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Proof step 15 from proof at-mod (previous step)

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])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
⊦ str ' + p ∈ dnode(md, sdt) ↔ str ∈ sdt ∧ p ∈ sdt[str]
Proof step 17:
⊦ str ∈ sdt → dnode(md, sdt)[str ' + p] = sdt[str][p]
Proof step 18:
sdt[str][p0] = dnode(md0, sdt0), str ∈ sdt, p0 ∈ sdt[str], q ∈ dnode(md, sdt)[str ' + p0, dnode(md1, sdt1)], ∀ 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

Additional information

Used simplifier rules: in-dnode: ⊦ str ' + p ∈ dnode(md, sdt) ↔ str ∈ sdt ∧ p ∈ sdt[str];%at-dnode: ⊦ str ∈ sdt → dnode(md, sdt)[str ' + p] = sdt[str][p];%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ str + p = str ' + p; (s)