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Proof step 47 from proof at-mod (previous step)

sdt[str][p0] = dnode(md0, sdt0), q0 ∈ dnode(md, sdt)[str ' + p0, dnode(md1, sdt1)][str0 '], str0 ' ∈ dnode(md, sdt[str, sdt[str][p0, dnode(md1, sdt1)]]), p0 ∈ sdt[str], str ∈ 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)][str0 '][q0] = (str0 ' + q0 ⊑ str ' + p0 ⊃ dnode(md, sdt)[str0 ' + q0][restn(# q0, p0), dnode(md1, sdt1)] ;dnode(md, sdt)[str ' + p0, dnode(md0, sdt1)][str0 ' + q0] ), p0 = ε

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 48:
p ≠ ε ⊦ str ∈ sdt → dnode(md, sdt)[str ' + p, tree0] = dnode(md, sdt[str, sdt[str][p, tree0]])
Proof step 49:
⊦ str ' ∈ dnode(md, sdt) ↔ str ∈ sdt
Proof step 50:
sdt[str][p0] = dnode(md0, sdt0), str ∈ sdt, p0 ∈ sdt[str], str0 ∈ sdt[str, sdt[str][p0, dnode(md1, sdt1)]], q0 ∈ dnode(md, sdt[str, sdt[str][p0, dnode(md1, sdt1)]])[str0 '], ∀ 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, sdt[str][p0, dnode(md1, sdt1)]])[str0 '][q0] = (str0 ' + q0 ⊑ str ' + p0 ⊃ dnode(md, sdt)[str0 ' + q0][restn(# q0, p0), dnode(md1, sdt1)] ;dnode(md, sdt[str, sdt[str][p0, dnode(md0, sdt1)]])[str0 ' + q0] ), p0 = ε

Rule argument

Additional information

Used simplifier rules: put-rec: p ≠ ε ⊦ str ∈ sdt → dnode(md, sdt)[str ' + p, tree0] = dnode(md, sdt[str, sdt[str][p, tree0]]);%one-ind: ⊦ str ' ∈ dnode(md, sdt) ↔ str ∈ sdt;%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]