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Proof step 141 from proof addarray-disjoint-nosharing-01 (previous step)

vals[i→n(i)] = refval(r1), vals[i→n(i0)] = refval(r2), st[r2 -- sks] = refval(r0), st[r1 -- sks1] = refval(r0), r2 -- sks ∈ st, st ⊆ addarray(r, ty, vals, st), is_referencevalue(addarray(r, ty, vals, st)[r -- (i0 ' ' + sks)]), 0 ≤ i0, fits(r2, sks, st), fits(r1, sks1, st), 0 ≤ i, i0 < n→i(# vals), disjoint(str, vals .nonnullrefs, st), validreflist(str, vals, ty, st), newref(r, st), i < n→i(# vals), is_referencevalue(addarray(r, ty, vals, st)[r -- (i ' ' + sks1)]), r1 -- sks1 ∈ st ⊦ r1 = r2, i = i0, i0 ' ' + sks = i ' ' + sks1, r = jvmref, vals = [], r0 = jvmref, junkref(str, r1, st), dups(vals .nonnullrefs), hasSharing(vals .nonnullrefs, st), junkref(str, r2, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 76:
⊦ validreflist(str, mkjavavalues(refs), mkclasstype(`java.lang.Object' .class), st) → ( disjoint(str, refs, st) ↔ ¬ (∃ r1, sks1, r2, sks2. r1 ∈ refs ∧ r2 ∈ refs ∧ r1 ≠ r2 ∧ fits(r1, sks1, st) ∧ fits(r2, sks2, st) ∧ st[r1 -- sks1] = st[r2 -- sks2] ∧ st[r1 -- sks1] .refval ≠ jvmref))
Proof step 110:
¬ validreflist(str, mkjavavalues(vals .nonnullrefs), mkclasstype(`java.lang.Object' .class), st), vals[i→n(i)] = refval(r1), vals[i→n(i0)] = refval(r2), st[r2 -- sks] = refval(r0), st[r1 -- sks1] = refval(r0), r2 -- sks ∈ st, st ⊆ addarray(r, ty, vals, st), is_referencevalue(addarray(r, ty, vals, st)[r -- (i0 ' ' + sks)]), 0 ≤ i0, fits(r2, sks, st), fits(r1, sks1, st), 0 ≤ i, i0 < n→i(# vals), disjoint(str, vals .nonnullrefs, st), validreflist(str, vals, ty, st), newref(r, st), i < n→i(# vals), is_referencevalue(addarray(r, ty, vals, st)[r -- (i ' ' + sks1)]), r1 -- sks1 ∈ st ⊦ r1 = r2, i = i0, i0 ' ' + sks = i ' ' + sks1, r = jvmref, vals = [], r0 = jvmref, junkref(str, r1, st), dups(vals .nonnullrefs), hasSharing(vals .nonnullrefs, st), junkref(str, r2, st)
Proof step 140:
validreflist(str, mkjavavalues(vals .nonnullrefs), mkclasstype(`java.lang.Object' .class), st), vals[i→n(i)] = refval(r1), vals[i→n(i0)] = refval(r2), st[r2 -- sks] = refval(r0), st[r1 -- sks1] = refval(r0), r2 -- sks ∈ st, st ⊆ addarray(r, ty, vals, st), is_referencevalue(addarray(r, ty, vals, st)[r -- (i0 ' ' + sks)]), 0 ≤ i0, fits(r2, sks, st), fits(r1, sks1, st), 0 ≤ i, i0 < n→i(# vals), ∀ r1, sks1, r2, sks2. ¬ ( r1 ∈ vals .nonnullrefs ∧ r2 ∈ vals .nonnullrefs ∧ r1 ≠ r2 ∧ fits(r1, sks1, st) ∧ fits(r2, sks2, st) ∧ st[r1 -- sks1] = st[r2 -- sks2] ∧ st[r1 -- sks1] .refval ≠ jvmref), validreflist(str, vals, ty, st), newref(r, st), i < n→i(# vals), is_referencevalue(addarray(r, ty, vals, st)[r -- (i ' ' + sks1)]), r1 -- sks1 ∈ st ⊦ r1 = r2, i = i0, i0 ' ' + sks = i ' ' + sks1, r = jvmref, vals = [], r0 = jvmref, junkref(str, r1, st), dups(vals .nonnullrefs), hasSharing(vals .nonnullrefs, st), junkref(str, r2, st)

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: disjoint-def-fits antecedent: succedent: validreflist (str, mkjavavalues(refs), mkclasstype(`java.lang.Object' .class), st) → ( disjoint(str, refs, st) ↔ ¬ (∃ r1, sks1, r2, sks2. r1 ∈ refs ∧ r2 ∈ refs ∧ r1 ≠ r2 ∧ fits(r1, sks1, st) ∧ fits(r2, sks2, st) ∧ st[r1 -- sks1] = st[r2 -- sks2] ∧ st[r1 -- sks1] .refval ≠ jvmref)) substlist: [st, refs, str] → [st, vals .nonnullrefs, str] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((0, 0, 12)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, vals .nonnullrefs, str)]]