KoAT2 Proof Output

Initial Problem

Start: eval_non_linear18_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: eval_non_linear18_bb0_in, eval_non_linear18_bb1_in, eval_non_linear18_bb2_in, eval_non_linear18_bb3_in, eval_non_linear18_bb4_in, eval_non_linear18_bb5_in, eval_non_linear18_start, eval_non_linear18_stop
Transitions:
t₁: eval_non_linear18_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb1_in(X₄, X₅, X₆, X₃, X₄, X₅, X₆)
t₂: eval_non_linear18_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂
t₃: eval_non_linear18_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb3_in(X₀, X₁, X₂, X₀, X₄, X₅, X₆) :|: X₂ ≤ 0
t₄: eval_non_linear18_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb1_in(X₀+2⋅X₁, 3⋅X₀+4⋅X₁, X₂-1, X₃, X₄, X₅, X₆)
t₅: eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₃
t₆: eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ 0
t₇: eval_non_linear18_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆)
t₈: eval_non_linear18_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_non_linear18_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear18_bb3_in

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear18_bb5_in

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear18_stop

Found invariant X₂ ≤ X₆ for location eval_non_linear18_bb1_in

Found invariant 1 ≤ X₆ ∧ 2 ≤ X₂+X₆ ∧ X₂ ≤ X₆ ∧ 1 ≤ X₂ for location eval_non_linear18_bb2_in

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_non_linear18_bb4_in

Problem after Preprocessing

Start: eval_non_linear18_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: eval_non_linear18_bb0_in, eval_non_linear18_bb1_in, eval_non_linear18_bb2_in, eval_non_linear18_bb3_in, eval_non_linear18_bb4_in, eval_non_linear18_bb5_in, eval_non_linear18_start, eval_non_linear18_stop
Transitions:
t₁: eval_non_linear18_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb1_in(X₄, X₅, X₆, X₃, X₄, X₅, X₆)
t₂: eval_non_linear18_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂ ∧ X₂ ≤ X₆
t₃: eval_non_linear18_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb3_in(X₀, X₁, X₂, X₀, X₄, X₅, X₆) :|: X₂ ≤ 0 ∧ X₂ ≤ X₆
t₄: eval_non_linear18_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb1_in(X₀+2⋅X₁, 3⋅X₀+4⋅X₁, X₂-1, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂ ∧ 1 ≤ X₆ ∧ 2 ≤ X₂+X₆ ∧ X₂ ≤ X₆
t₅: eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆
t₆: eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆
t₇: eval_non_linear18_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆
t₈: eval_non_linear18_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₆ ∧ X₃ ≤ 0
t₀: eval_non_linear18_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

MPRF for transition t₂: eval_non_linear18_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂ ∧ X₂ ≤ X₆ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• eval_non_linear18_bb1_in: [X₂]
• eval_non_linear18_bb2_in: [X₂-1]

MPRF for transition t₄: eval_non_linear18_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb1_in(X₀+2⋅X₁, 3⋅X₀+4⋅X₁, X₂-1, X₃, X₄, X₅, X₆) :|: 1 ≤ X₂ ∧ 1 ≤ X₆ ∧ 2 ≤ X₂+X₆ ∧ X₂ ≤ X₆ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• eval_non_linear18_bb1_in: [X₂]
• eval_non_linear18_bb2_in: [X₂]

MPRF for transition t₅: eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆ of depth 1:

new bound:

8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}

MPRF:

• eval_non_linear18_bb3_in: [X₃]
• eval_non_linear18_bb4_in: [X₃-1]

MPRF for transition t₇: eval_non_linear18_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_non_linear18_bb3_in(X₀, X₁, X₂, X₃-1, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₆ of depth 1:

new bound:

8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}

MPRF:

• eval_non_linear18_bb3_in: [X₃]
• eval_non_linear18_bb4_in: [X₃]

