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Proof step 55 from proof hasSharing-dd-help (previous step)

# refs + # refs1 = # refs2 + # refs0, # sks0 = 2 + # refs2 + # refs0, fits(r, sks0, st), validrefs(str, (r ' + refs0) + r2 ' + refs2 + r0 ', st[r -- _type] .type, st), validrefs(str, (r ' + refs) + r1 ' + refs1 + r0 ', st[r -- _type] .type, st), # refs < # refs0, ∀ n. n ≠ 0 ∧ ¬ # sks0 < n → st[r -- firstn(n, sks0)] .refval = ((r ' + refs0) + r2 ' + refs2 + r0 ')[n], ¬ # sks0 ≤ # refs → st[r -- firstn(1 + # refs, sks0)] .refval = r1 ⊦ r0 = jvmref, r1 = r0, r2 = r1, # refs = # refs0, refs0 = [], hasSharing(r, st), path∃(str, r1, r2, st), path∃(str, r2, r1, st)

User applied rule case distinction

And obtained the following new goals

Proof step 54:
st[r -- firstn(1 + # refs, sks0)] .refval = r1, # refs + # refs1 = # refs2 + # refs0, # sks0 = 2 + # refs2 + # refs0, fits(r, sks0, st), validrefs(str, (r ' + refs0) + r2 ' + refs2 + r0 ', st[r -- _type] .type, st), validrefs(str, (r ' + refs) + r1 ' + refs1 + r0 ', st[r -- _type] .type, st), # refs < # refs0, ∀ n. n ≠ 0 ∧ ¬ # sks0 < n → st[r -- firstn(n, sks0)] .refval = ((r ' + refs0) + r2 ' + refs2 + r0 ')[n] ⊦ r0 = jvmref, r1 = r0, r2 = r1, # refs = # refs0, refs0 = [], hasSharing(r, st), path∃(str, r1, r2, st), path∃(str, r2, r1, st)

Rule argument

[CASEDARG : pos: left 8, ()]

Additional information

[[CONCRETESIMPRULES : ([CSIMPFORWARD : ⊦ m ≤ n ∧ n < n0 → m < n0], [CSIMPFORWARD : ⊦ m < n ∧ m = n0 → n0 < n], [CSIMPSEQ : ⊦ ¬ m + n < m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])]]