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Proof step 141 from proof hasSharing-put-index-refval (previous step)

(st[r -- i ', refval(r2)][r0 -- i ']) = refval(r3), (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), fits(r2, sks, st), fits(r3, 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), 0 < st[r -- _length] .intval, is_arraytype(st[r0 -- _type] .type) → i < st[r0 -- _length] .intval ⊦ r = jvmref, r = r2, r1 = jvmref, r2 = jvmref, sks3 = [], sks + i ' ' + sks0 = sks3, r0 = jvmref, junkref(str, r, st), hasSharing(r2, st), hasSharing(r, st), is_classtype(st[r0 -- _type] .type), ∃ 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 134:
⊦ 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 137:
¬ validreflist(str, mkjavavalues(r ' + r2 '), mkclasstype(`java.lang.Object' .class), st), (st[r -- i ', refval(r2)][r0 -- i ']) = refval(r3), (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), fits(r2, sks, st), fits(r3, 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), 0 < st[r -- _length] .intval, is_arraytype(st[r0 -- _type] .type) → i < st[r0 -- _length] .intval ⊦ r = jvmref, r = r2, r1 = jvmref, r2 = jvmref, sks3 = [], sks + i ' ' + sks0 = sks3, r0 = jvmref, junkref(str, r, st), hasSharing(r2, st), hasSharing(r, st), is_classtype(st[r0 -- _type] .type), ∃ 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)
Proof step 140:
validreflist(str, mkjavavalues(r ' + r2 '), mkclasstype(`java.lang.Object' .class), st), (st[r -- i ', refval(r2)][r0 -- i ']) = refval(r3), (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), fits(r2, sks, st), fits(r3, sks0, st[r -- i ', refval(r2)]), i < st[r -- _length] .intval, 0 ≤ i, fits(r2, sks3, st[r -- i ', refval(r2)]), ∀ r1, sks1, r0, sks2. ¬ ( r1 ∈ r ' + r2 ' ∧ r0 ∈ r ' + r2 ' ∧ r1 ≠ r0 ∧ fits(r1, sks1, st) ∧ fits(r0, sks2, st) ∧ st[r1 -- sks1] = st[r0 -- sks2] ∧ st[r1 -- sks1] .refval ≠ jvmref), validref(str, r2, ty, st), 0 < st[r -- _length] .intval, is_arraytype(st[r0 -- _type] .type) → i < st[r0 -- _length] .intval ⊦ r = jvmref, r = r2, r1 = jvmref, r2 = jvmref, sks3 = [], sks + i ' ' + sks0 = sks3, r0 = jvmref, junkref(str, r, st), hasSharing(r2, st), hasSharing(r, st), is_classtype(st[r0 -- _type] .type), ∃ 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: 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, r ' + r2 ', str] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((0, 0, 11)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, r ' + r2 ', str)]]