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

sdt0[str0][p0] = dnode(md, sdt), q0 ∈ dnode(md0, sdt0)[str0 ' + p0, dnode(md, sdt[str, tree0])][str1 '], str1 ' ∈ dnode(md0, sdt0)[str0 ' + p0, dnode(md, sdt[str, tree0])], p0 ∈ sdt0[str0], str0 ∈ sdt0, ∀ md0, sdt0, md, sdt, q, str, tree0. dnode(md0, sdt0)[p0] = dnode(md, sdt) ∧ p0 ∈ dnode(md0, sdt0) ∧ q ∈ dnode(md0, sdt0)[p0, dnode(md, sdt[str, tree0])] → dnode(md0, sdt0)[p0, dnode(md, sdt[str, tree0])][q] = (p0 + str ' ⊑ q ⊃ tree0[restn(# p0, q.tail)];(q ⊑ p0 ⊃ dnode(md0, sdt0)[q][restn(# q, p0) + str ', tree0];dnode(md0, sdt0)[q])) ⊦ dnode(md0, sdt0)[str0 ' + p0, dnode(md, sdt[str, tree0])][str1 '][q0] = ((str0 ' + p0) + str ' ⊑ str1 ' + q0 ⊃ tree0[restn(# p0, q0.tail)] ;(str1 ' + q0 ⊑ str0 ' + p0 ⊃ dnode(md0, sdt0)[str1 ' + q0][restn(# q0, p0) + str ', tree0];dnode(md0, sdt0)[str1 ' + q0]) ), p0 ≠ ε ∧ str0 ∈ sdt0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 41:
⊦ tree[ε] = tree
Proof step 42:
⊦ dnode(md, sdt)[str ', tree0] = dnode(md, sdt[str, tree0])
Proof step 43:
⊦ str ' ∈ dnode(md, sdt) ↔ str ∈ sdt
Proof step 44:
⊦ ε ∈ tree
Proof step 45:
⊦ tree[ε, tree0] = tree0
Proof step 46:
sdt0[str0] = dnode(md, sdt), str0 ∈ sdt0, q0 ∈ dnode(md0, sdt0[str0, dnode(md, sdt[str, tree0])])[str1 '], str1 ∈ sdt0[str0, dnode(md, sdt[str, tree0])], ∀ md1, sdt1, md2, sdt2, q, str2, tree. dnode(md1, sdt1) = dnode(md2, sdt2) ∧ q ∈ dnode(md2, sdt2[str2, tree]) → dnode(md2, sdt2[str2, tree])[q] = (str2 ' ⊑ q ⊃ tree[q.tail];(q = ε ⊃ dnode(md1, sdt1[str2, tree]);dnode(md1, sdt1)[q])) ⊦ dnode(md0, sdt0[str0, dnode(md, sdt[str, tree0])])[str1 '][q0] = (str0 ' + str ' ⊑ str1 ' + q0 ⊃ tree0[q0.tail] ;(str1 ' + q0 = str0 ' ⊃ dnode(md0, sdt0)[str1 ' + q0][restn(# q0, ε) + str ', tree0];dnode(md0, sdt0)[str1 ' + 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;% ⊦ p ⊑ ε ↔ p = ε; (s)% ⊦ str ' + p ≠ ε; (s)% ⊦ p ⊑ str ' ↔ p = ε ∨ p = str '; (s)% ⊦ restn(0, p) = p; (s)% ⊦ # ε = 0; (s)% ⊦ ε + p = p; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ p + ε = p; (s)