KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(2⋅X₀, 1+X₁, X₂, X₃)
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₃ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₃ ≤ X₁ ∧ X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₁ ∧ X₂ ≤ X₀ for location eval_foo_stop

Found invariant X₃ ≤ X₁ ∧ X₂ ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(2⋅X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₀ ∧ X₃ ≤ X₁
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁ of depth 2:

new bound:

16⋅X₂+8⋅X₃+17 {O(n)}

MPRF:

• eval_foo_bb1_in: [2-X₀; X₁-X₀]
• eval_foo_bb2_in: [1-X₀; 1+X₁-2⋅X₀]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(2⋅X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁ of depth 2:

new bound:

16⋅X₃+24⋅X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₁-2⋅X₀; X₁-1-X₀]
• eval_foo_bb2_in: [1+X₁-2⋅X₀; 0]

All Bounds

Timebounds

Overall timebound:24⋅X₃+40⋅X₂+23 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₂+8⋅X₃+17 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 16⋅X₃+24⋅X₂+1 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 24⋅X₃+40⋅X₂+23 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₂+8⋅X₃+17 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 16⋅X₃+24⋅X₂+1 {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)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 2⋅2^(16⋅X₃)⋅2^(24⋅X₂)⋅X₂ {O(EXP)}
t₂, X₁: 16⋅X₂+9⋅X₃+17 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: 2^(2⋅X₂)⋅2^(X₃)⋅4⋅X₂+X₂ {O(EXP)}
t₄, X₁: 2⋅X₂+3⋅X₃+2 {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: 2^(2⋅X₂)⋅2^(X₃)⋅4⋅X₂ {O(EXP)}
t₅, X₁: 2⋅X₂+2⋅X₃+2 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: 2^(2⋅X₂)⋅2^(X₃)⋅4⋅X₂+2⋅X₂ {O(EXP)}
t₆, X₁: 2⋅X₂+4⋅X₃+2 {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}