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Proof step 7 from proof app (previous step)

unique_al((p0 ' + ax0) + p ' + ay), ∀ p, ay. unique_al(ax0 + p ' + ay) → ax0 + ay = [] ∨ ¬ (ax0 + ay).first.1 ∈ (ax0 + ay).rest ∧ unique_al((ax0 + ay).rest) ⊦ ¬ p0.1 ∈ ax0 ∧ ¬ p0.1 ∈ ay ∧ unique_al(ax0 + ay)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ unique_al(ay) ↔ ay = [] ∨ ¬ ay.first.1 ∈ ay.rest ∧ unique_al(ay.rest)
Proof step 9:
(p0 ' + ax0) + p ' + ay = [] ∨ ¬ ((p0 ' + ax0) + p ' + ay).first.1 ∈ ((p0 ' + ax0) + p ' + ay).rest ∧ unique_al(((p0 ' + ax0) + p ' + ay).rest), ∀ p, ay. unique_al(ax0 + p ' + ay) → ax0 + ay = [] ∨ ¬ (ax0 + ay).first.1 ∈ (ax0 + ay).rest ∧ unique_al((ax0 + ay).rest) ⊦ ¬ p0.1 ∈ ax0 ∧ ¬ p0.1 ∈ ay ∧ unique_al(ax0 + ay)

Rule argument

spec: inst: name: k-unique-r seq: ⊦ unique_al(ay) ↔ ay = [] ∨ ¬ ay.first.1 ∈ ay.rest ∧ unique_al(ay.rest) subst: [ay] → [(p0 ' + ax0) + p ' + ay] flags: rewrite, current

Additional information

Used terms: (p0 ' + ax0) + p ' + ay