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Proof step 45 from proof ex-validrefs-fits-stronger (previous step)

# sks = # refs1, st[r -- i '] = refval(r0), fits(r0, sks, st), validrefs(str, r0 ' + refs1, st[r -- _type] .type .type, st), i < st[r -- _length] .intval, st[r -- _type] .type ≤ ty, is_array(str, r, st), 0 < st[r -- _length] .intval, 0 ≤ i, ∀ 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, fits(r, i ' ' + sks, st) ∧ (n ≠ 0 ∧ n ≤ 1 + # refs1 → st[r -- firstn(n, i ' ' + sks)] .refval = (r ' + r0 ' + refs1)[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ 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 44:
# sks = # refs1, st[r -- i '] = refval(r0), fits(r0, sks, st), validrefs(str, r0 ' + refs1, st[r -- _type] .type .type, st), i < st[r -- _length] .intval, st[r -- _type] .type ≤ ty, is_array(str, r, st), 0 < st[r -- _length] .intval, 0 ≤ i, ∀ 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, r ≠ jvmref ∧ is_referencevalue(st[r -- i ']) ∧ (is_classtype(st[r -- _type] .type) → i ' ∈ instfields(st[r -- _type] .type .class)) ∧ (is_arraytype(st[r -- _type] .type) → is_indexkey(i ') ∧ 0 ≤ i ' .index ∧ i ' .index < st[r -- _length] .intval) ∧ fits(st[r -- i '] .refval, sks, st) ∧ (n ≠ 0 ∧ n ≤ 1 + # refs1 → st[r -- firstn(n, i ' ' + sks)] .refval = (r ' + r0 ' + refs1)[n])

Rule argument

[crewrite-arg: fits-storekeys-rec , F , [st, sks, sk, r] → [st, sks, i ', r] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, sks, i ', r)]]