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Proof step 4 from proof at-mod (previous step)

dnode(md, sdt)[p] = dnode(md0, sdt0), p ∈ dnode(md, sdt), q ∈ dnode(md, sdt)[p, dnode(md1, sdt1)] ⊦ dnode(md, sdt)[p, dnode(md1, sdt1)][q] = (q ⊑ p ⊃ dnode(md, sdt)[q][restn(# q, p), dnode(md1, sdt1)];dnode(md, sdt)[p, dnode(md0, sdt1)][q])

User applied rule structural induction

And obtained the following new goals

Proof step 5:
dnode(md, sdt)[ε] = dnode(md0, sdt0), q ∈ dnode(md, sdt)[ε, dnode(md1, sdt1)], ε ∈ dnode(md, sdt) ⊦ dnode(md, sdt)[ε, dnode(md1, sdt1)][q] = (q ⊑ ε ⊃ dnode(md, sdt)[q][restn(# q, ε), dnode(md1, sdt1)];dnode(md, sdt)[ε, dnode(md0, sdt1)][q])
Proof step 15:
dnode(md, sdt)[str + p0] = dnode(md0, sdt0), q ∈ dnode(md, sdt)[str + p0, dnode(md1, sdt1)], str + p0 ∈ dnode(md, sdt), ∀ md, sdt, md0, sdt0, q, md1, sdt1. dnode(md, sdt)[p0] = dnode(md0, sdt0) ∧ p0 ∈ dnode(md, sdt) ∧ q ∈ dnode(md, sdt)[p0, dnode(md1, sdt1)] → dnode(md, sdt)[p0, dnode(md1, sdt1)][q] = (q ⊑ p0 ⊃ dnode(md, sdt)[q][restn(# q, p0), dnode(md1, sdt1)];dnode(md, sdt)[p0, dnode(md0, sdt1)][q]) ⊦ dnode(md, sdt)[str + p0, dnode(md1, sdt1)][q] = (q ⊑ str + p0 ⊃ dnode(md, sdt)[q][restn(# q, str + p0), dnode(md1, sdt1)];dnode(md, sdt)[str + p0, dnode(md0, sdt1)][q])

Rule argument

indvar: p

Additional information

Used simplifier rules: path freely generated by ε, +;;