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Proof step 169 from proof abs-nodes-add-crash-aligned-h (previous step)

# buf1 = a.alignedsize + 2 * NODE_HEADER_SIZE, # buf2 = a0.alignedsize + 2 * NODE_HEADER_SIZE, n0 < # buf + # buf1, abs-node(buf2, a0), abs-node(buf1, a), abs-nodes(buf, m + # buf2, x0, lax0, false), abs-nodes (buf2 + subarray(buf + buf1, 0, n0), m, (a0 ' + x0) + a ', (@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ', boolvar), is-aligned(m, NODE_HEADER_SIZE), is-aligned(n0, NODE_HEADER_SIZE), ∀ buf3, buf1, a, n, m, x, lax, boolvar. # buf3 < #(buf2 + buf) → # buf1 = a.alignedsize + 2 * NODE_HEADER_SIZE ∧ n < # buf3 + # buf1 ∧ abs-node(buf1, a) ∧ abs-nodes(subarray(buf3 + buf1, 0, n), m, x + a ', lax + @(m + # buf3, # buf1) ', boolvar) ∧ abs-nodes(buf3, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) ∧ is-aligned(m, NODE_HEADER_SIZE) → n = # buf3 + # buf1 ⊦ n0 = # buf + # buf1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 170:
⊦ abs-nodes(buf, m, a ' + x, lax, weak) ↔ (∃ buf0, buf1. buf = buf0 + buf1 ∧ ( abs-padding(buf0) ∧ abs-nodes(buf1, m + # buf0, a ' + x, lax, weak) ∨ lax ≠ [] ∧ lax.head.off = m ∧ lax.head.size = # buf0 ∧ abs-node(buf0, a) ∧ abs-nodes(buf1, m + # buf0, x, lax.tail, weak)))
Proof step 171:
# buf2 = a0.alignedsize + 2 * NODE_HEADER_SIZE, # buf1 = a.alignedsize + 2 * NODE_HEADER_SIZE, n0 < # buf + # buf1, abs-node(buf1, a), abs-node(buf2, a0), abs-nodes(buf, m + # buf2, x0, lax0, false), is-aligned(n0, NODE_HEADER_SIZE), is-aligned(m, NODE_HEADER_SIZE), ∃ buf0, buf3. buf2 + subarray(buf + buf1, 0, n0) = buf0 + buf3 ∧ ( abs-padding(buf0) ∧ abs-nodes(buf3, m + # buf0, a0 ' + x0 + a ', (@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ', boolvar) ∨ (@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ' ≠ [] ∧ ((@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ').head.off = m ∧ ((@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ').head.size = # buf0 ∧ abs-node(buf0, a0) ∧ abs-nodes(buf3, m + # buf0, x0 + a ', ((@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ').tail, boolvar)), ∀ buf3, buf1, a, n, m, x, lax, boolvar. # buf3 < #(buf2 + buf) → # buf1 = a.alignedsize + 2 * NODE_HEADER_SIZE ∧ n < # buf3 + # buf1 ∧ abs-node(buf1, a) ∧ abs-nodes(subarray(buf3 + buf1, 0, n), m, x + a ', lax + @(m + # buf3, # buf1) ', boolvar) ∧ abs-nodes(buf3, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) ∧ is-aligned(m, NODE_HEADER_SIZE) → n = # buf3 + # buf1 ⊦ n0 = # buf + # buf1

Rule argument

spec: <toplevel> name: abs-nodes-rec seq: ⊦ abs-nodes(buf, m, a ' + x, lax, weak) ↔ (∃ buf0, buf1. buf = buf0 + buf1 ∧ ( abs-padding(buf0) ∧ abs-nodes(buf1, m + # buf0, a ' + x, lax, weak) ∨ lax ≠ [] ∧ lax.head.off = m ∧ lax.head.size = # buf0 ∧ abs-node(buf0, a) ∧ abs-nodes(buf1, m + # buf0, x, lax.tail, weak))) subst: [weak, lax, x, a, m, buf] → [boolvar, (@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ', x0 + a ', a0, m, buf2 + subarray(buf + buf1, 0, n0)] flags: <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (lax + lax0) + lax1 = lax + lax0 + lax1; from specification list[laddress-list] Used terms: boolvar, (@(m, # buf2) ' + lax0) + @(m + # buf2 + # buf, # buf1) ', x0 + a ', a0, m, buf2 + subarray(buf + buf1, 0, n0)