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Proof step 367 from proof hasSharing-put-index-refval (previous step)

st[r -- sks1] = refval(r2), (st[r2 -- i ', refval(r4)][r4 -- sks3]) = refval(r1), st[r2 -- _type] .type = mkarraytype(ty), (st[r2 -- i ', refval(r4)][r4 -- (sks + i ' ' + sks0)]) = refval(r1), st[r4 -- sks] = refval(r2), 0 < st[r2 -- _length] .intval, is_referencevalue(st[r2 -- i ', refval(r4)][r2 -- (i ' ' + sks3)]), 0 ≤ i, i < st[r2 -- _length] .intval, fits(r4, sks3, st[r2 -- i ', refval(r4)]), fits(r, sks1, st), disjoint(str, r ' + r4 ', st), validref(str, r4, ty, st), is_referencevalue(st[r2 -- i ', refval(r4)][r2 -- (i ' ' + sks + i ' ' + sks0)]), fits(r4, sks, st), fits(r4, sks0, st[r2 -- i ', refval(r4)]) ⊦ r4 = jvmref, sks1 = [], r2 = jvmref, r = r4, r1 = jvmref, sks + i ' ' + sks0 = sks3, junkref(str, r, st), hasSharing(r4, st), hasSharing(r, st), validreflist(str, refval(r) ' + refval(r4) ', mkclasstype(`java.lang.Object' .class), st), ∃ sks0, sks4. sks1 = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks1 → st[r -- sks4] .refval ≠ r2), ∃ sks0, sks2. sks1 = sks0 + i ' ' + sks2 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks0 → st[r -- sks4] .refval ≠ st[r -- sks0] .refval), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .refval ≠ r2), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r4 -- sks4] .refval ≠ r2), ∃ sks4, sks5. sks = sks4 + i ' ' + sks5 ∧ st[r4 -- sks4] .refval = r2 ∧ (∀ sks2. sks2 + i ' ' ⊑ sks4 → st[r4 -- sks2] .refval ≠ st[r4 -- sks4] .refval)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 362:
⊦ validreflist(str, val ' + vals, ty, st) ↔ is_referencevalue(val) ∧ validref(str, val .refval, ty, st) ∧ validreflist(str, vals, ty, st)
Proof step 366:
st[r -- sks1] = refval(r2), (st[r2 -- i ', refval(r4)][r4 -- sks3]) = refval(r1), st[r2 -- _type] .type = mkarraytype(ty), (st[r2 -- i ', refval(r4)][r4 -- (sks + i ' ' + sks0)]) = refval(r1), st[r4 -- sks] = refval(r2), 0 < st[r2 -- _length] .intval, is_referencevalue(st[r2 -- i ', refval(r4)][r2 -- (i ' ' + sks3)]), 0 ≤ i, i < st[r2 -- _length] .intval, fits(r4, sks3, st[r2 -- i ', refval(r4)]), fits(r, sks1, st), disjoint(str, r ' + r4 ', st), validref(str, r4, ty, st), is_referencevalue(st[r2 -- i ', refval(r4)][r2 -- (i ' ' + sks + i ' ' + sks0)]), fits(r4, sks, st), fits(r4, sks0, st[r2 -- i ', refval(r4)]) ⊦ r4 = jvmref, sks1 = [], r2 = jvmref, r = r4, r1 = jvmref, sks + i ' ' + sks0 = sks3, junkref(str, r, st), hasSharing(r4, st), hasSharing(r, st), is_referencevalue(refval(r)) ∧ validref(str, refval(r) .refval, mkclasstype(`java.lang.Object' .class), st) ∧ validreflist(str, refval(r4) ', mkclasstype(`java.lang.Object' .class), st), ∃ sks0, sks4. sks1 = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks1 → st[r -- sks4] .refval ≠ r2), ∃ sks0, sks2. sks1 = sks0 + i ' ' + sks2 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks0 → st[r -- sks4] .refval ≠ st[r -- sks0] .refval), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .refval ≠ r2), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r4 -- sks4] .refval ≠ r2), ∃ sks4, sks5. sks = sks4 + i ' ' + sks5 ∧ st[r4 -- sks4] .refval = r2 ∧ (∀ sks2. sks2 + i ' ' ⊑ sks4 → st[r4 -- sks2] .refval ≠ st[r4 -- sks4] .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(r4) ', 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(r4) ', refval(r), str)]]