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Proof step 412 from proof hasSharing-put-index-refval (previous step)

st[r -- sks1] = refval(r0), (st[r0 -- i ', refval(r3)][r3 -- (sks + i ' ' + sks0)]) = refval(r1), st[r0 -- _type] .type = mkarraytype(ty), st[r3 -- sks2] = refval(r1), st[r3 -- sks] = refval(r0), (st[r0 -- i ', refval(r3)][r3 -- sks]) = refval(r2), 0 < st[r0 -- _length] .intval, is_referencevalue(st[r0 -- i ', refval(r3)][r0 -- (i ' ' + sks2)]), fits(r3, sks2, st), is_referencevalue(st[r0 -- i ', refval(r3)][r0 -- (i ' ' + sks + i ' ' + sks0)]), 0 ≤ i, i < st[r0 -- _length] .intval, fits(r3, sks, st), fits(r2, i ' ' + sks0, st[r0 -- i ', refval(r3)]), fits(r, sks1, st), disjoint(str, r ' + r3 ', st), validref(str, r3, ty, st) ⊦ r3 = jvmref, sks1 = [], r0 = jvmref, r = r3, r1 = jvmref, sks2 = sks + i ' ' + sks0, junkref(str, r, st), hasSharing(r3, st), hasSharing(r, st), validreflist(str, refval(r) ' + refval(r3) ', mkclasstype(`java.lang.Object' .class), st), ∃ sks0, sks4. sks1 = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r0 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks1 → st[r -- sks4] .refval ≠ r0), ∃ sks0, sks2. sks1 = sks0 + i ' ' + sks2 ∧ st[r -- sks0] .refval = r0 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks0 → st[r -- sks4] .refval ≠ st[r -- sks0] .refval), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .refval ≠ r0), ∃ sks1, sks0. sks2 = sks1 + i ' ' + sks0 ∧ st[r3 -- sks1] .refval = r0 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks1 → st[r3 -- sks4] .refval ≠ st[r3 -- sks1] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r3 -- sks4] .refval ≠ r0), ∃ sks3, sks4. sks = sks3 + i ' ' + sks4 ∧ st[r3 -- sks3] .refval = r0 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks3 → st[r3 -- sks5] .refval ≠ st[r3 -- sks3] .refval)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 407:
⊦ validreflist(str, val ' + vals, ty, st) ↔ is_referencevalue(val) ∧ validref(str, val .refval, ty, st) ∧ validreflist(str, vals, ty, st)
Proof step 411:
st[r -- sks1] = refval(r0), (st[r0 -- i ', refval(r3)][r3 -- (sks + i ' ' + sks0)]) = refval(r1), st[r0 -- _type] .type = mkarraytype(ty), st[r3 -- sks2] = refval(r1), st[r3 -- sks] = refval(r0), (st[r0 -- i ', refval(r3)][r3 -- sks]) = refval(r2), 0 < st[r0 -- _length] .intval, is_referencevalue(st[r0 -- i ', refval(r3)][r0 -- (i ' ' + sks2)]), fits(r3, sks2, st), is_referencevalue(st[r0 -- i ', refval(r3)][r0 -- (i ' ' + sks + i ' ' + sks0)]), 0 ≤ i, i < st[r0 -- _length] .intval, fits(r3, sks, st), fits(r2, i ' ' + sks0, st[r0 -- i ', refval(r3)]), fits(r, sks1, st), disjoint(str, r ' + r3 ', st), validref(str, r3, ty, st) ⊦ r3 = jvmref, sks1 = [], r0 = jvmref, r = r3, r1 = jvmref, sks2 = sks + i ' ' + sks0, junkref(str, r, st), hasSharing(r3, st), hasSharing(r, st), is_referencevalue(refval(r)) ∧ validref(str, refval(r) .refval, mkclasstype(`java.lang.Object' .class), st) ∧ validreflist(str, refval(r3) ', mkclasstype(`java.lang.Object' .class), st), ∃ sks0, sks4. sks1 = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r0 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks1 → st[r -- sks4] .refval ≠ r0), ∃ sks0, sks2. sks1 = sks0 + i ' ' + sks2 ∧ st[r -- sks0] .refval = r0 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks0 → st[r -- sks4] .refval ≠ st[r -- sks0] .refval), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .refval ≠ r0), ∃ sks1, sks0. sks2 = sks1 + i ' ' + sks0 ∧ st[r3 -- sks1] .refval = r0 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks1 → st[r3 -- sks4] .refval ≠ st[r3 -- sks1] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r3 -- sks4] .refval ≠ r0), ∃ sks3, sks4. sks = sks3 + i ' ' + sks4 ∧ st[r3 -- sks3] .refval = r0 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks3 → st[r3 -- sks5] .refval ≠ st[r3 -- sks3] .refval)

Rule argument

[x0lemmaarg: spec-name: junkrefs instance-name: lemma-name: validreflist-rec antecedent: succedent: validreflist(str, val ' + vals, ty, st) ↔ is_referencevalue(val) ∧ validref(str, val .refval, ty, st) ∧ validreflist(str, vals, ty, st) substlist: [st, ty, vals, val, str] → [st, mkclasstype(`java.lang.Object' .class), refval(r3) ', refval(r), str] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 9)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, mkclasstype(`java.lang.Object' .class), refval(r3) ', refval(r), str)]]