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Proof step 9 from proof abs-nodes-append-h (previous step)

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)

User applied rule apply induction

And closed the goal!

Rule argument

subst: [buf3, m, x0, lax0, buf1, x1, lax1, weak] → [buf2, m + # buf, [], [], buf1, x1, lax1, weak]

Additional information

Used simplifier rules: ⊦ [] + x = x; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ [] + lax = lax; (s)%a: ⊦ (lax + lax0) + lax1 = lax + lax0 + lax1; from specification list[laddress-list]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ #(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)% ⊦ buf + buf0 = mkbuf(0) ↔ buf = mkbuf(0) ∧ buf0 = mkbuf(0); (ws)% ⊦ # mkbuf(n) = n; (s)% ⊦ n + 0 = n; (s)% ⊦ ¬ abs-padding(mkbuf(0)); (s) Used terms: buf2, m + # buf, [], [], buf1, x1, lax1, weak