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Proof step 20 from proof validsubset-trans (previous step)

∀ r, ty. validref(str, r, ty, st0) → validref(str, r, ty, st1), ∀ r. ¬ r ∈ st0 ∧ r ∈ st1 → ¬ junkref(str, r, st1), ∀ r. ¬ r ∈ st ∧ r ∈ st0 → ¬ junkref(str, r, st0), ∀ r, ty. validref(str, r, ty, st) → validref(str, r, ty, st0) ⊦ (validref(str, r, ty, st) → validref(str, r, ty, st1)) ∧ (¬ r0 ∈ st ∧ r0 ∈ st1 → ¬ junkref(str, r0, st1))

User applied rule case distinction

And obtained the following new goals

Proof step 7:
∀ r, ty. validref(str, r, ty, st0) → validref(str, r, ty, st1), ∀ r. ¬ r ∈ st0 ∧ r ∈ st1 → ¬ junkref(str, r, st1), ∀ r. ¬ r ∈ st ∧ r ∈ st0 → ¬ junkref(str, r, st0), ∀ r, ty. validref(str, r, ty, st) → validref(str, r, ty, st0) ⊦ validref(str, r, ty, st) → validref(str, r, ty, st1)
Proof step 19:
∀ r, ty. validref(str, r, ty, st0) → validref(str, r, ty, st1), ∀ r. ¬ r ∈ st0 ∧ r ∈ st1 → ¬ junkref(str, r, st1), ∀ r. ¬ r ∈ st ∧ r ∈ st0 → ¬ junkref(str, r, st0), ∀ r, ty. validref(str, r, ty, st) → validref(str, r, ty, st0) ⊦ ¬ r0 ∈ st ∧ r0 ∈ st1 → ¬ junkref(str, r0, st1)

Rule argument

[CASEDARG : pos: right 1, ()]

Additional information

[[CONCRETESIMPRULES : ()]]