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Proof step 51 from proof validrefs-put-badarrayfield-01 (previous step)

is_array(str, r, st), is_arraytype(st[r0 -- _type] .type), st[r -- _type] .type ≤ ty, ∀ st, r0, str, ty, fs, val. is_arraytype(st[r0 -- _type] .type) ∧ validrefs(str, acc, ty, st) → fs ' = _type ∨ fs ' = _length ∨ acc = [] ∨ junkref(str, acc .first, st) ∨ dups(acc .butlast) ∨ validrefs(str, acc, ty, st[r0 -- fs ', val]), acc = [] ∨ 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ acc .first = st[r -- i '] .refval ∧ validrefs(str, acc, st[r -- _type] .type .type, st) ⊦ fs ' = _type, fs ' = _length, r = jvmref, junkref(str, r, st[r0 -- fs ', val]), junkref(str, r, st), dups((r ' + acc) .butlast), validrefs(str, r ' + acc, ty, st[r0 -- fs ', val])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ refs .first = st[r -- i '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 50:
is_array(str, r, st[r0 -- fs ', val]) ∧ (st[r0 -- fs ', val][r -- _type]) .type ≤ ty, is_array(str, r, st), is_arraytype(st[r0 -- _type] .type), st[r -- _type] .type ≤ ty, ∀ st, r0, str, ty, fs, val. is_arraytype(st[r0 -- _type] .type) ∧ validrefs(str, acc, ty, st) → fs ' = _type ∨ fs ' = _length ∨ acc = [] ∨ junkref(str, acc .first, st) ∨ dups(acc .butlast) ∨ validrefs(str, acc, ty, st[r0 -- fs ', val]), acc = [] ∨ 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ acc .first = st[r -- i '] .refval ∧ validrefs(str, acc, st[r -- _type] .type .type, st) ⊦ fs ' = _type, fs ' = _length, r = jvmref, junkref(str, r, st[r0 -- fs ', val]), junkref(str, r, st), dups((r ' + acc) .butlast), acc = [] ∨ (∃ i. 0 ≤ i ∧ i < (st[r0 -- fs ', val][r -- _length]) .intval ∧ is_referencevalue(st[r0 -- fs ', val][r -- i ']) ∧ acc .first = (st[r0 -- fs ', val][r -- i ']) .refval ∧ validrefs(str, acc, (st[r0 -- fs ', val][r -- _type]) .type .type, st[r0 -- fs ', val]))

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st[r0 -- fs ', val], ty, acc, r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ fs ' ≠ _length ∧ fs ' ≠ _type → ( is_array (str, r, st[r0 -- fs ', val]) ↔ is_array(str, r, st))], [CSIMPSEQ : ⊦ sk ≠ sk0 → (st[r -- sk, val][r0 -- sk0]) = st[r0 -- sk0]])], [EXTRATERMS : (st[r0 -- fs ', val], ty, acc, r, str)]]