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Proof step 34 from proof abs-split-by-address-h (previous step)

abs-nodes(buf0, m + # buf1, nd ' + x2, @(m0, m1) ' + lax + ladr ' + lax1, boolvar0), abs-padding(buf1), ∀ buf2, m, x, lax0, ladr, lax1, boolvar0. # buf2 < #(buf1 + buf0) → abs-nodes(buf2, m, x, lax0 + ladr ' + lax1, boolvar0) → (∃ buf3, buf1, buf4, x0, a, x1. x = x0 + a ' + x1 ∧ # x0 = # lax0 ∧ buf2 = buf3 + buf1 + buf4 ∧ ladr = @(m + # buf3, 2 * NODE_HEADER_SIZE + a.alignedsize) ∧ # buf1 = 2 * NODE_HEADER_SIZE + a.alignedsize ∧ abs-node(buf1, a)) ⊦ ∃ buf, buf2, buf3, x, a, x0. nd ' + x2 = x + a ' + x0 ∧ # x = # lax + 1 ∧ buf1 + buf0 = buf + buf2 + buf3 ∧ ladr = @(m + # buf, 2 * NODE_HEADER_SIZE + a.alignedsize) ∧ # buf2 = 2 * NODE_HEADER_SIZE + a.alignedsize ∧ abs-node(buf2, a)

User applied rule apply induction

And obtained the following new goals

Proof step 35:
abs-nodes(buf0, m + # buf1, nd ' + x2, @(m0, m1) ' + lax + ladr ' + lax1, boolvar0), abs-padding(buf1), ∃ buf3, buf, buf4, x0, a, x1. nd ' + x2 = x0 + a ' + x1 ∧ # x0 = #(@(m0, m1) ' + lax) ∧ buf0 = buf3 + buf + buf4 ∧ ladr = @((m + # buf1) + # buf3, 2 * NODE_HEADER_SIZE + a.alignedsize) ∧ # buf = 2 * NODE_HEADER_SIZE + a.alignedsize ∧ abs-node(buf, a), ∀ buf2, m, x, lax0, ladr, lax1, boolvar0. # buf2 < #(buf1 + buf0) → abs-nodes(buf2, m, x, lax0 + ladr ' + lax1, boolvar0) → (∃ buf3, buf1, buf4, x0, a, x1. x = x0 + a ' + x1 ∧ # x0 = # lax0 ∧ buf2 = buf3 + buf1 + buf4 ∧ ladr = @(m + # buf3, 2 * NODE_HEADER_SIZE + a.alignedsize) ∧ # buf1 = 2 * NODE_HEADER_SIZE + a.alignedsize ∧ abs-node(buf1, a)) ⊦ ∃ buf, buf2, buf3, x, a, x0. nd ' + x2 = x + a ' + x0 ∧ # x = # lax + 1 ∧ buf1 + buf0 = buf + buf2 + buf3 ∧ ladr = @(m + # buf, 2 * NODE_HEADER_SIZE + a.alignedsize) ∧ # buf2 = 2 * NODE_HEADER_SIZE + a.alignedsize ∧ abs-node(buf2, a)

Rule argument

subst: [buf2, m, x, lax0, ladr, lax1, boolvar0] → [buf0, m + # buf1, nd ' + x2, @(m0, m1) ' + lax, ladr, lax1, boolvar0]

Additional information

Used simplifier rules: a: ⊦ (lax + lax0) + lax1 = lax + lax0 + lax1; from specification list[laddress-list]% ⊦ #(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)% ⊦ n + 0 = n; (s)% ⊦ ¬ abs-padding(mkbuf(0)); (s) Used terms: buf0, m + # buf1, nd ' + x2, @(m0, m1) ' + lax, ladr, lax1, boolvar0