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Proof step 230 from proof hasSharing-put-index-refval (previous step)

(st[r0 -- i ', refval(r4)][r0 -- (sks + i ' ' + sks1)]) = refval(r1), st[r0 -- _type] .type = mkarraytype(ty), (st[r0 -- i ', refval(r4)][r4 -- sks3]) = refval(r1), st[r0 -- sks] = refval(r0), (st[r0 -- i ', refval(r4)][r0 -- sks]) = refval(r), 0 < st[r0 -- _length] .intval, 0 ≤ i, i < st[r0 -- _length] .intval, fits(r0, sks, st), fits(r, i ' ' + sks1, st[r0 -- i ', refval(r4)]), validref(str, r4, ty, st), disjoint(str, r0 ' + r4 ', st), fits(r4, sks3, st[r0 -- i ', refval(r4)]) ⊦ r1 = jvmref, r0 = r4, r0 = jvmref, sks + i ' ' + sks1 = i ' ' + sks3, sks3 = [], r4 = jvmref, hasSharing(r0, st), hasSharing(r4, st), junkref(str, r0, st), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks → st[r0 -- sks2] .refval ≠ r0), ∃ sks0, sks4. sks = sks0 + i ' ' + sks4 ∧ st[r0 -- sks0] .refval = r0 ∧ (∀ sks2. sks2 + i ' ' ⊑ sks0 → st[r0 -- sks2] .refval ≠ r0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 121:
⊦ sks ≠ [] → ( fits(r, sks, st) ↔ r ≠ jvmref ∧ is_referencevalue(st[r -- sks .first]) ∧ (is_classtype(st[r -- _type] .type) → sks .first ∈ instfields(st[r -- _type] .type .class)) ∧ (is_arraytype(st[r -- _type] .type) → is_indexkey(sks .first) ∧ 0 ≤ sks .first .index ∧ sks .first .index < st[r -- _length] .intval) ∧ fits(st[r -- sks .first] .refval, sks .rest, st))
Proof step 229:
i ' ' + sks1 ≠ [], (st[r0 -- i ', refval(r4)][r0 -- (sks + i ' ' + sks1)]) = refval(r1), st[r0 -- _type] .type = mkarraytype(ty), (st[r0 -- i ', refval(r4)][r4 -- sks3]) = refval(r1), st[r0 -- sks] = refval(r0), (st[r0 -- i ', refval(r4)][r0 -- sks]) = refval(r), 0 < st[r0 -- _length] .intval, 0 ≤ i, i < st[r0 -- _length] .intval, fits(r0, sks, st), r ≠ jvmref ∧ is_referencevalue(st[r0 -- i ', refval(r4)][r -- (i ' ' + sks1) .first]) ∧ ( is_classtype((st[r0 -- i ', refval(r4)][r -- _type]) .type) → (i ' ' + sks1) .first ∈ instfields((st[r0 -- i ', refval(r4)][r -- _type]) .type .class)) ∧ ( is_arraytype((st[r0 -- i ', refval(r4)][r -- _type]) .type) → is_indexkey((i ' ' + sks1) .first) ∧ 0 ≤ (i ' ' + sks1) .first .index ∧ (i ' ' + sks1) .first .index < (st[r0 -- i ', refval(r4)][r -- _length]) .intval) ∧ fits((st[r0 -- i ', refval(r4)][r -- (i ' ' + sks1) .first]) .refval, (i ' ' + sks1) .rest, st[r0 -- i ', refval(r4)]), validref(str, r4, ty, st), disjoint(str, r0 ' + r4 ', st), fits(r4, sks3, st[r0 -- i ', refval(r4)]) ⊦ r1 = jvmref, r0 = r4, r0 = jvmref, sks + i ' ' + sks1 = i ' ' + sks3, sks3 = [], r4 = jvmref, hasSharing(r0, st), hasSharing(r4, st), junkref(str, r0, st), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks → st[r0 -- sks2] .refval ≠ r0), ∃ sks0, sks4. sks = sks0 + i ' ' + sks4 ∧ st[r0 -- sks0] .refval = r0 ∧ (∀ sks2. sks2 + i ' ' ⊑ sks0 → st[r0 -- sks2] .refval ≠ r0)

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: fits-rec antecedent: succedent: sks ≠ [] → ( fits(r, sks, st) ↔ r ≠ jvmref ∧ is_referencevalue(st[r -- sks .first]) ∧ ( is_classtype(st[r -- _type] .type) → sks .first ∈ instfields(st[r -- _type] .type .class)) ∧ ( is_arraytype(st[r -- _type] .type) → is_indexkey(sks .first) ∧ 0 ≤ sks .first .index ∧ sks .first .index < st[r -- _length] .intval) ∧ fits(st[r -- sks .first] .refval, sks .rest, st)) substlist: [st, sks, r] → [st[r0 -- i ', refval(r4)], i ' ' + sks1, r] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((0, 0, 9)) 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])], [EXTRATERMS : (st[r0 -- i ', refval(r4)], i ' ' + sks1, r)]]