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Proof step 452 from proof hasSharing-put-index-refval (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 450:
⊦ hasSharing(r, st) ↔ (∃ sks1, sks2. fits(r, sks1, st) ∧ fits(r, sks2, st) ∧ sks1 ≠ sks2 ∧ st[r -- sks1] .refval ≠ jvmref ∧ st[r -- sks1] = st[r -- sks2])
Proof step 451:
st[r -- sks] = refval(r2), st[r -- sks1] = refval(r2), (st[r2 -- i ', refval(r3)][r3 -- sks3]) = refval(r1), st[r2 -- _type] .type = mkarraytype(ty), (st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks2)]) = refval(r1), fits(r3, sks2, st[r2 -- i ', refval(r3)]), fits(r, sks, st), is_referencevalue(st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks3)]), 0 ≤ i, i < st[r2 -- _length] .intval, fits(r3, sks3, st[r2 -- i ', refval(r3)]), fits(r, sks1, st), disjoint(str, r ' + r3 ', st), validref(str, r3, ty, st), 0 < st[r2 -- _length] .intval, r3 = jvmref → is_reftype(ty) ⊦ sks = sks1, sks = [], r2 = jvmref, r = r3, r1 = jvmref, sks + i ' ' + sks2 = sks1 + i ' ' + sks3, sks1 = [], junkref(str, r, st), hasSharing(r3, st), ∃ sks1, sks2. fits(r, sks1, st) ∧ fits(r, sks2, st) ∧ sks1 ≠ sks2 ∧ st[r -- sks1] .refval ≠ jvmref ∧ st[r -- sks1] = st[r -- sks2], ∃ sks0, sks4. sks = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → 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)

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: hasSharing-def-better antecedent: succedent: hasSharing(r, st) ↔ (∃ sks1, sks2. fits(r, sks1, st) ∧ fits(r, sks2, st) ∧ sks1 ≠ sks2 ∧ st[r -- sks1] .refval ≠ jvmref ∧ st[r -- sks1] = st[r -- sks2]) substlist: [st, r] → [st, r] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((1, 0, 9)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, r)]]