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 for location eval_non_linear18_bb3_in

Found invariant X₂ ≤ X₆ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_non_linear18_bb3_in_v1

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear18_bb5_in

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_non_linear18_bb4_in_v1

Found invariant X₂ ≤ X₆ ∧ X₃ ≤ 0 ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 0 for location eval_non_linear18_stop

Found invariant X₂ ≤ X₆ for location eval_non_linear18_bb1_in

Found invariant X₂ ≤ X₆ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ 2+X₂ ≤ X₀ ∧ 2 ≤ X₀ for location eval_non_linear18_bb4_in_v2

Found invariant 1 ≤ X₆ ∧ 2 ≤ X₂+X₆ ∧ X₂ ≤ X₆ ∧ 1 ≤ X₂ for location eval_non_linear18_bb2_in

All Bounds

Timebounds

Overall timebound:2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₄+2⋅X₆+5 {O(EXP)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₆ {O(n)}
t₃: 1 {O(1)}
t₄: X₆ {O(n)}
t₅: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₆: 1 {O(1)}
t₇: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₄+2⋅X₆+5 {O(EXP)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₆ {O(n)}
t₃: 1 {O(1)}
t₄: X₆ {O(n)}
t₅: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₆: 1 {O(1)}
t₇: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
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)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₀, X₆: X₆ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₆ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₁, X₆: X₆ {O(n)}
t₂, X₀: 8^(X₆)⋅X₄+8^(X₆)⋅X₅ {O(EXP)}
t₂, X₁: 8^(X₆)⋅X₄+8^(X₆)⋅X₅ {O(EXP)}
t₂, X₂: X₆ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₂, X₆: X₆ {O(n)}
t₃, X₀: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₃, X₁: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₅ {O(EXP)}
t₃, X₂: 2⋅X₆ {O(n)}
t₃, X₃: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₃, X₆: 2⋅X₆ {O(n)}
t₄, X₀: 8^(X₆)⋅X₄+8^(X₆)⋅X₅ {O(EXP)}
t₄, X₁: 8^(X₆)⋅X₄+8^(X₆)⋅X₅ {O(EXP)}
t₄, X₂: X₆ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₄, X₆: X₆ {O(n)}
t₅, X₀: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₅, X₁: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₅ {O(EXP)}
t₅, X₂: 2⋅X₆ {O(n)}
t₅, X₃: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₅, X₄: 2⋅X₄ {O(n)}
t₅, X₅: 2⋅X₅ {O(n)}
t₅, X₆: 2⋅X₆ {O(n)}
t₆, X₀: 2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₄ {O(EXP)}
t₆, X₁: 2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₅ {O(EXP)}
t₆, X₂: 4⋅X₆ {O(n)}
t₆, X₃: 2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₄ {O(EXP)}
t₆, X₄: 4⋅X₄ {O(n)}
t₆, X₅: 4⋅X₅ {O(n)}
t₆, X₆: 4⋅X₆ {O(n)}
t₇, X₀: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₇, X₁: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₅ {O(EXP)}
t₇, X₂: 2⋅X₆ {O(n)}
t₇, X₃: 8^(X₆)⋅X₄+8^(X₆)⋅X₅+X₄ {O(EXP)}
t₇, X₄: 2⋅X₄ {O(n)}
t₇, X₅: 2⋅X₅ {O(n)}
t₇, X₆: 2⋅X₆ {O(n)}
t₈, X₀: 2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₄ {O(EXP)}
t₈, X₁: 2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₅ {O(EXP)}
t₈, X₂: 4⋅X₆ {O(n)}
t₈, X₃: 2⋅8^(X₆)⋅X₄+2⋅8^(X₆)⋅X₅+2⋅X₄ {O(EXP)}
t₈, X₄: 4⋅X₄ {O(n)}
t₈, X₅: 4⋅X₅ {O(n)}
t₈, X₆: 4⋅X₆ {O(n)}