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Proof step 46 from proof hassharing-validrefs-dups (previous step)

st[r1 -- sks] = refval(r2), st[r -- i '] = refval(r1), st[r1 -- sks0] = refval(r2), fits(r1, sks0, st), fits(r1, sks, st), validrefs(str, r1 ' + refs1, st[r1 -- _type] .type, st), is_array(str, r, st), i < st[r -- _length] .intval, validrefs(str, r1 ' + refs1, st[r -- _type] .type .type, st), 0 ≤ i, 0 < st[r -- _length] .intval, ∀ st0, str, refs, r0. # (r0 ' + refs) < # (r ' + (r1 + refs1)) → (validrefs(str, r0 ' + refs, st0[r0 -- _type] .type, st0) → hasSharing(r0, st0) ∨ ¬ r0 ∈ refs ∧ ¬ dups(refs)) ⊦ r = jvmref, r2 = jvmref, sks = [], sks0 = [], sks0 = sks, is_classtype(st[r -- _type] .type), r ∈ refs1, ¬ r1 ∈ refs1 ∧ ¬ dups(refs1), st[r -- (i ' ' + sks)] = st[r -- (i ' ' + sks0)] ∧ st[r -- (i ' ' + sks)] .refval ≠ jvmref ∧ fits(r, i ' ' + sks0, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ 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 45:
st[r1 -- sks] = refval(r2), st[r -- i '] = refval(r1), st[r1 -- sks0] = refval(r2), fits(r1, sks0, st), fits(r1, sks, st), validrefs(str, r1 ' + refs1, st[r1 -- _type] .type, st), is_array(str, r, st), i < st[r -- _length] .intval, validrefs(str, r1 ' + refs1, st[r -- _type] .type .type, st), 0 ≤ i, 0 < st[r -- _length] .intval, ∀ st0, str, refs, r0. # (r0 ' + refs) < # (r ' + (r1 + refs1)) → (validrefs(str, r0 ' + refs, st0[r0 -- _type] .type, st0) → hasSharing(r0, st0) ∨ ¬ r0 ∈ refs ∧ ¬ dups(refs)) ⊦ r = jvmref, r2 = jvmref, sks = [], sks0 = [], sks0 = sks, is_classtype(st[r -- _type] .type), r ∈ refs1, ¬ r1 ∈ refs1 ∧ ¬ dups(refs1), st[r -- (i ' ' + sks)] = st[r -- (i ' ' + sks0)] ∧ st[r -- (i ' ' + sks)] .refval ≠ 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, sks0, st)

Rule argument

[crewrite-arg: fits-storekeys-rec , F , [st, sks, sk, r] → [st, sks0, i ', r] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, sks0, i ', r)]]