KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
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, X₄, X₅) → eval_foo_bb1_in(X₃, X₄, X₅, 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
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀+X₁, X₁+X₂, X₂-1, X₃, X₄, X₅)
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Found invariant X₂ ≤ X₅ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₂ ≤ X₅ for location eval_foo_bb1_in

Found invariant X₂ ≤ X₅ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant X₂ ≤ X₅ ∧ X₀ ≤ 0 for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
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, X₄, X₅) → eval_foo_bb1_in(X₃, X₄, X₅, 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 ∧ X₂ ≤ X₅
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₂ ≤ X₅
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀+X₁, X₁+X₂, X₂-1, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₂ ≤ X₅
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₂ ≤ X₅
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₂ ≤ X₅ of depth 3:

new bound:

27⋅X₃+27⋅X₄+27⋅X₅+55 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₂; 1+X₁; X₀]
• eval_foo_bb2_in: [X₂; 1+X₁+X₂; X₀+X₁]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀+X₁, X₁+X₂, X₂-1, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₂ ≤ X₅ of depth 3:

new bound:

27⋅X₃+27⋅X₄+27⋅X₅+55 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₂; 1+X₁; X₀]
• eval_foo_bb2_in: [1+X₂; 1+X₁; X₀]

All Bounds

Timebounds

Overall timebound:54⋅X₃+54⋅X₄+54⋅X₅+114 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 27⋅X₃+27⋅X₄+27⋅X₅+55 {O(n)}
t₃: 1 {O(1)}
t₄: 27⋅X₃+27⋅X₄+27⋅X₅+55 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 54⋅X₃+54⋅X₄+54⋅X₅+114 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 27⋅X₃+27⋅X₄+27⋅X₅+55 {O(n)}
t₃: 1 {O(1)}
t₄: 27⋅X₃+27⋅X₄+27⋅X₅+55 {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)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₅ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: 119556⋅X₃⋅X₄⋅X₅+19683⋅X₃⋅X₃⋅X₃+19683⋅X₄⋅X₄⋅X₄+20412⋅X₅⋅X₅⋅X₅+59049⋅X₃⋅X₃⋅X₄+59049⋅X₃⋅X₄⋅X₄+59778⋅X₃⋅X₃⋅X₅+59778⋅X₄⋅X₄⋅X₅+60507⋅X₃⋅X₅⋅X₅+60507⋅X₄⋅X₅⋅X₅+121014⋅X₃⋅X₃+121041⋅X₄⋅X₄+124011⋅X₅⋅X₅+242055⋅X₃⋅X₄+245025⋅X₃⋅X₅+245052⋅X₄⋅X₅+248024⋅X₃+248078⋅X₄+251102⋅X₅+169455 {O(n^3)}
t₂, X₁: 1458⋅X₃⋅X₄+1485⋅X₃⋅X₅+1485⋅X₄⋅X₅+729⋅X₃⋅X₃+729⋅X₄⋅X₄+756⋅X₅⋅X₅+2997⋅X₃+2998⋅X₄+3052⋅X₅+3080 {O(n^2)}
t₂, X₂: 27⋅X₃+27⋅X₄+28⋅X₅+55 {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: 119556⋅X₃⋅X₄⋅X₅+19683⋅X₃⋅X₃⋅X₃+19683⋅X₄⋅X₄⋅X₄+20412⋅X₅⋅X₅⋅X₅+59049⋅X₃⋅X₃⋅X₄+59049⋅X₃⋅X₄⋅X₄+59778⋅X₃⋅X₃⋅X₅+59778⋅X₄⋅X₄⋅X₅+60507⋅X₃⋅X₅⋅X₅+60507⋅X₄⋅X₅⋅X₅+121014⋅X₃⋅X₃+121041⋅X₄⋅X₄+124011⋅X₅⋅X₅+242055⋅X₃⋅X₄+245025⋅X₃⋅X₅+245052⋅X₄⋅X₅+248025⋅X₃+248078⋅X₄+251102⋅X₅+169455 {O(n^3)}
t₃, X₁: 1458⋅X₃⋅X₄+1485⋅X₃⋅X₅+1485⋅X₄⋅X₅+729⋅X₃⋅X₃+729⋅X₄⋅X₄+756⋅X₅⋅X₅+2997⋅X₃+2999⋅X₄+3052⋅X₅+3080 {O(n^2)}
t₃, X₂: 27⋅X₃+27⋅X₄+29⋅X₅+55 {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₄, X₀: 119556⋅X₃⋅X₄⋅X₅+19683⋅X₃⋅X₃⋅X₃+19683⋅X₄⋅X₄⋅X₄+20412⋅X₅⋅X₅⋅X₅+59049⋅X₃⋅X₃⋅X₄+59049⋅X₃⋅X₄⋅X₄+59778⋅X₃⋅X₃⋅X₅+59778⋅X₄⋅X₄⋅X₅+60507⋅X₃⋅X₅⋅X₅+60507⋅X₄⋅X₅⋅X₅+121014⋅X₃⋅X₃+121041⋅X₄⋅X₄+124011⋅X₅⋅X₅+242055⋅X₃⋅X₄+245025⋅X₃⋅X₅+245052⋅X₄⋅X₅+248024⋅X₃+248078⋅X₄+251102⋅X₅+169455 {O(n^3)}
t₄, X₁: 1458⋅X₃⋅X₄+1485⋅X₃⋅X₅+1485⋅X₄⋅X₅+729⋅X₃⋅X₃+729⋅X₄⋅X₄+756⋅X₅⋅X₅+2997⋅X₃+2998⋅X₄+3052⋅X₅+3080 {O(n^2)}
t₄, X₂: 27⋅X₃+27⋅X₄+28⋅X₅+55 {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: 119556⋅X₃⋅X₄⋅X₅+19683⋅X₃⋅X₃⋅X₃+19683⋅X₄⋅X₄⋅X₄+20412⋅X₅⋅X₅⋅X₅+59049⋅X₃⋅X₃⋅X₄+59049⋅X₃⋅X₄⋅X₄+59778⋅X₃⋅X₃⋅X₅+59778⋅X₄⋅X₄⋅X₅+60507⋅X₃⋅X₅⋅X₅+60507⋅X₄⋅X₅⋅X₅+121014⋅X₃⋅X₃+121041⋅X₄⋅X₄+124011⋅X₅⋅X₅+242055⋅X₃⋅X₄+245025⋅X₃⋅X₅+245052⋅X₄⋅X₅+248025⋅X₃+248078⋅X₄+251102⋅X₅+169455 {O(n^3)}
t₅, X₁: 1458⋅X₃⋅X₄+1485⋅X₃⋅X₅+1485⋅X₄⋅X₅+729⋅X₃⋅X₃+729⋅X₄⋅X₄+756⋅X₅⋅X₅+2997⋅X₃+2999⋅X₄+3052⋅X₅+3080 {O(n^2)}
t₅, X₂: 27⋅X₃+27⋅X₄+29⋅X₅+55 {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}
t₅, X₅: 2⋅X₅ {O(n)}