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Proof step 6 from proof app (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
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)

Rule argument

Additional information

Used simplifier rules: inapp: ⊦ a ∈ ax + ay ↔ ¬ ¬(a ∈ ax ∨ a ∈ ay);%c-07: ⊦ (p ' + ax).rest = ax; from specification list[assoclista]%c-08: ⊦ (p ' + ax).first = p; from specification list[assoclista]%e-01: ⊦ p ' + ax ≠ []; from specification list[assoclista]%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[assoclista]%c: ⊦ p + ax = p ' + ax; from specification list[assoclista]