KoAT2 Proof Output

Initial Problem

Start: evalSimpleSinglestart
Program_Vars: X₀, X₁
Temp_Vars:
Locations: evalSimpleSinglebb3in, evalSimpleSinglebbin, evalSimpleSingleentryin, evalSimpleSinglereturnin, evalSimpleSinglestart, evalSimpleSinglestop
Transitions:
t₂: evalSimpleSinglebb3in(X₀, X₁) → evalSimpleSinglebbin(X₀, X₁) :|: 1+X₀ ≤ X₁
t₃: evalSimpleSinglebb3in(X₀, X₁) → evalSimpleSinglereturnin(X₀, X₁) :|: X₁ ≤ X₀
t₄: evalSimpleSinglebbin(X₀, X₁) → evalSimpleSinglebb3in(1+X₀, X₁)
t₁: evalSimpleSingleentryin(X₀, X₁) → evalSimpleSinglebb3in(0, X₁)
t₅: evalSimpleSinglereturnin(X₀, X₁) → evalSimpleSinglestop(X₀, X₁)
t₀: evalSimpleSinglestart(X₀, X₁) → evalSimpleSingleentryin(X₀, X₁)

Preprocessing

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalSimpleSinglebbin

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalSimpleSinglereturnin

Found invariant 0 ≤ X₀ for location evalSimpleSinglebb3in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalSimpleSinglestop

Problem after Preprocessing

Start: evalSimpleSinglestart
Program_Vars: X₀, X₁
Temp_Vars:
Locations: evalSimpleSinglebb3in, evalSimpleSinglebbin, evalSimpleSingleentryin, evalSimpleSinglereturnin, evalSimpleSinglestart, evalSimpleSinglestop
Transitions:
t₂: evalSimpleSinglebb3in(X₀, X₁) → evalSimpleSinglebbin(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₃: evalSimpleSinglebb3in(X₀, X₁) → evalSimpleSinglereturnin(X₀, X₁) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₄: evalSimpleSinglebbin(X₀, X₁) → evalSimpleSinglebb3in(1+X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₁: evalSimpleSingleentryin(X₀, X₁) → evalSimpleSinglebb3in(0, X₁)
t₅: evalSimpleSinglereturnin(X₀, X₁) → evalSimpleSinglestop(X₀, X₁) :|: 0 ≤ X₀ ∧ X₁ ≤ X₀
t₀: evalSimpleSinglestart(X₀, X₁) → evalSimpleSingleentryin(X₀, X₁)

MPRF for transition t₂: evalSimpleSinglebb3in(X₀, X₁) → evalSimpleSinglebbin(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalSimpleSinglebb3in: [X₁-X₀]
• evalSimpleSinglebbin: [X₁-1-X₀]

MPRF for transition t₄: evalSimpleSinglebbin(X₀, X₁) → evalSimpleSinglebb3in(1+X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalSimpleSinglebb3in: [X₁-X₀]
• evalSimpleSinglebbin: [X₁-X₀]

All Bounds

Timebounds

Overall timebound:2⋅X₁+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: X₁ {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: X₁ {O(n)}
t₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}