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Proof step 58 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 = [], reachable-n(H, m, r, x2 .last)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ n ≠ 0 → (reachable-n(H, n, r, r0) ↔ (∃ x. path(x, H) ∧ hd x = r ∧ x .last = r0 ∧ # x = n + 1))
Proof step 57:
m ≠ 0, (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 = [], ∃ x. path(x, H) ∧ hd x = r ∧ x .last = x2 .last ∧ # x = m + 1

Rule argument

[x0lemmaarg: spec-name: path-reachable instance-name: lemma-name: reach-n-path-n antecedent: succedent: n ≠ 0 → ( reachable-n(H, n, r, r0) ↔ (∃ x. path(x, H) ∧ hd x = r ∧ x .last = r0 ∧ # x = n + 1)) substlist: [r0, r, n, H] → [x2 .last, r, m, H] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 2)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (r ' + x0)[0] = r], [CSIMPSEQ : ⊦ (x0 + x1) + x2 = x0 + x1 + x2], [CSIMPSEQ : ⊦ ¬ m < 0])], [EXTRATERMS : (x2 .last, r, m, H)]]