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Proof step 36 from proof range-eq-validtrailer (previous step)

n0 < NODE_HEADER_SIZE, is-aligned(m, NODE_HEADER_SIZE), is-aligned(# buf, NODE_HEADER_SIZE), m + NODE_HEADER_SIZE ≤ # buf + n, m + NODE_HEADER_SIZE ≤ # buf, ∀ n1. n1 < NODE_HEADER_SIZE → buf[m + n1] = (validtrailer)[n1] ⊦ (buf + mkempbuf(n))[m + n0] = (validtrailer)[n0]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 37:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 38:
n0 < NODE_HEADER_SIZE, m + n0 < # buf + # mkempbuf(n), is-aligned(# buf, NODE_HEADER_SIZE), is-aligned(m, NODE_HEADER_SIZE), m + NODE_HEADER_SIZE ≤ # buf, m + NODE_HEADER_SIZE ≤ # buf + n, ∀ n1. n1 < NODE_HEADER_SIZE → buf[m + n1] = (validtrailer)[n1] ⊦ (m + n0 < # buf ⊃ buf[m + n0];mkempbuf(n)[m + n0 - # buf]) = (validtrailer)[n0]

Rule argument

spec: array-splice, inst: buffer name: append-at seq: ⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf]) subst: [n, buf0, buf] → [m + n0, mkempbuf(n), buf] flags: <all paths>

Additional information

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