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Proof step 8 from proof del1 (previous step)

ordered<(a0 ' + x0), ∀ a1. ordered<(x0 -1l a1) ⊦ a = a0, ordered<(a0 ' + (x0 -1l a))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ x ≠ [] ∧ x.tail ≠ [] → (ordered<(x) ↔ x.head < x.tail.head ∧ ordered<(x.tail))
Proof step 10:
ordered<(a0 ' + x0), ∀ a1. ordered<(x0 -1l a1) ⊦ a0 ' + (x0 -1l a) = [], (a0 ' + (x0 -1l a)).tail = [], a = a0, (a0 ' + (x0 -1l a)).head < (a0 ' + (x0 -1l a)).tail.head ∧ ordered<((a0 ' + (x0 -1l a)).tail)

Rule argument

spec: <toplevel> name: ols-rec seq: ⊦ x ≠ [] ∧ x.tail ≠ [] → ( ordered<(x) ↔ x.head < x.tail.head ∧ ordered<(x.tail)) subst: [x] → [a0 ' + (x0 -1l a)] flags: <all paths>

Additional information

Used simplifier rules: ls-o: ⊦ ordered<(a ');% ⊦ a ' + x ≠ []; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ (a ' + x).tail = x; (s)% ⊦ x + [] = x; (s) Used terms: a0 ' + (x0 -1l a)