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Proof step 457 from proof hasSharing-put-index-refval (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 337:
⊦ sks ≠ [] ∧ sks0 ≠ [] → st[r -- (sks + sks0)] = st[st[r -- sks] .refval -- sks0]
Proof step 338:
⊦ fits(r, [], st)
Proof step 339:
⊦ st[r -- sks] = refval(r) ∧ fits(r, sks, st) ∧ sks ≠ [] ∧ r ≠ jvmref → hasSharing(r, st)
Proof step 456:
sks ≠ [] ∧ i ' ' + sks2 ≠ [], (st[r2 -- i ', refval(r3)][(st[r2 -- i ', refval(r3)][r -- sks]) .refval -- (i ' ' + sks2)]) = refval(r1), st[r2 -- _type] .type = mkarraytype(ty), (st[r2 -- i ', refval(r3)][r3 -- sks3]) = refval(r1), st[r -- sks1] = refval(r2), st[r -- sks] = refval(r2), 0 < st[r2 -- _length] .intval, validref(str, r3, ty, st), disjoint(str, r ' + r3 ', st), fits(r, sks1, st), fits(r3, sks3, st[r2 -- i ', refval(r3)]), i < st[r2 -- _length] .intval, 0 ≤ i, is_referencevalue(st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks3)]), fits(r, sks, st), fits(r3, sks2, st[r2 -- i ', refval(r3)]), r3 = jvmref → is_reftype(ty) ⊦ sks1 = [], sks + i ' ' + sks2 = sks1 + i ' ' + sks3, r1 = jvmref, r = r3, r2 = jvmref, hasSharing(r, st), hasSharing(r3, st), junkref(str, r, st), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .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), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r -- sks4] .refval ≠ r2), ∃ sks0, sks4. sks = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .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 ' ' + sks2, sks, r, st[r2 -- i ', refval(r3)]] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 0, 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 : storefuns, , lookup-empty, ⊦ st[r -- []] = refval(r)], [CSIMPNAMEDSEQ : javavalue, , inj-03, ⊦ refval(r) = refval(r0) ↔ r = r0], [CSIMPNAMEDSEQ : list, storekeys, Nil, ⊦ [] + sks = sks], [CSIMPNAMEDSEQ : list, storekeys, e-03, ⊦ sks + sk ' + sks0 ≠ []], [CSIMPNAMEDSEQ : list-last, storekeys, eq-01-01, ⊦ sks ⊑ [] ↔ sks = []], [CSIMPNAMEDSEQ : list, storekeys, e-02, ⊦ sks + sk ' ≠ []])], [EXTRATERMS : (i ' ' + sks2, sks, r, st[r2 -- i ', refval(r3)])]]