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Proof step 8 from proof abs-nodes-split-h (previous step)

abs-nodes(buf, m, (nd ' + x) + x1, lax, weak), ∀ buf0, m, x0, x1, lax, weak. # buf0 < # buf → abs-nodes(buf0, m, x0 + x1, lax, weak) → (∃ buf2, buf1, lax0, lax1. lax = lax0 + lax1 ∧ buf0 = buf2 + buf1 ∧ abs-nodes(buf2, m, x0, lax0, false) ∧ abs-nodes(buf1, m + # buf2, x1, lax1, weak)) ⊦ ∃ buf0, buf1, lax0, lax1. lax = lax0 + lax1 ∧ buf = buf0 + buf1 ∧ abs-nodes(buf0, m, nd ' + x, lax0, false) ∧ abs-nodes(buf1, m + # buf0, x1, lax1, weak)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ 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 10:
∃ buf0, buf1. buf = buf0 + buf1 ∧ ( abs-padding(buf0) ∧ abs-nodes(buf1, m + # buf0, nd ' + x + x1, lax, weak) ∨ lax ≠ [] ∧ lax.head.off = m ∧ lax.head.size = # buf0 ∧ abs-node(buf0, nd) ∧ abs-nodes(buf1, m + # buf0, x + x1, lax.tail, weak)), ∀ buf0, m, x0, x1, lax, weak. # buf0 < # buf → abs-nodes(buf0, m, x0 + x1, lax, weak) → (∃ buf2, buf1, lax0, lax1. lax = lax0 + lax1 ∧ buf0 = buf2 + buf1 ∧ abs-nodes(buf2, m, x0, lax0, false) ∧ abs-nodes(buf1, m + # buf2, x1, lax1, weak)) ⊦ ∃ buf0, buf1, lax0, lax1. lax = lax0 + lax1 ∧ buf = buf0 + buf1 ∧ abs-nodes(buf0, m, nd ' + x, lax0, false) ∧ abs-nodes(buf1, m + # buf0, 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, lax, x + x1, nd, m, buf] flags: <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: weak, lax, x + x1, nd, m, buf