[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 28 from proof completeness-rec (previous step)

〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = some(t0)), ∀ j0, i0, x, x0, to. i→n(j0 - i0) < i→n(i1) → (x = x0 ∧ x0 ≠ [] ∧ i0 ≤ j0 ∧ max(x0) ≤ j0 → 〈build_rec#(i0; x, to)〉 (to = fail → (∀ t, x1. x0 ≠ depths(t, i0) + x1))), x = d ' + y, to2 = fail, d ' + y = depths(t0, i + 1) + x2, 0 ≠ i1, d ∈ depths(t0, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 < i1, ¬ max(x2) ≤ i1 + i ∨ x2 = [], x2 ≠ [] ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(t0, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 27:
〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = some(t0)), ∀ j0, i0, x, x0, to. i→n(j0 - i0) < i→n(i1) → (x = x0 ∧ x0 ≠ [] ∧ i0 ≤ j0 ∧ max(x0) ≤ j0 → 〈build_rec#(i0; x, to)〉 (to = fail → (∀ t, x1. x0 ≠ depths(t, i0) + x1))), x = d ' + y, to2 = fail, d ' + y = depths(t0, i + 1) + x2, 0 ≠ i1, x2 ≠ [], d ∈ depths(t0, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 < i1, i1 + i < max(x2), max(d ' + y) < max(x2) ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(t0, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPFORWARD : ⊦ i ≤ j ∧ j < j0 → i < j0], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i])]]