KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfreturnin

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbb1in

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalfbb2in

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfbbin

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfbb3in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfstop

Problem after Preprocessing

Start: evalfstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalfbb1in, evalfbb2in, evalfbb3in, evalfbbin, evalfentryin, evalfreturnin, evalfstart, evalfstop
Transitions:
t₇: evalfbb1in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₂
t₈: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂-1) :|: 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ X₀ ≤ 0
t₂: evalfbb3in(X₀, X₁, X₂) → evalfbbin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₃: evalfbb3in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₄: evalfbb3in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₅: evalfbbin(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂
t₆: evalfbbin(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb3in(X₂, X₁, X₀) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁
t₉: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂) :|: 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

Cut unsatisfiable transition [t₃: evalfbb3in→evalfreturnin; t₄: evalfbb3in→evalfreturnin; t₅₇: evalfbb3in→evalfreturnin; t₅₈: evalfbb3in→evalfreturnin]

Found invariant 2+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 3 ≤ X₁ ∧ 3+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalfbbin_v3

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfreturnin

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfbbin_v1

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfbb3in

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbb3in_v1

Found invariant 2+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 3 ≤ X₁ ∧ 3+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalfbb2in_v2

Found invariant 2+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalfbb3in_v2

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location evalfbb2in_v1

Found invariant 1+X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbbin_v2

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbb1in_v1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁ for location evalfstop

Found invariant 1+X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalfbb1in_v2

Analysing control-flow refined program

MPRF for transition t₆₉: evalfbb3in_v2(X₀, X₁, X₂) → evalfbbin_v3(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 2+X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalfbb2in_v2: [X₂]
• evalfbb3in_v2: [1+X₂]
• evalfbbin_v3: [X₂]

MPRF for transition t₇₀: evalfbbin_v3(X₀, X₁, X₂) → evalfbb2in_v2(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 3+X₀ ≤ X₁ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₁+X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb2in_v2: [X₂-1]
• evalfbb3in_v2: [X₂]
• evalfbbin_v3: [X₂]

MPRF for transition t₇₁: evalfbb2in_v2(X₀, X₁, X₂) → evalfbb3in_v2(X₀, X₁, X₂-1) :|: 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 3+X₀ ≤ X₁ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₀ ≤ 0 of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb2in_v2: [X₂]
• evalfbb3in_v2: [X₂]
• evalfbbin_v3: [X₂]

MPRF for transition t₆₄: evalfbb3in_v1(X₀, X₁, X₂) → evalfbbin_v2(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀+X₁+2 {O(n)}

MPRF:

• evalfbb1in_v2: [X₁-X₂]
• evalfbb3in_v1: [1+X₁-X₂]
• evalfbbin_v2: [X₁-X₂]

MPRF for transition t₆₅: evalfbbin_v2(X₀, X₁, X₂) → evalfbb1in_v2(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ 5 ≤ X₁+X₂ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

• evalfbb1in_v2: [X₁-1-X₂]
• evalfbb3in_v1: [X₁-X₂]
• evalfbbin_v2: [X₁-X₂]

MPRF for transition t₆₆: evalfbb1in_v2(X₀, X₁, X₂) → evalfbb3in_v1(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ 5 ≤ X₁+X₂ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

• evalfbb1in_v2: [X₁-X₂]
• evalfbb3in_v1: [X₁-X₂]
• evalfbbin_v2: [X₁-X₂]

CFR: Improvement to new bound with the following program:

method: PartialEvaluation new bound:

O(n)

cfr-program:

Start: evalfstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalfbb1in_v1, evalfbb1in_v2, evalfbb2in_v1, evalfbb2in_v2, evalfbb3in, evalfbb3in_v1, evalfbb3in_v2, evalfbbin_v1, evalfbbin_v2, evalfbbin_v3, evalfentryin, evalfreturnin, evalfstart, evalfstop
Transitions:
t₆₂: evalfbb1in_v1(X₀, X₁, X₂) → evalfbb3in_v1(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₂
t₆₆: evalfbb1in_v2(X₀, X₁, X₂) → evalfbb3in_v1(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ 5 ≤ X₁+X₂
t₆₇: evalfbb2in_v1(X₀, X₁, X₂) → evalfbb3in_v2(X₀, X₁, X₂-1) :|: 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ X₀ ≤ 0
t₇₁: evalfbb2in_v2(X₀, X₁, X₂) → evalfbb3in_v2(X₀, X₁, X₂-1) :|: 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 3+X₀ ≤ X₁ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₀ ≤ 0
t₅₉: evalfbb3in(X₀, X₁, X₂) → evalfbbin_v1(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₆₄: evalfbb3in_v1(X₀, X₁, X₂) → evalfbbin_v2(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₆₃: evalfbb3in_v1(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₆₉: evalfbb3in_v2(X₀, X₁, X₂) → evalfbbin_v3(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 2+X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₆₈: evalfbb3in_v2(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 2+X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₆₁: evalfbbin_v1(X₀, X₁, X₂) → evalfbb1in_v1(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂
t₆₀: evalfbbin_v1(X₀, X₁, X₂) → evalfbb2in_v1(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂
t₆₅: evalfbbin_v2(X₀, X₁, X₂) → evalfbb1in_v2(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ 5 ≤ X₁+X₂
t₇₀: evalfbbin_v3(X₀, X₁, X₂) → evalfbb2in_v2(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 3+X₀ ≤ X₁ ∧ 3 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₁+X₂
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb3in(X₂, X₁, X₀) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁
t₉: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂) :|: 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

