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Proof step 25 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 ' + 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 ∧ str ∈ sdt

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 26:
⊦ tree[ε] = tree
Proof step 27:
⊦ dnode(md, sdt)[str ', tree0] = dnode(md, sdt[str, tree0])
Proof step 28:
⊦ str ' ∈ dnode(md, sdt) ↔ str ∈ sdt
Proof step 29:
⊦ ε ∈ tree
Proof step 30:
⊦ tree[ε, tree0] = tree0
Proof step 31:
sdt[str] = dnode(md0, sdt0), str ∈ sdt, q0 ∈ dnode(md, sdt[str, dnode(md1, sdt1)])[str0 '], str0 ∈ sdt[str, dnode(md1, sdt1)], ∀ 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])

Rule argument

Additional information

Used simplifier rules: nil-at: ⊦ tree[ε] = tree;%put-here: ⊦ dnode(md, sdt)[str ', tree0] = dnode(md, sdt[str, tree0]);%one-ind: ⊦ str ' ∈ dnode(md, sdt) ↔ str ∈ sdt;%nil-in: ⊦ ε ∈ tree;%nil-put: ⊦ tree[ε, tree0] = tree0;% ⊦ restn(0, p) = p; (s)% ⊦ # ε = 0; (s)% ⊦ p ⊑ ε ↔ p = ε; (s)% ⊦ str ' + p ≠ ε; (s)% ⊦ p ⊑ str ' ↔ p = ε ∨ p = str '; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ p + ε = p; (s)