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Proof step 19 from proof unchNorec-validrefs-pathex (previous step)

validrefs(str, r1 ' + refs1 + r0 ' + refs, ty, st), validrefs(str, r1 ' + refs1 + r0 ', ty, st), ∀ r. r = r1 ∨ r ∈ refs1 ∨ r = r0 ∨ r ∈ refs → unchangedNorec(r, st, st0) ⊦ r0 = r1, validrefs(str, r1 ' + refs1 + r0 ', ty, st0), validrefs(str, r1 ' + refs1 + r0 ', mkclasstype(`java.lang.Object' .class), st0), validrefs(str, r1 ' + refs1 + r0 ', mkclasstype(`java.lang.Object' .class), st) ∧ (r = r1 ∨ r ∈ refs1 ∨ r = r0 → unchangedNorec(r, st, st0))

User applied rule case distinction

And obtained the following new goals

Proof step 16:
validrefs(str, r1 ' + refs1 + r0 ' + refs, ty, st), validrefs(str, r1 ' + refs1 + r0 ', ty, st), ∀ r. r = r1 ∨ r ∈ refs1 ∨ r = r0 ∨ r ∈ refs → unchangedNorec(r, st, st0) ⊦ validrefs(str, r1 ' + refs1 + r0 ', mkclasstype(`java.lang.Object' .class), st), r0 = r1, validrefs(str, r1 ' + refs1 + r0 ', ty, st0), validrefs(str, r1 ' + refs1 + r0 ', mkclasstype(`java.lang.Object' .class), st0)
Proof step 18:
validrefs(str, r1 ' + refs1 + r0 ' + refs, ty, st), validrefs(str, r1 ' + refs1 + r0 ', ty, st), ∀ r. r = r1 ∨ r ∈ refs1 ∨ r = r0 ∨ r ∈ refs → unchangedNorec(r, st, st0) ⊦ r = r1 ∨ r ∈ refs1 ∨ r = r0 → unchangedNorec(r, st, st0), r0 = r1, validrefs(str, r1 ' + refs1 + r0 ', ty, st0), validrefs(str, r1 ' + refs1 + r0 ', mkclasstype(`java.lang.Object' .class), st0)

Rule argument

[CASEDARG : pos: right 4, ()]

Additional information

[[CONCRETESIMPRULES : ()]]