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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ X₀ ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(2⋅X₀+X₁, 1+X₁, X₂, X₃, X₄)
t₆: eval_foo_bb3_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

Found invariant 0 ≤ X₄ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₃ ≤ X₁ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₁ for location eval_foo_stop

Found invariant X₃ ≤ X₁ for location eval_foo_bb3_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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ 0 ∧ X₃ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₀ ∧ X₃ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(2⋅X₀+X₁, 1+X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₄
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₁
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, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ of depth 2:

new bound:

16⋅X₄+8⋅X₂+8⋅X₃+9 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₄-X₁; X₄-X₀]
• eval_foo_bb2_in: [X₄-X₁; X₄-X₁]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(2⋅X₀+X₁, 1+X₁, X₂, X₃, X₄) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₄ of depth 2:

new bound:

16⋅X₄+8⋅X₂+8⋅X₃+9 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₄-X₁; X₄-X₀]
• eval_foo_bb2_in: [1+X₄-X₁; 0]

All Bounds

Timebounds

Overall timebound:16⋅X₂+16⋅X₃+32⋅X₄+23 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₄+8⋅X₂+8⋅X₃+9 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 16⋅X₄+8⋅X₂+8⋅X₃+9 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₂+16⋅X₃+32⋅X₄+23 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₄+8⋅X₂+8⋅X₃+9 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 16⋅X₄+8⋅X₂+8⋅X₃+9 {O(n)}
t₆: 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: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2^(8⋅X₂)⋅2^(8⋅X₃)⋅2^(8⋅X₄)⋅512+2^(8⋅X₂)⋅2^(8⋅X₃)⋅2^(8⋅X₄)⋅512⋅X₂+2^(8⋅X₂)⋅2^(8⋅X₃)⋅2^(8⋅X₄)⋅512⋅X₃+8⋅X₂+8⋅X₄+9⋅X₃+10 {O(EXP)}
t₂, X₁: 8⋅X₂+8⋅X₄+9⋅X₃+9 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2^(X₃)⋅2^(X₄)⋅4+2^(X₃)⋅2^(X₄)⋅4⋅X₂+2^(X₃)⋅2^(X₄)⋅4⋅X₃+2⋅X₃+X₂+X₄+3 {O(EXP)}
t₃, X₁: 3⋅X₃+X₄+2 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 2^(X₃)⋅2^(X₄)⋅4+2^(X₃)⋅2^(X₄)⋅4⋅X₂+2^(X₃)⋅2^(X₄)⋅4⋅X₃+2⋅X₃+X₂+X₄+3 {O(EXP)}
t₄, X₁: 3⋅X₃+X₄+2 {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₀: 2^(X₃)⋅2^(X₄)⋅4+2^(X₃)⋅2^(X₄)⋅4⋅X₂+2^(X₃)⋅2^(X₄)⋅4⋅X₃+2⋅X₃+X₄+3 {O(EXP)}
t₅, X₁: 2⋅X₃+X₄+2 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 2^(X₃)⋅2^(X₄)⋅8+2^(X₃)⋅2^(X₄)⋅8⋅X₂+2^(X₃)⋅2^(X₄)⋅8⋅X₃+2⋅X₂+2⋅X₄+4⋅X₃+6 {O(EXP)}
t₆, X₁: 2⋅X₄+6⋅X₃+4 {O(n)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₆, X₄: 4⋅X₄ {O(n)}