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Proof step 116 from proof hasSharing-dd-help (previous step)

# sks = 2 + # refs0 + # refs2, validrefs(str, (r ' + refs) + r1 ' + refs1 + r0 ', st[r -- _type] .type, st), validrefs(str, (r ' + refs0) + r2 ' + refs2 + r0 ', st[r -- _type] .type, st), fits(r, sks, st), ∀ n. n ≠ 0 ∧ ¬ # sks < n → st[r -- firstn(n, sks)] .refval = ((r ' + refs0) + r2 ' + refs2 + r0 ')[n] ⊦ r2 = r1, r1 = r0, r0 = jvmref, path∃(str, r2, r1, st), path∃(str, r1, r2, st), hasSharing(r, st)

User applied rule insert lemma

And obtained the following new goals

Proof step 14:
validrefs(str, refs, ty, st), refs ≠ [] ⊦ ∃ sks. fits(refs .first, sks, st) ∧ 1 + # sks = # refs ∧ (∀ n. 0 < n ∧ n ≤ # sks → st[refs .first -- firstn(n, sks)] .refval = refs[n])
Proof step 115:
∃ sks. fits(((r ' + refs) + r1 ' + refs1 + r0 ') .first, sks, st) ∧ 1 + # sks = # ((r ' + refs) + r1 ' + refs1 + r0 ') ∧ (∀ n. 0 < n ∧ n ≤ # sks → st[((r ' + refs) + r1 ' + refs1 + r0 ') .first -- firstn(n, sks)] .refval = ((r ' + refs) + r1 ' + refs1 + r0 ')[n]), # sks = 2 + # refs0 + # refs2, validrefs(str, (r ' + refs) + r1 ' + refs1 + r0 ', st[r -- _type] .type, st), validrefs(str, (r ' + refs0) + r2 ' + refs2 + r0 ', st[r -- _type] .type, st), fits(r, sks, st), ∀ n. n ≠ 0 ∧ ¬ # sks < n → st[r -- firstn(n, sks)] .refval = ((r ' + refs0) + r2 ' + refs2 + r0 ')[n] ⊦ r2 = r1, r1 = r0, r0 = jvmref, path∃(str, r2, r1, st), path∃(str, r1, r2, st), hasSharing(r, st)

Rule argument

[LEMMAARG : ex-validrefs-fits-stronger, [str, refs, ty, st] → [str, (r ' + refs) + r1 ' + refs1 + r0 ', st[r -- _type] .type, st]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ refs + refs0 ≠ [] ↔ refs = [] ∧ refs0 = []], [CSIMPSEQ : ⊦ r ' ≠ []], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (str, (r ' + refs) + r1 ' + refs1 + r0 ', st[r -- _type] .type, st)]]