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Proof step 147 from proof hasSharing-put-index-refval (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 132:
⊦ 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 146:
i ' ' + sks0 ≠ [], (st[r -- i ', refval(r2)][r2 -- sks3]) = refval(r1), st[r -- _type] .type = mkarraytype(ty), (st[r -- i ', refval(r2)][r2 -- (sks + i ' ' + sks0)]) = refval(r1), st[r2 -- sks] = refval(r), (st[r -- i ', refval(r2)][r2 -- sks]) = refval(r0), 0 < st[r -- _length] .intval, is_referencevalue(st[r -- i ', refval(r2)][r -- (i ' ' + sks + i ' ' + sks0)]), fits(r2, sks, st), r0 ≠ jvmref ∧ is_referencevalue(st[r -- i ', refval(r2)][r0 -- (i ' ' + sks0) .first]) ∧ ( is_classtype((st[r -- i ', refval(r2)][r0 -- _type]) .type) → (i ' ' + sks0) .first ∈ instfields((st[r -- i ', refval(r2)][r0 -- _type]) .type .class)) ∧ ( is_arraytype((st[r -- i ', refval(r2)][r0 -- _type]) .type) → is_indexkey((i ' ' + sks0) .first) ∧ 0 ≤ (i ' ' + sks0) .first .index ∧ (i ' ' + sks0) .first .index < (st[r -- i ', refval(r2)][r0 -- _length]) .intval) ∧ fits((st[r -- i ', refval(r2)][r0 -- (i ' ' + sks0) .first]) .refval, (i ' ' + sks0) .rest, st[r -- i ', refval(r2)]), i < st[r -- _length] .intval, 0 ≤ i, fits(r2, sks3, st[r -- i ', refval(r2)]), disjoint(str, r ' + r2 ', st), validref(str, r2, ty, st) ⊦ r = jvmref, r = r2, r1 = jvmref, r2 = jvmref, sks3 = [], sks + i ' ' + sks0 = sks3, junkref(str, r, st), hasSharing(r2, st), hasSharing(r, st), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks → st[r2 -- sks2] .refval ≠ r), ∃ sks1, sks4. sks = sks1 + i ' ' + sks4 ∧ st[r2 -- sks1] .refval = r ∧ (∀ sks2. sks2 + i ' ' ⊑ sks1 → st[r2 -- sks2] .refval ≠ st[r2 -- sks1] .refval)

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[r -- i ', refval(r2)], i ' ' + sks0, r0] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((0, 0, 8)) 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[r -- i ', refval(r2)], i ' ' + sks0, r0)]]