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Proof step 22 from proof exvalidrefs (previous step)

st[r -- mkfs(class, st[r -- _type] .type, fieldvar) '] .refval = jvmref, mkfs(class, st[r -- _type] .type, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), ∀ ty. type∃(str, ty) → (ty ≤ st[r -- _type] .type → ty = st[r -- _type] .type), ∀ r, str, st0. # st0 .refs < # st .refs → ( ¬ junkref(str, r, st0) ∧ ¬ cyclic(str, r, st0) ∧ is_obj(str, r, st0) ∧ r ≠ jvmref ∧ (∀ ty. type∃(str, ty) → (ty ≤ st0[r -- _type] .type ↔ ty = st0[r -- _type] .type)) ∧ (∃ fs. fs ' ∈ instfields(st0[r -- _type] .type .class) ∧ fs .type = st0[r -- _type] .type) → (∃ refs. validrefs(str, refs, st0[r -- _type] .type, st0) ∧ refs ≠ [] ∧ refs .first = r ∧ refs .last = jvmref)) ⊦ r = jvmref, cyclic(str, r, st), validrefs(str, r ' + jvmref ', st[r -- _type] .type, st), ∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ (is_reftype(fs .type) ∨ fs .type = abstract_type) ∧ st[r -- fs '] .refval ≠ jvmref ∧ (¬ is_referencevalue(st[r -- fs ']) ∨ ¬ st[st[r -- fs '] .refval -- _type] .type ≤ fs .type ∨ junkref(str, st[r -- fs '] .refval, st))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ 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 21:
is_obj(str, r, st) ∧ st[r -- _type] .type ≤ st[r -- _type] .type, st[r -- mkfs(class, st[r -- _type] .type, fieldvar) '] .refval = jvmref, mkfs(class, st[r -- _type] .type, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), ∀ ty. type∃(str, ty) → (ty ≤ st[r -- _type] .type → ty = st[r -- _type] .type), ∀ r, str, st0. # st0 .refs < # st .refs → ( ¬ junkref(str, r, st0) ∧ ¬ cyclic(str, r, st0) ∧ is_obj(str, r, st0) ∧ r ≠ jvmref ∧ (∀ ty. type∃(str, ty) → (ty ≤ st0[r -- _type] .type ↔ ty = st0[r -- _type] .type)) ∧ (∃ fs. fs ' ∈ instfields(st0[r -- _type] .type .class) ∧ fs .type = st0[r -- _type] .type) → (∃ refs. validrefs(str, refs, st0[r -- _type] .type, st0) ∧ refs ≠ [] ∧ refs .first = r ∧ refs .last = jvmref)) ⊦ r = jvmref, cyclic(str, r, st), jvmref ' = [] ∨ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ jvmref ' .first = st[r -- fs '] .refval ∧ validrefs(str, jvmref ', fs .type, st)), ∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ (is_reftype(fs .type) ∨ fs .type = abstract_type) ∧ st[r -- fs '] .refval ≠ jvmref ∧ (¬ is_referencevalue(st[r -- fs ']) ∨ ¬ st[st[r -- fs '] .refval -- _type] .type ≤ fs .type ∨ junkref(str, st[r -- fs '] .refval, st))

Rule argument

[crewrite-arg: validrefs-class , F , [st, ty, refs, r, str] → [st, st[r -- _type] .type, jvmref ', r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ty ≤ ty])], [EXTRATERMS : (st, st[r -- _type] .type, jvmref ', r, str)]]