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+X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, -2⋅X₁, X₂, X₃)
t₅: 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 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 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+X₁ ∧ X₀ ≤ X₂
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, -2⋅X₁, X₂, X₃) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂
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+X₁ ∧ X₀ ≤ X₂ of depth 2:

new bound:

32⋅X₂+8⋅X₃+23 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+2⋅X₂-3; X₀+X₁]
• eval_foo_bb2_in: [X₀+2⋅X₂-4; 2⋅X₂-1-X₀-2⋅X₁]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, -2⋅X₁, X₂, X₃) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ of depth 2:

new bound:

32⋅X₂+8⋅X₃+31 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+2⋅X₂-3; X₀+X₁-1]
• eval_foo_bb2_in: [X₀+2⋅X₂-3; 0]

All Bounds

Timebounds

Overall timebound:16⋅X₃+64⋅X₂+58 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 32⋅X₂+8⋅X₃+23 {O(n)}
t₃: 1 {O(1)}
t₄: 32⋅X₂+8⋅X₃+31 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₃+64⋅X₂+58 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 32⋅X₂+8⋅X₃+23 {O(n)}
t₃: 1 {O(1)}
t₄: 32⋅X₂+8⋅X₃+31 {O(n)}
t₅: 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: 33⋅X₂+8⋅X₃+25 {O(n)}
t₂, X₁: 2147483648⋅2^(32⋅X₂)⋅2^(8⋅X₃)⋅X₃ {O(EXP)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 5⋅X₂+1 {O(n)}
t₃, X₁: 2⋅2^(3⋅X₂)⋅X₃+X₃ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 4⋅X₂+1 {O(n)}
t₄, X₁: 2⋅2^(3⋅X₂)⋅X₃ {O(EXP)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 5⋅X₂+1 {O(n)}
t₅, X₁: 2⋅2^(3⋅X₂)⋅X₃+X₃ {O(EXP)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}