All Bounds

Timebounds

Overall timebound:3⋅X₁+6⋅X₀+15 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₉: 1 {O(1)}
t₅₉: 1 {O(1)}
t₆₀: 1 {O(1)}
t₆₁: 1 {O(1)}
t₆₂: 1 {O(1)}
t₆₃: 1 {O(1)}
t₆₄: X₀+X₁+2 {O(n)}
t₆₅: X₀+X₁+1 {O(n)}
t₆₆: X₀+X₁+1 {O(n)}
t₆₇: 1 {O(1)}
t₆₈: 1 {O(1)}
t₆₉: X₀+1 {O(n)}
t₇₀: X₀ {O(n)}
t₇₁: X₀ {O(n)}

Costbounds

Overall costbound: 3⋅X₁+6⋅X₀+15 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₉: 1 {O(1)}
t₅₉: 1 {O(1)}
t₆₀: 1 {O(1)}
t₆₁: 1 {O(1)}
t₆₂: 1 {O(1)}
t₆₃: 1 {O(1)}
t₆₄: X₀+X₁+2 {O(n)}
t₆₅: X₀+X₁+1 {O(n)}
t₆₆: X₀+X₁+1 {O(n)}
t₆₇: 1 {O(1)}
t₆₈: 1 {O(1)}
t₆₉: X₀+1 {O(n)}
t₇₀: X₀ {O(n)}
t₇₁: X₀ {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₀ {O(n)}
t₉, X₀: 4⋅X₂ {O(n)}
t₉, X₁: 4⋅X₁ {O(n)}
t₉, X₂: 3⋅X₀+X₁+3 {O(n)}
t₅₉, X₀: X₂ {O(n)}
t₅₉, X₁: X₁ {O(n)}
t₅₉, X₂: X₀ {O(n)}
t₆₀, X₀: X₂ {O(n)}
t₆₀, X₁: X₁ {O(n)}
t₆₀, X₂: X₀ {O(n)}
t₆₁, X₀: X₂ {O(n)}
t₆₁, X₁: X₁ {O(n)}
t₆₁, X₂: X₀ {O(n)}
t₆₂, X₀: X₂ {O(n)}
t₆₂, X₁: X₁ {O(n)}
t₆₂, X₂: X₀+1 {O(n)}
t₆₃, X₀: 2⋅X₂ {O(n)}
t₆₃, X₁: 2⋅X₁ {O(n)}
t₆₃, X₂: 3⋅X₀+X₁+3 {O(n)}
t₆₄, X₀: X₂ {O(n)}
t₆₄, X₁: X₁ {O(n)}
t₆₄, X₂: 2⋅X₀+X₁+2 {O(n)}
t₆₅, X₀: X₂ {O(n)}
t₆₅, X₁: X₁ {O(n)}
t₆₅, X₂: 2⋅X₀+X₁+2 {O(n)}
t₆₆, X₀: X₂ {O(n)}
t₆₆, X₁: X₁ {O(n)}
t₆₆, X₂: 2⋅X₀+X₁+2 {O(n)}
t₆₇, X₀: X₂ {O(n)}
t₆₇, X₁: X₁ {O(n)}
t₆₇, X₂: X₀ {O(n)}
t₆₈, X₀: 2⋅X₂ {O(n)}
t₆₈, X₁: 2⋅X₁ {O(n)}
t₆₈, X₂: 0 {O(1)}
t₆₉, X₀: X₂ {O(n)}
t₆₉, X₁: X₁ {O(n)}
t₆₉, X₂: X₀ {O(n)}
t₇₀, X₀: X₂ {O(n)}
t₇₀, X₁: X₁ {O(n)}
t₇₀, X₂: X₀ {O(n)}
t₇₁, X₀: X₂ {O(n)}
t₇₁, X₁: X₁ {O(n)}
t₇₁, X₂: X₀ {O(n)}