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Proof step 13 from proof abs-nodes-no-padding-last (previous step)

# buf = (2 * NODE_HEADER_SIZE + a0.alignedsize) + (x0 + a ').flashsize, abs-nodes(buf, m, (a0 ' + x0) + a ', lax + ladr ', false), ∀ buf0, x, a, m, lax, ladr. # buf0 < # buf → # buf0 = (x + a ').flashsize ∧ abs-nodes(buf0, m, x + a ', lax + ladr ', false) → (∃ buf2, buf1. buf0 = buf2 + buf1 ∧ abs-nodes(buf2, m, x, lax, false) ∧ abs-node(buf1, a) ∧ ladr = @(# buf2 + m, 2 * NODE_HEADER_SIZE + a.alignedsize)) ⊦ ∃ buf0, buf1. buf = buf0 + buf1 ∧ abs-nodes(buf0, m, a0 ' + x0, lax, false) ∧ abs-node(buf1, a) ∧ ladr = @(# buf0 + m, 2 * NODE_HEADER_SIZE + a.alignedsize)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ # buf = (a ' + x).flashsize → ( abs-nodes(buf, m, a ' + x, lax, false) ↔ (∃ buf0, buf1. buf = buf0 + buf1 ∧ lax ≠ [] ∧ lax.head.off = m ∧ lax.head.size = # buf0 ∧ abs-node(buf0, a) ∧ abs-nodes(buf1, m + # buf0, x, lax.tail, false)))
Proof step 15:
# buf = (2 * NODE_HEADER_SIZE + a0.alignedsize) + (x0 + a ').flashsize, # buf = (a0 ' + x0 + a ').flashsize, ∃ buf0, buf1. buf = buf0 + buf1 ∧ lax + ladr ' ≠ [] ∧ (lax + ladr ').head.off = m ∧ (lax + ladr ').head.size = # buf0 ∧ abs-node(buf0, a0) ∧ abs-nodes(buf1, m + # buf0, x0 + a ', (lax + ladr ').tail, false), ∀ buf0, x, a, m, lax, ladr. # buf0 < # buf → # buf0 = (x + a ').flashsize ∧ abs-nodes(buf0, m, x + a ', lax + ladr ', false) → (∃ buf2, buf1. buf0 = buf2 + buf1 ∧ abs-nodes(buf2, m, x, lax, false) ∧ abs-node(buf1, a) ∧ ladr = @(# buf2 + m, 2 * NODE_HEADER_SIZE + a.alignedsize)) ⊦ ∃ buf0, buf1. buf = buf0 + buf1 ∧ abs-nodes(buf0, m, a0 ' + x0, lax, false) ∧ abs-node(buf1, a) ∧ ladr = @(# buf0 + m, 2 * NODE_HEADER_SIZE + a.alignedsize)

Rule argument

spec: <toplevel> name: abs-nodes-no-padding-rec seq: ⊦ # buf = (a ' + x).flashsize → ( abs-nodes(buf, m, a ' + x, lax, false) ↔ (∃ buf0, buf1. buf = buf0 + buf1 ∧ lax ≠ [] ∧ lax.head.off = m ∧ lax.head.size = # buf0 ∧ abs-node(buf0, a) ∧ abs-nodes(buf1, m + # buf0, x, lax.tail, false))) subst: [lax, x, a, m, buf] → [lax + ladr ', x0 + a ', a0, m, buf] flags: <all paths>

Additional information

Used simplifier rules: size-rec: ⊦ (a ' + x).flashsize = a.flashsize + x.flashsize;% ⊦ nd.flashsize = 2 * NODE_HEADER_SIZE + nd.alignedsize; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: lax + ladr ', x0 + a ', a0, m, buf