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Proof step 22 from proof unchNorec-validrefs (previous step)

st[r -- _type] = st0[r -- _type], is_obj(str, r, st), st0[r -- _type] .type ≤ ty, unchangedNorec(r, st, st0), ∀ r0. r0 ∈ refs0 → unchangedNorec(r0, st, st0), refs0 = [] ∨ mkfs(class, ty0, fieldvar) ' ∈ instfields(st0[r -- _type] .type .class) ∧ is_referencevalue(st[r -- mkfs(class, ty0, fieldvar) ']) ∧ refs0 .first = st[r -- mkfs(class, ty0, fieldvar) '] .refval ∧ validrefs(str, refs0, ty0, st), ∀ str, ty, st, st0. validrefs(str, refs0, ty, st) ∧ (∀ r. r ∈ refs0 → unchangedNorec(r, st, st0)) → validrefs(str, refs0, ty, st0) ⊦ r = jvmref, validrefs(str, r ' + refs0, ty, st0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ is_obj(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ refs .first = st[r -- fs '] .refval ∧ validrefs(str, refs, fs .type, st)))
Proof step 6:
⊦ unchangedNorec(r, st, st0) ∧ is_obj(str, r, st) → (¬ is_obj(str, r, st0) ↔ false)
Proof step 21:
is_obj(str, r, st0) ∧ st0[r -- _type] .type ≤ ty, st[r -- _type] = st0[r -- _type], is_obj(str, r, st), st0[r -- _type] .type ≤ ty, unchangedNorec(r, st, st0), ∀ r0. r0 ∈ refs0 → unchangedNorec(r0, st, st0), refs0 = [] ∨ mkfs(class, ty0, fieldvar) ' ∈ instfields(st0[r -- _type] .type .class) ∧ is_referencevalue(st[r -- mkfs(class, ty0, fieldvar) ']) ∧ refs0 .first = st[r -- mkfs(class, ty0, fieldvar) '] .refval ∧ validrefs(str, refs0, ty0, st), ∀ str, ty, st, st0. validrefs(str, refs0, ty, st) ∧ (∀ r. r ∈ refs0 → unchangedNorec(r, st, st0)) → validrefs(str, refs0, ty, st0) ⊦ r = jvmref, refs0 = [] ∨ (∃ fs. fs ' ∈ instfields(st0[r -- _type] .type .class) ∧ is_referencevalue(st0[r -- fs ']) ∧ refs0 .first = st0[r -- fs '] .refval ∧ validrefs(str, refs0, fs .type, st0))

Rule argument

[x0lemmaarg: spec-name: cyclic instance-name: lemma-name: validrefs-class antecedent: succedent: is_obj(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ refs .first = st[r -- fs '] .refval ∧ validrefs(str, refs, fs .type, st))) substlist: [st, ty, refs, r, str] → [st0, ty, refs0, r, str] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 1)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st0, ty, refs0, r, str)]]