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Proof step 13 from proof refine-add (previous step)

prefixn(# xs1 + 1, xs1 + ys) = prefixn(# xs1 + 1, xs0), # xs1 + # ys ≤ # xs0, # xs1 ≤ # xs0, ys = [] ∨ # ys = # xs1 + 1 ⊦ xs1 = prefixn(# xs1, xs0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ # x0 ≤ n → prefixn(n, x0 + x1) = x0 + prefixn(n - # x0, x1)
Proof step 15:
xs1 + prefixn(# xs1 + 1 - # xs1, ys) = prefixn(# xs1 + 1, xs0), # xs1 ≤ # xs0, # xs1 + # ys ≤ # xs0, # xs1 ≤ # xs1 + 1, ys = [] ∨ # ys = # xs1 + 1 ⊦ xs1 = prefixn(# xs1, xs0)

Rule argument

spec: list-prefixn name: prefixn-shift-right seq: ⊦ # x0 ≤ n → prefixn(n, x0 + x1) = x0 + prefixn(n - # x0, x1) subst: [x1, x0, n] → [ys, xs1, # xs1 + 1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%# x ≤ m ⊦ prefixn(m, x) = x; (s) Used terms: ys, xs1, # xs1 + 1