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Proof step 43 from proof validrefs-putbyte (previous step)

st[r0 -- i0 '] = refval(r1), i0 < st[r0 -- _length] .intval, is_array(str, r0, st[r -- i ', byteval(by)]), st[r0 -- _type] .type ≤ ty, 0 < st[r0 -- _length] .intval, 0 ≤ i0, okbytearray(r, st), is_array(str, r0, st), validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st), ∀ by, i, ty, str, st, r, refs1. # refs1 < # (r0 ' + (r1 + refs)) → (okbytearray(r, st) ∧ validrefs(str, refs1, ty, st) → validrefs(str, refs1, ty, st[r -- i ', byteval(by)])) ⊦ r0 = jvmref, is_referencevalue(st[r -- i ', byteval(by)][r0 -- i0 ']) ∧ r1 = (st[r -- i ', byteval(by)][r0 -- i0 ']) .refval ∧ validrefs(str, (st[r -- i ', byteval(by)][r0 -- i0 ']) .refval ' + refs, st[r0 -- _type] .type .type, st[r -- i ', byteval(by)])

User applied rule apply induction

And obtained the following new goals

Rule argument

[substlist-arg: [by, i, ty, str, st, r, refs1] → [by, i, st[r0 -- _type] .type .type, str, st, r, (st[r -- i ', byteval(by)][r0 -- i0 ']) .refval ' + refs] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ # (refs + refs0) = # refs + # refs0], [CSIMPSEQ : ⊦ # r ' = 0 +1], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ r + refs = r ' + refs], [CSIMPSEQ : ⊦ m + n < m + n0 ↔ n < n0], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (by, i, st[r0 -- _type] .type .type, str, st, r, (st[r -- i ', byteval(by)][r0 -- i0 ']) .refval ' + refs)]]