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Proof step 7 from proof abs-nodes-subarray-failed (previous step)

# buf = (2 * NODE_HEADER_SIZE + nd.alignedsize) + x0.flashsize, n < # buf, abs-nodes(buf, m, nd ' + x0, lax, false), is-aligned(n, NODE_HEADER_SIZE), ∀ buf0, x, n, m, lax. # buf0 < # buf → # buf0 = x.flashsize ∧ n < x.flashsize ∧ abs-nodes(buf0, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) → n = 0 ∨ x = [] ∨ (∃ n0. n0 < # lax ∧ abs-nodes(subarray(buf0, 0, n), m, prefixn(n0, x), prefixn(n0, lax), true)) ⊦ buf = mkbuf(0), n = 0, ∃ n0. n0 < # lax ∧ abs-nodes(subarray(buf, 0, n), m, prefixn(n0, nd ' + x0), prefixn(n0, lax), true)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ # 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 9:
# buf = (2 * NODE_HEADER_SIZE + nd.alignedsize) + x0.flashsize, # buf = (nd ' + x0).flashsize, n < # buf, is-aligned(n, NODE_HEADER_SIZE), ∃ buf0, buf1. buf = buf0 + buf1 ∧ lax ≠ [] ∧ lax.head.off = m ∧ lax.head.size = # buf0 ∧ abs-node(buf0, nd) ∧ abs-nodes(buf1, m + # buf0, x0, lax.tail, false), ∀ buf0, x, n, m, lax. # buf0 < # buf → # buf0 = x.flashsize ∧ n < x.flashsize ∧ abs-nodes(buf0, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) → n = 0 ∨ x = [] ∨ (∃ n0. n0 < # lax ∧ abs-nodes(subarray(buf0, 0, n), m, prefixn(n0, x), prefixn(n0, lax), true)) ⊦ buf = mkbuf(0), n = 0, ∃ n0. n0 < # lax ∧ abs-nodes(subarray(buf, 0, n), m, prefixn(n0, nd ' + x0), prefixn(n0, lax), true)

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, x0, nd, 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: ⊦ (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 Used terms: lax, x0, nd, m, buf