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Proof step 48 from proof validrefs-put-statfield (previous step)

is_array(str, r, st), st[r -- _type] .type ≤ ty, ∀ str, ty, st, sk, val. validrefs(str, acc, ty, st[jvmref -- sk, val]) ↔ validrefs(str, acc, ty, st) ⊦ r = jvmref, validrefs(str, r ' + acc, ty, st[jvmref -- sk, val]) ↔ acc = [] ∨ (∃ i. ¬ i < 0 ∧ ¬ st[r -- _length] .intval ≤ i ∧ is_referencevalue(st[r -- i ']) ∧ acc .first = st[r -- i '] .refval ∧ validrefs(str, acc, st[r -- _type] .type .type, st))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ 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 47:
is_array(str, r, st[jvmref -- sk, val]) ∧ (st[jvmref -- sk, val][r -- _type]) .type ≤ ty, is_array(str, r, st), st[r -- _type] .type ≤ ty, ∀ str, ty, st, sk, val. validrefs(str, acc, ty, st[jvmref -- sk, val]) ↔ validrefs(str, acc, ty, st) ⊦ r = jvmref, acc = [] ∨ (∃ i. 0 ≤ i ∧ i < (st[jvmref -- sk, val][r -- _length]) .intval ∧ is_referencevalue(st[jvmref -- sk, val][r -- i ']) ∧ acc .first = (st[jvmref -- sk, val][r -- i ']) .refval ∧ validrefs(str, acc, (st[jvmref -- sk, val][r -- _type]) .type .type, st[jvmref -- sk, val])) ↔ acc = [] ∨ (∃ i. ¬ i < 0 ∧ ¬ st[r -- _length] .intval ≤ i ∧ is_referencevalue(st[r -- i ']) ∧ acc .first = st[r -- i '] .refval ∧ validrefs(str, acc, st[r -- _type] .type .type, st))

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st[jvmref -- sk, val], ty, acc, r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ is_array(str, r, st) → is_array (str, r, st[jvmref -- sk, val])], [CSIMPSEQ : ⊦ r ≠ r0 → (st[r0 -- sk0, val][r -- sk]) = st[r -- sk]])], [EXTRATERMS : (st[jvmref -- sk, val], ty, acc, r, str)]]