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Proof step 14 from proof validrefs-applast-class (previous step)

validrefs(str, refs + r ', ty, st), is_referencevalue(st[r -- mkfs(class, ty0, fieldvar) ']), is_obj(str, r, st), mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), st[st[r -- mkfs(class, ty0, fieldvar) '] .refval -- _type] .type ≤ ty0, is_valid_ref(str, st[r -- mkfs(class, ty0, fieldvar) '] .refval, st), type∃(str, st[st[r -- mkfs(class, ty0, fieldvar) '] .refval -- _type] .type) ⊦ validrefs(str, refs + r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ', ty, st)

User applied rule insert lemma

And obtained the following new goals

Proof step 1:
⊦ validrefs(str, refs, ty, st) ∧ validrefs(str, refs0, ty0, st) ∧ refs .last = refs0 .first ∧ refs ≠ [] ∧ refs0 ≠ [] → validrefs(str, refs + refs0 .rest, ty, st)
Proof step 13:
validrefs(str, refs + r ', ty, st) ∧ validrefs(str, r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ', st[r -- _type] .type, st) ∧ (refs + r ') .last = (r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ') .first ∧ refs + r ' ≠ [] ∧ r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ' ≠ [] → validrefs(str, (refs + r ') + (r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ') .rest, ty, st), validrefs(str, refs + r ', ty, st), is_referencevalue(st[r -- mkfs(class, ty0, fieldvar) ']), is_obj(str, r, st), mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), st[st[r -- mkfs(class, ty0, fieldvar) '] .refval -- _type] .type ≤ ty0, is_valid_ref(str, st[r -- mkfs(class, ty0, fieldvar) '] .refval, st), type∃(str, st[st[r -- mkfs(class, ty0, fieldvar) '] .refval -- _type] .type) ⊦ validrefs(str, refs + r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ', ty, st)

Rule argument

[LEMMAARG : validrefs-append-01, [str, refs, ty, st, refs0, ty0] → [str, refs + r ', ty, st, r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ', st[r -- _type] .type]]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (str, refs + r ', ty, st, r ' + st[r -- mkfs(class, ty0, fieldvar) '] .refval ', st[r -- _type] .type)]]