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Proof step 33 from proof datasize-further (previous step)

n + 1 = datasize(buf, n1), datasize(buf, n) = datasize(buf, n1) ↔ (∀ n2. n ≤ n2 ∧ n2 < n1 → buf[n2] = empty), n < n0, n0 < n1, n < # buf, n < n1, datasize(buf, n1) < n1, datasize(buf, n1) ≤ n0, datasize(buf, n1) ≤ # buf ⊦ buf[n0] = empty, buf[n] = empty

User applied rule structural induction

And obtained the following new goals

Proof step 34:
⊦ datasize(buf, 0) = 0
Proof step 35:
n + 1 = datasize(buf, n2 +1), datasize(buf, n) = datasize(buf, n2 +1) ↔ (∀ n0. n ≤ n0 ∧ n0 < n2 +1 → buf[n0] = empty), datasize(buf, n2 +1) < n2 +1, n < n2 +1, n < # buf, n0 < n2 +1, n < n0, datasize(buf, n2 +1) ≤ # buf, datasize(buf, n2 +1) ≤ n0, ∀ n, buf, n0. n + 1 = datasize(buf, n2) ∧ (datasize(buf, n) = datasize(buf, n2) ↔ (∀ n3. n ≤ n3 ∧ n3 < n2 → buf[n3] = empty)) ∧ n < n0 ∧ n0 < n2 ∧ n < # buf ∧ n < n2 ∧ datasize(buf, n2) < n2 ∧ datasize(buf, n2) ≤ n0 ∧ datasize(buf, n2) ≤ # buf → buf[n0] = empty ∨ buf[n] = empty ⊦ buf[n] = empty, buf[n0] = empty

Rule argument

indvar: n1

Additional information

Used simplifier rules: datasize-base: ⊦ datasize(buf, 0) = 0;%nat freely generated by 0, +1;;% ⊦ ¬ m < 0; (s)% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic