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Proof step 24 from proof abs-nodes-subarray-failed (previous step)

# buf0 = x0.flashsize, # buf1 = nd.alignedsize + 2 * NODE_HEADER_SIZE, n1 < x0.flashsize, abs-node(buf1, nd), abs-nodes(buf0, m0 + # buf1, x0, lax0, false), is-aligned(n1, NODE_HEADER_SIZE), ∀ buf2, x, n, m, lax. # buf2 < #(buf1 + buf0) → # buf2 = x.flashsize ∧ n < x.flashsize ∧ abs-nodes(buf2, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) → n = 0 ∨ x = [] ∨ (∃ n0. n0 < # lax ∧ abs-nodes(subarray(buf2, 0, n), m, prefixn(n0, x), prefixn(n0, lax), true)) ⊦ buf1 + buf0 = mkbuf(0), n1 + # buf1 = 0, x0 = [], ∃ n0. ¬ # lax0 < n0 ∧ abs-nodes(buf1 + subarray(buf0, 0, n1), m0, prefixn(n0, nd ' + x0), prefixn(n0, @(m0, # buf1) ' + lax0), true)

User applied rule apply induction

And obtained the following new goals

Proof step 25:
# buf1 = nd.alignedsize + 2 * NODE_HEADER_SIZE, # buf0 = x0.flashsize, n1 < x0.flashsize, abs-node(buf1, nd), abs-nodes(buf0, m0 + # buf1, x0, lax0, false), is-aligned(n1, NODE_HEADER_SIZE), n1 = 0 ∨ x0 = [] ∨ (∃ n0. n0 < # lax0 ∧ abs-nodes(subarray(buf0, 0, n1), m0 + nd.alignedsize + 2 * NODE_HEADER_SIZE, prefixn(n0, x0), prefixn(n0, lax0), true)), ∀ buf2, x, n, m, lax. # buf2 < #(buf1 + buf0) → # buf2 = x.flashsize ∧ n < x.flashsize ∧ abs-nodes(buf2, m, x, lax, false) ∧ is-aligned(n, NODE_HEADER_SIZE) → n = 0 ∨ x = [] ∨ (∃ n0. n0 < # lax ∧ abs-nodes(subarray(buf2, 0, n), m, prefixn(n0, x), prefixn(n0, lax), true)) ⊦ n1 + # buf1 = 0, buf1 + buf0 = mkbuf(0), x0 = [], ∃ n0. ¬ # lax0 < n0 ∧ abs-nodes(buf1 + subarray(buf0, 0, n1), m0, prefixn(n0, nd ' + x0), prefixn(n0, @(m0, # buf1) ' + lax0), true)

Rule argument

subst: [buf2, x, n, m, lax] → [buf0, x0, n1, m0 + nd.alignedsize + 2 * NODE_HEADER_SIZE, lax0]

Additional information

Used simplifier rules: ⊦ #(buf + buf0) = # buf + # buf0; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)% ⊦ # buf = 0 ↔ buf = mkbuf(0); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ buf + buf0 = mkbuf(0) ↔ buf = mkbuf(0) ∧ buf0 = mkbuf(0); (ws)% ⊦ # mkbuf(n) = n; (s)% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws)% ⊦ 2 * m = 0 ↔ m = 0; (s)% ⊦ NODE_HEADER_SIZE ≠ 0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: buf0, x0, n1, m0 + nd.alignedsize + 2 * NODE_HEADER_SIZE, lax0