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Proof step 18 from proof abs-nodes-add-crash-aligned-h (previous step)

# buf0 = NODE_HEADER_SIZE, # buf = a.alignedsize, subarray(buf0 + buf + validtrailer, 0, n) = buf2 + buf1, n < 2 * NODE_HEADER_SIZE + a.alignedsize, is-aligned(n, NODE_HEADER_SIZE), is-aligned(m, NODE_HEADER_SIZE), is-encoded(a, subarray(buf, 0, node-size-headerless(a))), is-encoded(nodeheader(node-size-headerless(a), false), buf0), abs-padding(buf2) ∧ abs-nodes(buf1, m + # buf2, a ', @(m, 2 * NODE_HEADER_SIZE + a.alignedsize) ', boolvar) ∨ abs-node(buf2, a) ∧ abs-nodes(buf1, m + # buf2, [], [], boolvar), ∀ buf, buf1, a, n, m, x, lax, boolvar. # buf < # mkbuf(0) → # buf1 = a.alignedsize + 2 * NODE_HEADER_SIZE ∧ n < # buf + # buf1 ∧ abs-node(buf1, a) ∧ abs-nodes(subarray(buf + buf1, 0, n), m, x + a ', lax + @(m + # buf, # buf1) ', boolvar) ∧ abs-nodes(buf, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) ∧ is-aligned(m, NODE_HEADER_SIZE) → n = # buf + # buf1 ⊦ buf0 + buf + validtrailer = mkbuf(0), n = 2 * NODE_HEADER_SIZE + a.alignedsize

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 19:
⊦ # buf ≤ m ∧ m ≤ # buf + # buf0 → subarray(buf + buf0, 0, m) = buf + subarray(buf0, 0, m - # buf)
Proof step 20:
# buf = a.alignedsize, # buf0 = NODE_HEADER_SIZE, buf0 + subarray(buf + validtrailer, 0, n - # buf0) = buf2 + buf1, n < 2 * NODE_HEADER_SIZE + a.alignedsize, is-aligned(m, NODE_HEADER_SIZE), is-aligned(n, NODE_HEADER_SIZE), is-encoded(nodeheader(node-size-headerless(a), false), buf0), is-encoded(a, subarray(buf, 0, node-size-headerless(a))), # buf0 ≤ n, n ≤ # buf0 + #(buf + validtrailer), abs-padding(buf2) ∧ abs-nodes(buf1, m + # buf2, a ', @(m, 2 * NODE_HEADER_SIZE + a.alignedsize) ', boolvar) ∨ abs-node(buf2, a) ∧ abs-nodes(buf1, m + # buf2, [], [], boolvar), ∀ buf, buf1, a, n, m, x, lax, boolvar. # buf < # mkbuf(0) → # buf1 = a.alignedsize + 2 * NODE_HEADER_SIZE ∧ n < # buf + # buf1 ∧ abs-node(buf1, a) ∧ abs-nodes(subarray(buf + buf1, 0, n), m, x + a ', lax + @(m + # buf, # buf1) ', boolvar) ∧ abs-nodes(buf, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) ∧ is-aligned(m, NODE_HEADER_SIZE) → n = # buf + # buf1 ⊦ buf0 + buf + validtrailer = mkbuf(0), n = 2 * NODE_HEADER_SIZE + a.alignedsize

Rule argument

spec: array-splice, inst: buffer name: subarray-append-shift-right seq: ⊦ # buf ≤ m ∧ m ≤ # buf + # buf0 → subarray(buf + buf0, 0, m) = buf + subarray(buf0, 0, m - # buf) subst: [m, buf0, buf] → [n, buf + validtrailer, buf0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # validtrailer = NODE_HEADER_SIZE; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ NODE_HEADER_SIZE ≠ 0; (s)%m ≠ 0 ⊦ is-aligned(n, m) → (m ≤ n ↔ n ≠ 0); (s)% ⊦ m + m = 2 * m; (s)%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n → m ≤ n; (s)% ⊦ 0 ≤ m; (s)%n + m ≤ # buf ⊦ subarray(buf + buf0, n, m) = subarray(buf, n, m); (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ subarray(buf, n, 0) = mkbuf(0); (s)% ⊦ buf + buf0 = mkbuf(0) ↔ buf = mkbuf(0) ∧ buf0 = mkbuf(0); (ws)% ⊦ 2 * NODE_HEADER_SIZE + n ≠ 0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ 0 < m ↔ m ≠ 0; (s)%n ≠ 0 ⊦ is-aligned(0, n); (s)% ⊦ ¬ abs-padding(mkbuf(0)); (s)% ⊦ ¬ abs-node(mkbuf(0), nd); (s) Used terms: n, buf + validtrailer, buf0