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Proof step 6 from proof get-infto (previous step)

m ≤ n0, ∀ m, nf. m < n0 → (nf to n0)[m] = nf(m) ⊦ (nf(0) ' + ((λ n. nf(n + 1)) to n0))[m] = nf(m)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ ¬ n < # x → (x + y)[n] = y[n - # x]
Proof step 8:
¬ m < # nf(0) ', m ≤ n0, ∀ m, nf. m < n0 → (nf to n0)[m] = nf(m) ⊦ ((λ n. nf(n + 1)) to n0)[m - # nf(0) '] = nf(m)

Rule argument

spec: list-del inst: name: get-append-grt seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x] subst: [n, y, x] → [m, (λ n. nf(n + 1)) to n0, nf(0) '] flags: rewrite

Additional information

Used simplifier rules: get-append-less: ⊦ n < # x → (x + y)[n] = x[n]; from specification list-del%a: ⊦ (x + y) + z = x + y + z; from specification list%len-02: ⊦ # a ' = 0 +1; from specification list%one-01: ⊦ m < 1 ↔ m = 0; from specification nat-basic1%one-03: ⊦ a '[0] = a; from specification list-del Used terms: m, (λ n. nf(n + 1)) to n0, nf(0) '