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Proof step 6 from proof fill-stream-append (previous step)

m + n = # buf, n0 < m + n ⊦ ⌜fill(buf, zero, n, m)⌝ [n0] = ((⌜buf⌝ to n) + (σ⊥ to m))[n0]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ n < # buf → ⌜buf⌝ [n] = buf[n]
Proof step 8:
m + n = # buf, n0 < m + n, n0 < # fill(buf, zero, n, m) ⊦ fill(buf, zero, n, m)[n0] = ((⌜buf⌝ to n) + (σ⊥ to m))[n0]

Rule argument

spec: buffer-inf-conv name: tostream-at seq: ⊦ n < # buf → ⌜buf⌝ [n] = buf[n] subst: [n, buf] → [n0, fill(buf, zero, n, m)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # fill(buf, d, m, n) = # buf; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n0, fill(buf, zero, n, m)