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Proof step 23 from proof abs-nodes-append-h (previous step)

abs-nodes(buf, m + # buf2, nd ' + x, @(m2, m3) ' + lax, false), abs-padding(buf2), ∀ buf3, m, x0, lax0, buf1, x1, lax1, weak. # buf3 < #(buf2 + buf) → abs-nodes(buf3, m, x0, lax0, false) → (abs-nodes(buf3 + buf1, m, x0 + x1, lax0 + lax1, weak) ↔ abs-nodes(buf1, m + # buf3, x1, lax1, weak)) ⊦ (∃ buf0, buf3. (buf2 + buf) + buf1 = buf0 + buf3 ∧ ( abs-padding(buf0) ∧ abs-nodes(buf3, m + # buf0, nd ' + x + x1, (@(m2, m3) ' + lax) + lax1, weak) ∨ m2 = m ∧ m3 = # buf0 ∧ abs-node(buf0, nd) ∧ abs-nodes(buf3, m + # buf0, x + x1, lax + lax1, weak))) ↔ abs-nodes(buf1, m + # buf2 + # buf, x1, lax1, weak)

User applied rule apply induction

And obtained the following new goals

Proof step 24:
abs-nodes(buf + buf1, m + # buf2, (nd ' + x) + x1, (@(m2, m3) ' + lax) + lax1, weak) ↔ abs-nodes(buf1, (m + # buf2) + # buf, x1, lax1, weak), abs-nodes(buf, m + # buf2, nd ' + x, @(m2, m3) ' + lax, false), abs-padding(buf2), ∀ buf3, m, x0, lax0, buf1, x1, lax1, weak. # buf3 < #(buf2 + buf) → abs-nodes(buf3, m, x0, lax0, false) → (abs-nodes(buf3 + buf1, m, x0 + x1, lax0 + lax1, weak) ↔ abs-nodes(buf1, m + # buf3, x1, lax1, weak)) ⊦ (∃ buf0, buf3. (buf2 + buf) + buf1 = buf0 + buf3 ∧ ( abs-padding(buf0) ∧ abs-nodes(buf3, m + # buf0, nd ' + x + x1, (@(m2, m3) ' + lax) + lax1, weak) ∨ m2 = m ∧ m3 = # buf0 ∧ abs-node(buf0, nd) ∧ abs-nodes(buf3, m + # buf0, x + x1, lax + lax1, weak))) ↔ abs-nodes(buf1, m + # buf2 + # buf, x1, lax1, weak)

Rule argument

subst: [buf3, m, x0, lax0, buf1, x1, lax1, weak] → [buf, m + # buf2, nd ' + x, @(m2, m3) ' + lax, buf1, x1, lax1, weak]

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% ⊦ mkbuf(0) + buf = buf; (s)% ⊦ # mkbuf(n) = n; (s)% ⊦ n + 0 = n; (s)% ⊦ ¬ abs-padding(mkbuf(0)); (s) Used terms: buf, m + # buf2, nd ' + x, @(m2, m3) ' + lax, buf1, x1, lax1, weak