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Proof step 25 from proof abs-nodes-no-padding-last (previous step)

# buf0 = (x0 + a ').flashsize, # buf1 = a0.alignedsize + 2 * NODE_HEADER_SIZE, abs-node(buf1, a0), abs-nodes(buf0, m0 + # buf1, x0 + a ', lax1 + ladr ', false), ∀ buf2, x, a, m, lax, ladr. # buf2 < #(buf1 + buf0) → # buf2 = (x + a ').flashsize ∧ abs-nodes(buf2, m, x + a ', lax + ladr ', false) → (∃ buf3, buf1. buf2 = buf3 + buf1 ∧ abs-nodes(buf3, m, x, lax, false) ∧ abs-node(buf1, a) ∧ ladr = @(# buf3 + m, 2 * NODE_HEADER_SIZE + a.alignedsize)) ⊦ ∃ buf, buf2. buf1 + buf0 = buf + buf2 ∧ abs-nodes(buf, m0, a0 ' + x0, @(m0, # buf1) ' + lax1, false) ∧ abs-node(buf2, a) ∧ ladr = @(# buf + m0, 2 * NODE_HEADER_SIZE + a.alignedsize)

User applied rule apply induction

And obtained the following new goals

Proof step 26:
# buf1 = a0.alignedsize + 2 * NODE_HEADER_SIZE, # buf0 = (x0 + a ').flashsize, abs-node(buf1, a0), abs-nodes(buf0, m0 + # buf1, x0 + a ', lax1 + ladr ', false), ∃ buf3, buf. buf0 = buf3 + buf ∧ abs-nodes(buf3, m0 + # buf1, x0, lax1, false) ∧ abs-node(buf, a) ∧ ladr = @(# buf3 + m0 + # buf1, 2 * NODE_HEADER_SIZE + a.alignedsize), ∀ buf2, x, a, m, lax, ladr. # buf2 < #(buf1 + buf0) → # buf2 = (x + a ').flashsize ∧ abs-nodes(buf2, m, x + a ', lax + ladr ', false) → (∃ buf3, buf1. buf2 = buf3 + buf1 ∧ abs-nodes(buf3, m, x, lax, false) ∧ abs-node(buf1, a) ∧ ladr = @(# buf3 + m, 2 * NODE_HEADER_SIZE + a.alignedsize)) ⊦ ∃ buf, buf2. buf1 + buf0 = buf + buf2 ∧ abs-nodes(buf, m0, a0 ' + x0, @(m0, # buf1) ' + lax1, false) ∧ abs-node(buf2, a) ∧ ladr = @(# buf + m0, 2 * NODE_HEADER_SIZE + a.alignedsize)

Rule argument

subst: [buf2, x, a, m, lax, ladr] → [buf0, x0, a, m0 + # buf1, lax1, ladr]

Additional information

Used simplifier rules: abs-nodes-unique-strong-h: ⊦ abs-nodes(buf, m, x0, lax0, false) → (abs-nodes(buf, m, x1, lax1, boolvar1) ↔ x0 = x1 ∧ lax0 = lax1);% ⊦ #(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% ⊦ # 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, a, m0 + # buf1, lax1, ladr