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Proof step 60 from proof validrefs-put-array-badindex (previous step)

i < 0, is_obj(str, r0, st) ∨ is_array(str, r0, st), st[r0 -- _type] .type ≤ ty, is_valid_ref(str, r0, st[r -- i ', val]), ∀ i0, str0, ty0, st0, r1, val0. ¬ 0 ≤ i0 → (validrefs(str0, refs0, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, refs0, ty0, st0)) ⊦ validrefs(str, r0 ' + refs0, ty, st[r -- i ', val]) ↔ validrefs(str, r0 ' + refs0, ty, st), r0 = jvmref

User applied rule case distinction

And obtained the following new goals

Proof step 32:
is_obj(str, r0, st), i < 0, st[r0 -- _type] .type ≤ ty, is_valid_ref(str, r0, st[r -- i ', val]), ∀ i0, str0, ty0, st0, r1, val0. ¬ 0 ≤ i0 → (validrefs(str0, refs0, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, refs0, ty0, st0)) ⊦ validrefs(str, r0 ' + refs0, ty, st[r -- i ', val]) ↔ validrefs(str, r0 ' + refs0, ty, st), r0 = jvmref
Proof step 59:
is_array(str, r0, st), i < 0, st[r0 -- _type] .type ≤ ty, is_valid_ref(str, r0, st[r -- i ', val]), ∀ i0, str0, ty0, st0, r1, val0. ¬ 0 ≤ i0 → (validrefs(str0, refs0, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, refs0, ty0, st0)) ⊦ validrefs(str, r0 ' + refs0, ty, st[r -- i ', val]) ↔ validrefs(str, r0 ' + refs0, ty, st), r0 = jvmref

Rule argument

[CASEDARG : pos: left 2, ()]

Additional information

[[CONCRETESIMPRULES : ()]]