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Proof step 16 from proof abs-nodes-append-h (previous step)

lax0.head = @(m0, m1), abs-padding(buf2) ∧ abs-nodes(buf, m + # buf2, nd ' + x, lax0, false) ∨ lax0 ≠ [] ∧ m0 = m ∧ m1 = # buf2 ∧ abs-node(buf2, nd) ∧ abs-nodes(buf, m + # buf2, x, lax0.tail, false), ∀ buf3, m, x0, lax0, buf1, x1, lax1, weak. # buf3 < #(buf2 + buf) → abs-nodes(buf3, m, x0, lax0, false) → (abs-nodes(buf3 + buf1, m, x0 + x1, lax0 + lax1, weak) ↔ abs-nodes(buf1, m + # buf3, x1, lax1, weak)) ⊦ abs-nodes((buf2 + buf) + buf1, m, (nd ' + x) + x1, lax0 + lax1, weak) ↔ abs-nodes(buf1, m + # buf2 + # buf, x1, lax1, weak)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ 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 18:
lax0.head = @(m0, m1), abs-padding(buf2) ∧ abs-nodes(buf, m + # buf2, nd ' + x, lax0, false) ∨ lax0 ≠ [] ∧ m0 = m ∧ m1 = # buf2 ∧ abs-node(buf2, nd) ∧ abs-nodes(buf, m + # buf2, x, lax0.tail, false), ∀ buf3, m, x0, lax0, buf1, x1, lax1, weak. # buf3 < #(buf2 + buf) → abs-nodes(buf3, m, x0, lax0, false) → (abs-nodes(buf3 + buf1, m, x0 + x1, lax0 + lax1, weak) ↔ abs-nodes(buf1, m + # buf3, x1, lax1, weak)) ⊦ (∃ buf0, buf3. (buf2 + buf) + buf1 = buf0 + buf3 ∧ ( abs-padding(buf0) ∧ abs-nodes(buf3, m + # buf0, nd ' + x + x1, lax0 + lax1, weak) ∨ lax0 + lax1 ≠ [] ∧ (lax0 + lax1).head.off = m ∧ (lax0 + lax1).head.size = # buf0 ∧ abs-node(buf0, nd) ∧ abs-nodes(buf3, m + # buf0, x + x1, (lax0 + lax1).tail, weak))) ↔ abs-nodes(buf1, m + # buf2 + # buf, x1, lax1, weak)

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] → [weak, lax0 + lax1, x + x1, nd, m, (buf2 + buf) + buf1] flags: <all paths>

Additional information

Used simplifier rules: append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: weak, lax0 + lax1, x + x1, nd, m, (buf2 + buf) + buf1