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Proof step 9 from proof pos-get (previous step)

m < # x0, ∀ n. n < # x0 → pos(x0[n], x0) = n ⊦ pos(x0[m], a ' + x0) = m + 1, x0 = [], dups(x0), a ∈ x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ a ≠ b → pos(a, b ' + x) = pos(a, x) + 1
Proof step 11:
x0[m] ≠ a, m < # x0, ∀ n. n < # x0 → pos(x0[n], x0) = n ⊦ pos(x0[m], x0) + 1 = m + 1, x0 = [], dups(x0), a ∈ x0

Rule argument

spec: inst: name: pos-n seq: ⊦ a ≠ b → pos(a, b ' + x) = pos(a, x) + 1 subst: [x, b, a] → [x0, a, x0[m]] flags: rewrite, current

Additional information

Used simplifier rules: pos-y: ⊦ pos(a, a ' + x) = 0;%get-in: ⊦ m < # x → (¬ x[m] ∈ x ↔ false);%a: ⊦ (x + y) + z = x + y + z; from specification list%oz: ⊦ m + 1 ≠ 0; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2 Used terms: x0, a, x0[m]