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Proof step 9 from proof diffone-nodupsget (previous step)

n < # x0, pos(a, x0) < n, pos(a, x0) + 1 < # x0, a ∈ x0, ∀ n0, a1. n0 + 1 < # x0 ∧ a1 ∈ x0 → (x0 \ a1 ')[n0] = (n0 < pos(a1, x0) ⊃ x0[n0];x0[n0 + 1]) ⊦ (a0 ' + (x0 \ a '))[n] = x0[n], x0 = [], dups(x0), a0 ∈ x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ¬ n < # x → (x + y)[n] = y[n - # x]
Proof step 11:
pos(a, x0) + 1 < # x0, pos(a, x0) < n, n < # x0, a ∈ x0, ∀ n0, a1. n0 + 1 < # x0 ∧ a1 ∈ x0 → (x0 \ a1 ')[n0] = (n0 < pos(a1, x0) ⊃ x0[n0];x0[n0 + 1]) ⊦ (x0 \ a ')[n - # a0 '] = x0[n], x0 = [], n < # a0 ', dups(x0), a0 ∈ x0

Rule argument

spec: list-del name: get-append-grt seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x] subst: [n, y, x] → [n, x0 \ a ', a0 '] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n < # x → (x + y)[n] = x[n]; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ # a ' = 0 +1; (s)% ⊦ m < 1 ↔ m = 0; (s)% ⊦ m < n → n ≠ 0; (s) Used terms: n, x0 \ a ', a0 '