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Proof step 10 from proof dec-nat (previous step)

⊦ n % 2 = 0, decode(~ n→i((n + 1) / 2)) = n

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ i < 0 → decode(i) = 2 * i→n(i) - 1
Proof step 12:
~ n→i((n + 1) / 2) < 0 ⊦ n % 2 = 0, 2 * i→n(~ n→i((n + 1) / 2)) - 1 = n

Rule argument

spec: <toplevel> name: dec-nat-neg seq: ⊦ i < 0 → decode(i) = 2 * i→n(i) - 1 subst: [i] → [~ n→i((n + 1) / 2)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ~ i < 0 ↔ 0 < i; (s)%0 ≤ i ⊦ (* i) < n→i(n) ↔ i→n(i) < n; (s)% ⊦ 0 < n ↔ n ≠ 0; (s)% ⊦ m / 2 = 0 ↔ m = 0 ∨ m = 1; (s)% ⊦ m + 1 ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ 1 = m + 1 ↔ m = 0; (s) Used terms: ~ n→i((n + 1) / 2)