KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₁ ≤ X₅ for location eval_foo_bb5_in

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_0

Found invariant X₁ ≤ X₅ for location eval_foo_stop

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_5

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_1

Found invariant X₁ ≤ X₅ for location eval_foo_bb1_in

Found invariant 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_4

Found invariant 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: nondef.0, nondef.1
Locations: eval_foo_0, eval_foo_1, eval_foo_4, eval_foo_5, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₇: eval_foo_0(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_1(X₀, X₁, nondef.0, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅
t₈: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅
t₉: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅
t₁₀: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅
t₁₄: eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_5(X₀, X₁, X₂, nondef.1, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₁₅: eval_foo_5(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₃, X₁-1, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₅
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₁ ≤ X₅
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 0 ∧ X₁ ≤ X₅
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_0(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅
t₁₁: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅
t₁₂: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₁₆: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₅
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₁₀: eval_foo_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₁]
• eval_foo_1: [1+X₁]
• eval_foo_4: [X₁]
• eval_foo_5: [X₁]
• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [1+X₁]
• eval_foo_bb3_in: [1+X₁]
• eval_foo_bb4_in: [X₁]

MPRF for transition t₁₂: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

MPRF for transition t₁₄: eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_5(X₀, X₁, X₂, nondef.1, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

• eval_foo_0: [1+X₁]
• eval_foo_1: [1+X₁]
• eval_foo_4: [1+X₁]
• eval_foo_5: [X₁]
• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [1+X₁]
• eval_foo_bb3_in: [1+X₁]
• eval_foo_bb4_in: [1+X₁]

MPRF for transition t₁₅: eval_foo_5(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₃, X₁-1, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 of depth 1:

new bound:

X₅ {O(n)}

MPRF:

• eval_foo_0: [X₁]
• eval_foo_1: [X₁]
• eval_foo_4: [X₁]
• eval_foo_5: [X₁]
• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁]
• eval_foo_bb4_in: [X₁]

Cut unreachable locations [eval_foo_1] from the program graph

Found invariant X₁ ≤ X₅ for location eval_foo_bb5_in

Found invariant 1 ≤ X₅ ∧ 2+X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1+X₂ ≤ 0 ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v1

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v2

Found invariant X₁ ≤ X₅ for location eval_foo_stop

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_0_v1

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_1_v1

Found invariant 1 ≤ X₅ ∧ 2+X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1+X₂ ≤ 0 ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v2

Found invariant 1 ≤ X₅ ∧ 2+X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1+X₂ ≤ 0 ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_0_v2

Found invariant X₁ ≤ X₅ for location eval_foo_bb1_in

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_0_v3

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v3

Found invariant 1 ≤ X₅ ∧ 2+X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1+X₂ ≤ 0 ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in_v2

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in_v1

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: inf {Infinity}
t₇: inf {Infinity}
t₈: inf {Infinity}
t₉: inf {Infinity}
t₁₀: X₅+1 {O(n)}
t₁₁: inf {Infinity}
t₁₂: X₅+1 {O(n)}
t₁₄: X₅+1 {O(n)}
t₁₅: X₅ {O(n)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: inf {Infinity}
t₇: inf {Infinity}
t₈: inf {Infinity}
t₉: inf {Infinity}
t₁₀: X₅+1 {O(n)}
t₁₁: inf {Infinity}
t₁₂: X₅+1 {O(n)}
t₁₄: X₅+1 {O(n)}
t₁₅: X₅ {O(n)}
t₁₆: 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)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₁: X₅ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₁: 3⋅X₅ {O(n)}
t₃, X₄: 3⋅X₄ {O(n)}
t₃, X₅: 3⋅X₅ {O(n)}
t₄, X₁: 2⋅X₅ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₅, X₁: X₅ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₇, X₁: X₅ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₁: X₅ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₅ {O(n)}
t₉, X₁: X₅ {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: X₅ {O(n)}
t₁₀, X₁: X₅ {O(n)}
t₁₀, X₂: 0 {O(1)}
t₁₀, X₄: X₄ {O(n)}
t₁₀, X₅: X₅ {O(n)}
t₁₁, X₁: X₅ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₁, X₅: X₅ {O(n)}
t₁₂, X₁: X₅ {O(n)}
t₁₂, X₂: 0 {O(1)}
t₁₂, X₄: X₄ {O(n)}
t₁₂, X₅: X₅ {O(n)}
t₁₄, X₁: X₅ {O(n)}
t₁₄, X₂: 0 {O(1)}
t₁₄, X₄: X₄ {O(n)}
t₁₄, X₅: X₅ {O(n)}
t₁₅, X₁: X₅ {O(n)}
t₁₅, X₂: 0 {O(1)}
t₁₅, X₄: X₄ {O(n)}
t₁₅, X₅: X₅ {O(n)}
t₁₆, X₁: 5⋅X₅ {O(n)}
t₁₆, X₄: 5⋅X₄ {O(n)}
t₁₆, X₅: 5⋅X₅ {O(n)}