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Proof step 29 from proof validrefs-putbyte-field (previous step)

st[r0 -- i '] = refval(r1), 0 ≤ i, 0 < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, is_array(str, r0, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), st[r0 -- _type] .type ≤ ty, i < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), is_array(str, r0, st), ∀ by, fieldvar, class, r, st0, ty, str, refs0. # refs0 < # (r0 + r1 + refs) → ( validrefs(str, refs0, ty, st0[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]) → refs0 = [] ∨ junkref(str, refs0 .first, st0) ∨ validrefs(str, refs0, ty, st0)) ⊦ validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st), r0 = jvmref, junkref(str, r0, st)

User applied rule apply induction

And obtained the following new goals

Proof step 28:
r1 ' + refs = [] ∨ junkref(str, (r1 ' + refs) .first, st) ∨ validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st), st[r0 -- i '] = refval(r1), 0 ≤ i, 0 < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, is_array(str, r0, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), st[r0 -- _type] .type ≤ ty, i < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), is_array(str, r0, st), ∀ by, fieldvar, class, r, st0, ty, str, refs0. # refs0 < # (r0 + r1 + refs) → ( validrefs(str, refs0, ty, st0[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]) → refs0 = [] ∨ junkref(str, refs0 .first, st0) ∨ validrefs(str, refs0, ty, st0)) ⊦ validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st), r0 = jvmref, junkref(str, r0, st)

Rule argument

[substlist-arg: [by, fieldvar, class, r, st0, ty, str, refs0] → [by, fieldvar, class, r, st, st[r0 -- _type] .type .type, str, r1 ' + refs] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ # (refs + refs0) = # refs + # refs0], [CSIMPSEQ : ⊦ # r ' = 0 +1], [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, fieldvar, class, r, st, st[r0 -- _type] .type .type, str, r1 ' + refs)]]