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Proof step 188 from proof hasSharing-put-index-refval (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 175:
⊦ sks ≠ [] ∧ sks0 ≠ [] → st[r -- (sks + sks0)] = st[st[r -- sks] .refval -- sks0]
Proof step 187:
sks ≠ [] ∧ i ' ' + sks0 ≠ [], (st[r -- i ', refval(r2)][r2 -- sks]) = refval(r0), st[r2 -- sks] = refval(r), st[r2 -- sks1] = refval(r1), st[r -- _type] .type = mkarraytype(ty), (st[r -- i ', refval(r2)][(st[r -- i ', refval(r2)][r2 -- sks]) .refval -- (i ' ' + sks0)]) = refval(r1), (st[r -- i ', refval(r2)][r0 -- i ']) = refval(r3), fits(r3, sks0, st[r -- i ', refval(r2)]), fits(r2, sks, st), 0 ≤ i, i < st[r -- _length] .intval, fits(r2, sks1, st), is_referencevalue(st[r -- i ', refval(r2)][r -- (i ' ' + sks1)]), 0 < st[r -- _length] .intval, validref(str, r2, ty, st), disjoint(str, r ' + r2 ', st), is_arraytype(st[r0 -- _type] .type) → i < st[r0 -- _length] .intval ⊦ r0 = jvmref, sks1 = sks + i ' ' + sks0, r2 = jvmref, r1 = jvmref, r = r2, r = jvmref, is_classtype(st[r0 -- _type] .type), hasSharing(r, st), hasSharing(r2, st), junkref(str, r, st), ∃ sks3, sks4. sks = sks3 + i ' ' + sks4 ∧ st[r2 -- sks3] .refval = r ∧ (∀ sks2. sks2 + i ' ' ⊑ sks3 → st[r2 -- sks2] .refval ≠ st[r2 -- sks3] .refval), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks → st[r2 -- sks2] .refval ≠ r), ∃ sks2, sks0. sks1 = sks2 + i ' ' + sks0 ∧ st[r2 -- sks2] .refval = r ∧ (∀ sks3. sks3 + i ' ' ⊑ sks2 → st[r2 -- sks3] .refval ≠ st[r2 -- sks2] .refval)

Rule argument

[x0lemmaarg: spec-name: storefuns instance-name: lemma-name: lookup-append antecedent: succedent: sks ≠ [] ∧ sks0 ≠ [] → st[r -- (sks + sks0)] = st[st[r -- sks] .refval -- sks0] substlist: [sks0, sks, r, st] → [i ' ' + sks0, sks, r2, st[r -- i ', refval(r2)]] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 4, 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 : list, storekeys, Nil, ⊦ [] + sks = sks], [CSIMPNAMEDSEQ : storefuns, , lookup-empty, ⊦ st[r -- []] = refval(r)], [CSIMPNAMEDSEQ : javavalue, , inj-03, ⊦ refval(r) = refval(r0) ↔ r = r0])], [EXTRATERMS : (i ' ' + sks0, sks, r2, st[r -- i ', refval(r2)])]]