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Proof step 17 from proof ex-validrefs-fits-stronger (previous step)

# sks = # refs1, st[r -- mkfs(class, ty0, fieldvar) '] = refval(r0), fits(r0, sks, st), validrefs(str, r0 ' + refs1, ty0, st), mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), st[r -- _type] .type ≤ ty, is_obj(str, r, st), ∀ n. n ≠ 0 ∧ ¬ # refs1 < n → st[r0 -- firstn(n, sks)] .refval = (r0 ' + refs1)[n], ∀ st0, ty, r1, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, r1 ' + refs, ty, st0) → (∃ sks. fits(r1, sks, st0) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st0[r1 -- firstn(n, sks)] .refval = (r1 ' + refs)[n]))) ⊦ r = jvmref, ∃ sks. fits(r, sks, st) ∧ # sks = 1 + # refs1 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + r0 ' + refs1)[n]), fits(r, mkfs(class, ty0, fieldvar) ' ' + sks, st) ∧ (n ≠ 0 ∧ n ≤ 1 + # refs1 → st[r -- firstn(n, mkfs(class, ty0, fieldvar) ' ' + sks)] .refval = (r ' + r0 ' + refs1)[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ fits(r, sk ' + sks, st) ↔ r ≠ jvmref ∧ is_referencevalue(st[r -- sk]) ∧ (is_classtype(st[r -- _type] .type) → sk ∈ instfields(st[r -- _type] .type .class)) ∧ (is_arraytype(st[r -- _type] .type) → is_indexkey(sk) ∧ 0 ≤ sk .index ∧ sk .index < st[r -- _length] .intval) ∧ fits(st[r -- sk] .refval, sks, st)
Proof step 16:
# sks = # refs1, st[r -- mkfs(class, ty0, fieldvar) '] = refval(r0), fits(r0, sks, st), validrefs(str, r0 ' + refs1, ty0, st), mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), st[r -- _type] .type ≤ ty, is_obj(str, r, st), ∀ n. n ≠ 0 ∧ ¬ # refs1 < n → st[r0 -- firstn(n, sks)] .refval = (r0 ' + refs1)[n], ∀ st0, ty, r1, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, r1 ' + refs, ty, st0) → (∃ sks. fits(r1, sks, st0) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st0[r1 -- firstn(n, sks)] .refval = (r1 ' + refs)[n]))) ⊦ r = jvmref, ∃ sks. fits(r, sks, st) ∧ # sks = 1 + # refs1 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + r0 ' + refs1)[n]), r ≠ jvmref ∧ is_referencevalue(st[r -- mkfs(class, ty0, fieldvar) ']) ∧ (is_classtype(st[r -- _type] .type) → mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class)) ∧ ( is_arraytype(st[r -- _type] .type) → is_indexkey(mkfs(class, ty0, fieldvar) ') ∧ 0 ≤ mkfs(class, ty0, fieldvar) ' .index ∧ mkfs(class, ty0, fieldvar) ' .index < st[r -- _length] .intval) ∧ fits(st[r -- mkfs(class, ty0, fieldvar) '] .refval, sks, st) ∧ (n ≠ 0 ∧ n ≤ 1 + # refs1 → st[r -- firstn(n, mkfs(class, ty0, fieldvar) ' ' + sks)] .refval = (r ' + r0 ' + refs1)[n])

Rule argument

[crewrite-arg: fits-storekeys-rec , F , [st, sks, sk, r] → [st, sks, mkfs(class, ty0, fieldvar) ', r] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, sks, mkfs(class, ty0, fieldvar) ', r)]]