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Proof step 15 from proof to-to-leq (previous step)

n0 < # x, ∀ m, x. m ≤ n0 ∧ n0 ≤ # x ∧ m ≤ # x → x to n0 to m = x to m ⊦ ((x to n0) + x[n0] ' to n0) + ((x to n0) + x[n0] ')[n0] ' = x to n0 + 1, x = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
⊦ ¬ n < # x → (x + y)[n] = y[n - # x]
Proof step 17:
n0 < # x, ∀ m, x. m ≤ n0 ∧ n0 ≤ # x ∧ m ≤ # x → x to n0 to m = x to m ⊦ ((x to n0) + x[n0] ' to n0) + x[n0] '[n0 - #(x to n0)] ' = x to n0 + 1, x = [], n0 < #(x to n0)

Rule argument

spec: list-del name: get-append-grt seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x] subst: [n, y, x] → [n0, x[n0] ', x to n0] flags: <all paths>

Additional information

Used simplifier rules: length-to: ¬ # x < n ⊦ #(x to n) = n;% ⊦ n < # x → (x + y)[n] = x[n]; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ m < n → ¬ n < m; (s)% ⊦ ¬ n < n; (s) Used terms: n0, x[n0] ', x to n0