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Proof step 7 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 8:
⊦ x ≠ [] ∧ x.rest ≠ [] → (ordered<(x) ↔ x.first < x.rest.first ∧ ordered<(x.rest))
Proof step 9:
a0 ' + (x0 -1l a) ≠ [] ∧ (a0 ' + (x0 -1l a)).rest ≠ [], ordered<(a0 ' + x0), ∀ a1. ordered<(x0 -1l a1) ⊦ a = a0, (a0 ' + (x0 -1l a)).first < (a0 ' + (x0 -1l a)).rest.first ∧ ordered<((a0 ' + (x0 -1l a)).rest)

Rule argument

spec: inst: name: ols-rec seq: ⊦ x ≠ [] ∧ x.rest ≠ [] → ( ordered<(x) ↔ x.first < x.rest.first ∧ ordered<(x.rest)) subst: [x] → [a0 ' + (x0 -1l a)] flags: rewrite, current

Additional information

Used simplifier rules: ls-o: ⊦ ordered<(a ');%e-01: ⊦ a ' + x ≠ []; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%c-07: ⊦ (a ' + x).rest = x; from specification list%app: ⊦ x + [] = x; from specification list Used terms: a0 ' + (x0 -1l a)