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Proof step 3 from proof unique-01 (previous step)

∀ a, d. ¬(a ∈ ax0 ∧ unique_al(ax0 + (a × d)') ∧ unique_al(ax0)) ⊦ ¬(a ∈ p + ax0 ∧ unique_al((p + ax0) + (a × d)') ∧ unique_al(p + ax0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
unique_al((p ' + ax0) + (a × d)'), unique_al(p ' + ax0), ∀ a0, d0. ¬ a0 ∈ ax0 ∨ ¬ unique_al(ax0 + (a0 × d0)') ∨ ¬ unique_al(ax0) ⊦ ¬ a ∈ p ' ∧ ¬ a ∈ ax0

Rule argument

Additional information

Used simplifier rules: inapp: ⊦ a ∈ ax + ay ↔ ¬ ¬(a ∈ ax ∨ a ∈ ay);%unique: ⊦ unique_al(ax + p ') → unique_al(ax);% ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1;% ⊦ p + ax = p ' + ax;