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Proof step 42 from proof pathex-put-cyclic (previous step)

st[r0 -- _type] .type = mkclasstype(class0), validrefs(str, (r ' + refs0) + r0 ', st[r -- _type] .type, st), is_obj(str, r0, st[r0 -- mkfs(class, ty, fieldvar) ', refval(r)]), validrefs(str, r ' + refs0 + r0 ', st[r -- _type] .type, st), st[r -- _type] .type ≤ ty, mkfs(class, ty, fieldvar) ' ∈ instfields(class0) ⊦ _type = mkfs(class, ty, fieldvar) ', r = r0, r0 = jvmref, r = jvmref, r ∈ refs0, dups(refs0), r0 ∈ refs0, validrefs(str, r ' + refs0 + r0 ', ty, st[r0 -- mkfs(class, ty, fieldvar) ', refval(r)]), ∃ refs. dups(refs) ∧ refs .first = r0 ∧ validrefs(str, refs, st[refs .first -- _type] .type, st[refs .first -- mkfs(class, ty, fieldvar) ', refval(r)]), mkfs(class, ty, fieldvar) ' ≠ _length ∧ (ty = void_type ∨ is_reftype(ty))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 36:
⊦ ¬ is_reftype(ty) → (ty ≤ ty0 ↔ ty = ty0)
Proof step 40:
¬ ¬ is_reftype(st[r -- _type] .type), st[r0 -- _type] .type = mkclasstype(class0), validrefs(str, (r ' + refs0) + r0 ', st[r -- _type] .type, st), is_obj(str, r0, st[r0 -- mkfs(class, ty, fieldvar) ', refval(r)]), validrefs(str, r ' + refs0 + r0 ', st[r -- _type] .type, st), st[r -- _type] .type ≤ ty, mkfs(class, ty, fieldvar) ' ∈ instfields(class0) ⊦ _type = mkfs(class, ty, fieldvar) ', r = r0, r0 = jvmref, r = jvmref, r ∈ refs0, dups(refs0), r0 ∈ refs0, validrefs(str, r ' + refs0 + r0 ', ty, st[r0 -- mkfs(class, ty, fieldvar) ', refval(r)]), ∃ refs. dups(refs) ∧ refs .first = r0 ∧ validrefs(str, refs, st[refs .first -- _type] .type, st[refs .first -- mkfs(class, ty, fieldvar) ', refval(r)]), mkfs(class, ty, fieldvar) ' ≠ _length ∧ (ty = void_type ∨ is_reftype(ty))
Proof step 41:
⊦ ¬ is_reftype(ty) → (validrefs(str, refs, ty, st) ↔ refs = [] ∨ refs = jvmref ')

Rule argument

[speclemmaarg: ,¬ is_reftype(ty) → (ty ≤ ty0 ↔ ty = ty0) ,[ty0, ty] → [ty, st[r -- _type] .type] ,F ,typefuns , ,le-noreftype ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r ' + refs ≠ []], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ ¬ refs + refs0 ≠ [] ↔ refs = [] ∧ refs0 = []], [CSIMPSEQ : ⊦ r ' ≠ []])], [EXTRATERMS : (ty, st[r -- _type] .type)]]