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Proof step 31 from proof ex-validrefs-fits-stronger (previous step)

is_obj(str, r, st), validrefs(str, r ' + refs0, ty, st), st[r -- _type] .type ≤ ty, ∀ st, ty, r, str, refs. # refs < # refs0 → ( validrefs(str, r ' + refs, ty, st) → (∃ sks. fits(r, sks, st) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + refs)[n]))) ⊦ r = jvmref, ∃ sks. fits(r, sks, st) ∧ # sks = # refs0 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + refs0)[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ 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 30:
is_obj(str, r, st) ∧ st[r -- _type] .type ≤ ty, is_obj(str, r, st), refs0 = [] ∨ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ refs0 .first = st[r -- fs '] .refval ∧ validrefs(str, refs0, fs .type, st)), st[r -- _type] .type ≤ ty, ∀ st, ty, r, str, refs. # refs < # refs0 → ( validrefs(str, r ' + refs, ty, st) → (∃ sks. fits(r, sks, st) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + refs)[n]))) ⊦ r = jvmref, ∃ sks. fits(r, sks, st) ∧ # sks = # refs0 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + refs0)[n])

Rule argument

[speclemmaarg: , 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))) ,[st, ty, refs, r, str] → [st, ty, refs0, r, str] ,F ,cyclic , ,validrefs-class ]

Additional information

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