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Proof step 8 from proof abs-nodes-append-h (previous step)

# buf0 = 0 ∧ [] = [] ∨ (∃ buf2, buf1. buf0 = buf2 + buf1 ∧ abs-padding(buf2) ∧ abs-nodes(buf1, m + # buf2, [], [], false)) ∨ false ∧ abs-failed(buf0) ∧ [] = [], ∀ buf, m, x0, lax0, buf1, x1, lax1, weak. # buf < # buf0 → abs-nodes(buf, m, x0, lax0, false) → (abs-nodes(buf + buf1, m, x0 + x1, lax0 + lax1, weak) ↔ abs-nodes(buf1, m + # buf, x1, lax1, weak)) ⊦ abs-nodes(buf0 + buf1, m, x1, lax1, weak) ↔ abs-nodes(buf1, m + # buf0, x1, lax1, weak)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
abs-nodes(buf2, m + # buf, [], [], false), abs-padding(buf), ∀ buf3, m, x0, lax0, buf1, x1, lax1, weak. # buf3 < #(buf + buf2) → 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)) ⊦ buf + buf2 = mkbuf(0), abs-nodes(buf2 + buf1, m + # buf, x1, lax1, weak) ↔ abs-nodes(buf1, m + # buf + # buf2, x1, lax1, weak)

Rule argument

Additional information

Used simplifier rules: abs-nodes-padding-prefix: ⊦ abs-padding(buf0) → ( abs-nodes(buf0 + buf1, m, x, lax, weak) ↔ abs-nodes(buf1, m + # buf0, x, lax, weak));% ⊦ #(buf + buf0) = # buf + # buf0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)% ⊦ # mkbuf(n) = n; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ mkbuf(0) + buf = buf; (s)% ⊦ # buf = 0 ↔ buf = mkbuf(0); (s)