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Proof step 448 from proof hasSharing-put-index-refval (previous step)

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)]), 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) ⊦ sks1 = [], r2 = jvmref, r = r3, r1 = jvmref, sks2 = sks3, junkref(str, r, st), hasSharing(r3, st), hasSharing(r, 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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 345:
⊦ is_referencevalue(st[r -- sks]) ∧ sks ≠ [] → st[r -- sks] = st[st[r -- sks .first] .refval -- sks .rest]
Proof step 447:
is_referencevalue(st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks2)]) ∧ i ' ' + sks2 ≠ [], 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)][(st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks2) .first]) .refval -- (i ' ' + sks2) .rest]) = refval(r1), fits(r3, sks2, st[r2 -- i ', refval(r3)]), 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) ⊦ sks1 = [], r2 = jvmref, r = r3, r1 = jvmref, sks2 = sks3, junkref(str, r, st), hasSharing(r3, st), hasSharing(r, 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)

Rule argument

[x0lemmaarg: spec-name: storefuns instance-name: lemma-name: lookup-rec-refval antecedent: succedent: is_referencevalue(st[r -- sks]) ∧ sks ≠ [] → st[r -- sks] = st[st[r -- sks .first] .refval -- sks .rest] substlist: [sks, r, st] → [i ' ' + sks2, r2, st[r2 -- i ', refval(r3)]] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 3, 1, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : list, storekeys, app-02-ant, ⊦ ¬ sks + sks0 ≠ [] ↔ sks = [] ∧ sks0 = []], [CSIMPNAMEDSEQ : list, storekeys, e, ⊦ sk ' ≠ []], [CSIMPNAMEDSEQ : list, storekeys, a, ⊦ (sks + sks0) + sks1 = sks + sks0 + sks1], [CSIMPNAMEDSEQ : javavalue, , test-19, ⊦ is_referencevalue(refval(r))])], [EXTRATERMS : (i ' ' + sks2, r2, st[r2 -- i ', refval(r3)])]]