KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₁-5-X₀, 2⋅X₁, X₂, X₃, X₄)
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₁-5-X₀, 2⋅X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ of depth 2:

new bound:

16⋅X₂+8⋅X₄+41 {O(n)}

MPRF:

• eval_foo_bb1_in: [5+X₀+X₂-X₁; X₂-X₀]
• eval_foo_bb2_in: [X₀+X₂-2-X₁; 5+X₀+X₂-X₁]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₁-5-X₀, 2⋅X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ of depth 2:

new bound:

32⋅X₂+8⋅X₄+49 {O(n)}

MPRF:

• eval_foo_bb1_in: [7+X₀+3⋅X₂-X₁; X₂-1-X₀]
• eval_foo_bb2_in: [3+3⋅X₂-X₀-X₁; 0]

All Bounds

Timebounds

Overall timebound:16⋅X₄+48⋅X₂+95 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₂+8⋅X₄+41 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 32⋅X₂+8⋅X₄+49 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₄+48⋅X₂+95 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₂+8⋅X₄+41 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 32⋅X₂+8⋅X₄+49 {O(n)}
t₆: 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)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2199023255552⋅2^(16⋅X₂)⋅2^(16⋅X₃)⋅2^(8⋅X₄)⋅X₄+X₃+X₄+6 {O(EXP)}
t₂, X₁: 2^(32⋅X₂)⋅2^(8⋅X₄)⋅562949953421312⋅X₄ {O(EXP)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2^(2⋅X₂)⋅2^(X₄)⋅32⋅X₄+2⋅X₃+X₄+6 {O(EXP)}
t₃, X₁: 2^(2⋅X₂)⋅2^(X₄)⋅32⋅X₄+X₄ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 2^(2⋅X₂)⋅2^(X₄)⋅32⋅X₄+2⋅X₃+X₄+6 {O(EXP)}
t₄, X₁: 2^(2⋅X₂)⋅2^(X₄)⋅32⋅X₄+X₄ {O(EXP)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₀: 2^(2⋅X₂)⋅2^(X₄)⋅32⋅X₄+X₃+X₄+6 {O(EXP)}
t₅, X₁: 2^(2⋅X₂)⋅2^(X₄)⋅32⋅X₄ {O(EXP)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 2^(2⋅X₂)⋅2^(X₄)⋅64⋅X₄+2⋅X₄+4⋅X₃+12 {O(EXP)}
t₆, X₁: 2^(2⋅X₂)⋅2^(X₄)⋅64⋅X₄+2⋅X₄ {O(EXP)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₆, X₄: 4⋅X₄ {O(n)}