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Proof step 54 from proof path-dups (previous step)

(r ' + x1)[m] = x2 .last, (r ' + x1)[# x2] = x2 .last, path(r ' + x2, H), reachable-n(H, # x2, r, x2 .last), # x2 < m, # x2 < # x1, m ≤ # x1, path(r ' + x1, H) ⊦ x1 = [], x2 = [], path(r ' + firstn(m, x1), H) ∧ (r ' + firstn(m, x1)) .last = x2 .last

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ x0 ≠ [] → (r ' + x0) .last = x0 .last
Proof step 53:
firstn(m, x1) ≠ [], (r ' + x1)[m] = x2 .last, (r ' + x1)[# x2] = x2 .last, path(r ' + x2, H), reachable-n(H, # x2, r, x2 .last), # x2 < m, # x2 < # x1, m ≤ # x1, path(r ' + x1, H) ⊦ x1 = [], x2 = [], path(r ' + firstn(m, x1), H) ∧ firstn(m, x1) .last = x2 .last

Rule argument

[x0lemmaarg: spec-name: list-last instance-name: reflist lemma-name: l-02 antecedent: succedent: x0 ≠ [] → (r ' + x0) .last = x0 .last substlist: [x0, r] → [firstn(m, x1), r] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 2, 1, 1, 1, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ firstn(n, x0) = [] ↔ n = 0], [CSIMPSEQ : ⊦ m < n → n ≠ 0])], [EXTRATERMS : (firstn(m, x1), r)]]