₉, X₂: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₇ {O(EXP)}
t₁₉, X₃: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₆ {O(EXP)}
t₁₉, X₄: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₇ {O(EXP)}
t₁₉, X₅: 2⋅X₅ {O(n)}
t₁₉, X₆: 2⋅X₆ {O(n)}
t₁₉, X₇: 2⋅X₇ {O(n)}
t₂₀, X₀: X₅ {O(n)}
t₂₀, X₁: 10^(X₅)⋅X₆+10^(X₅)⋅X₇ {O(EXP)}
t₂₀, X₂: 10^(X₅)⋅X₆+10^(X₅)⋅X₇ {O(EXP)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₀, X₅: X₅ {O(n)}
t₂₀, X₆: X₆ {O(n)}
t₂₀, X₇: X₇ {O(n)}
t₂₁, X₀: 2⋅X₅ {O(n)}
t₂₁, X₁: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₆ {O(EXP)}
t₂₁, X₂: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₇ {O(EXP)}
t₂₁, X₃: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+10^(X₅)⋅X₆+10^(X₅)⋅X₇+2⋅X₆+X₇ {O(EXP)}
t₂₁, X₄: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+10^(X₅)⋅X₆+10^(X₅)⋅X₇+2⋅X₇+X₆ {O(EXP)}
t₂₁, X₅: 2⋅X₅ {O(n)}
t₂₁, X₆: 2⋅X₆ {O(n)}
t₂₁, X₇: 2⋅X₇ {O(n)}
t₂₂, X₀: 4⋅X₅ {O(n)}
t₂₂, X₁: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+2⋅X₆ {O(EXP)}
t₂₂, X₂: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+2⋅X₇ {O(EXP)}
t₂₂, X₃: 10^(X₅)⋅4⋅X₆+10^(X₅)⋅4⋅X₇+3⋅X₆+X₇ {O(EXP)}
t₂₂, X₄: 10^(X₅)⋅4⋅X₆+10^(X₅)⋅4⋅X₇+3⋅X₇+X₆ {O(EXP)}
t₂₂, X₅: 4⋅X₅ {O(n)}
t₂₂, X₆: 4⋅X₆ {O(n)}
t₂₂, X₇: 4⋅X₇ {O(n)}
t₂₃, X₀: 2⋅X₅ {O(n)}
t₂₃, X₁: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₆ {O(EXP)}
t₂₃, X₂: 10^(X₅)⋅X₆+10^(X₅)⋅X₇+X₇ {O(EXP)}
t₂₃, X₃: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+10^(X₅)⋅X₆+10^(X₅)⋅X₇+2⋅X₆+X₇ {O(EXP)}
t₂₃, X₄: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+10^(X₅)⋅X₆+10^(X₅)⋅X₇+2⋅X₇+X₆ {O(EXP)}
t₂₃, X₅: 2⋅X₅ {O(n)}
t₂₃, X₆: 2⋅X₆ {O(n)}
t₂₃, X₇: 2⋅X₇ {O(n)}
t₂₄, X₀: 4⋅X₅ {O(n)}
t₂₄, X₁: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+2⋅X₆ {O(EXP)}
t₂₄, X₂: 10^(X₅)⋅2⋅X₆+10^(X₅)⋅2⋅X₇+2⋅X₇ {O(EXP)}
t₂₄, X₃: 10^(X₅)⋅4⋅X₆+10^(X₅)⋅4⋅X₇+3⋅X₆+X₇ {O(EXP)}
t₂₄, X₄: 10^(X₅)⋅4⋅X₆+10^(X₅)⋅4⋅X₇+3⋅X₇+X₆ {O(EXP)}
t₂₄, X₅: 4⋅X₅ {O(n)}
t₂₄, X₆: 4⋅X₆ {O(n)}
t₂₄, X₇: 4⋅X₇ {O(n)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₅, X₄: X₄ {O(n)}
t₂₅, X₅: X₅ {O(n)}
t₂₅, X₆: X₆ {O(n)}
t₂₅, X₇: X₇ {O(n)}