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Proof step 106 from proof hasSharing-put-index-refval (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 99:
⊦ 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 102:
¬ validreflist(str, mkjavavalues(r0 ' + r1 '), mkclasstype(`java.lang.Object' .class), st), st[r0 -- _type] .type = mkarraytype(ty), st[r0 -- sks0] = refval(r1), disjoint(str, r0 ' + r1 ', st), validref(str, r1, ty, st), fits(r0, sks0, st), i < st[r0 -- _length] .intval, 0 ≤ i, 0 < st[r0 -- _length] .intval ⊦ sks0 = i ' ', r0 = jvmref, r0 = r1, r1 = jvmref, sks0 = [], junkref(str, r0, st), hasSharing(r1, st), hasSharing(r0, st), ∃ sks, sks1. sks0 = sks + i ' ' + sks1 ∧ st[r0 -- sks] .refval = r0 ∧ (∀ sks2. sks2 + i ' ' ⊑ sks → st[r0 -- sks2] .refval ≠ r0)
Proof step 105:
validreflist(str, mkjavavalues(r0 ' + r1 '), mkclasstype(`java.lang.Object' .class), st), st[r0 -- _type] .type = mkarraytype(ty), st[r0 -- sks0] = refval(r1), ∀ r, sks1, r2, sks2. ¬ ( r ∈ r0 ' + r1 ' ∧ r2 ∈ r0 ' + r1 ' ∧ r ≠ r2 ∧ fits(r, sks1, st) ∧ fits(r2, sks2, st) ∧ st[r -- sks1] = st[r2 -- sks2] ∧ st[r -- sks1] .refval ≠ jvmref), validref(str, r1, ty, st), fits(r0, sks0, st), i < st[r0 -- _length] .intval, 0 ≤ i, 0 < st[r0 -- _length] .intval ⊦ sks0 = i ' ', r0 = jvmref, r0 = r1, r1 = jvmref, sks0 = [], junkref(str, r0, st), hasSharing(r1, st), hasSharing(r0, st), ∃ sks, sks1. sks0 = sks + i ' ' + sks1 ∧ st[r0 -- sks] .refval = r0 ∧ (∀ sks2. sks2 + i ' ' ⊑ sks → st[r0 -- sks2] .refval ≠ r0)

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, r0 ' + r1 ', str] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((0, 0, 2)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, r0 ' + r1 ', str)]]