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Proof step 40 from proof at-mod (previous step)

sdt[str] = dnode(md0, sdt0), str0 ∈ sdt[str, dnode(md1, sdt1)], str ∈ sdt, q0 ∈ sdt[str0], ∀ md2, sdt2, md3, sdt3, q, md4, sdt4. dnode(md2, sdt2) = dnode(md3, sdt3) ∧ q ∈ dnode(md4, sdt4) → dnode(md4, sdt4)[q] = (q = ε ⊃ dnode(md4, sdt4);dnode(md3, sdt4)[q]) ⊦ dnode(md, sdt[str, dnode(md1, sdt1)])[str0 '][q0] = (str0 ' + q0 = str ' ⊃ dnode(md, sdt)[str0 ' + q0][restn(# q0, ε), dnode(md1, sdt1)];dnode(md, sdt[str, dnode(md0, sdt1)])[str0 ' + q0]), str0 = str

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 41:
str ∈ sdt ⊦ dnode(md, sdt)[str '] = sdt[str]
Proof step 42:
⊦ str ∈ sdt → dnode(md, sdt)[str ' + p] = sdt[str][p]
Proof step 43:
sdt[str] = dnode(md0, sdt0), q0 ∈ sdt[str0], str ∈ sdt, str0 ∈ sdt, ∀ md2, sdt2, md3, sdt3, q, md4, sdt4. dnode(md2, sdt2) = dnode(md3, sdt3) ∧ q ∈ dnode(md4, sdt4) → dnode(md4, sdt4)[q] = (q = ε ⊃ dnode(md4, sdt4);dnode(md3, sdt4)[q]) ⊦ sdt[str0][q0] = (str0 ' + q0 = str ' ⊃ sdt[str0][q0][restn(# q0, ε), dnode(md1, sdt1)];dnode(md, sdt[str, dnode(md0, sdt1)])[str0 ' + q0]), str0 = str

Rule argument

Additional information

Used simplifier rules: one-at: str ∈ sdt ⊦ dnode(md, sdt)[str '] = 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 ≠ str0 → sdt[str0, tree][str] = sdt[str]; (s)% ⊦ str ≠ str0 → (str ∈ sdt[str0, tree] ↔ str ∈ sdt); (s)% ⊦ str ∈ sdt[str0, tree] ↔ str = str0 ∨ str ∈ sdt; (ws)