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Proof step 88 from proof validrefs-subset-equiv (previous step)

refs = r0 + refs1, r0 = st0[r -- i '] .refval, is_array(str, r, st), st[r -- _type] .type ≤ ty, st ⊆ st0, st0[r -- _type] .type ≤ ty, i < st0[r -- _length] .intval, is_referencevalue(st0[r -- i ']), validrefs(str, refs, st0[r -- _type] .type .type, st0), 0 ≤ i, 0 < st0[r -- _length] .intval, ∀ st, st0, ty, r, str, refs0. # refs0 < # refs → (validrefs(str, r ' + refs0, ty, st0) ∧ st ⊆ st0 ∧ validref(str, r, st[r -- _type] .type, st) → validrefs(str, r ' + refs0, ty, st)) ⊦ r = jvmref, refs = [], junkref(str, r, st), ∃ i0. ¬ i0 < 0 ∧ ¬ st[r -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r -- i0 ']) ∧ r0 = st[r -- i0 '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 87:
0 ≤ i, 0 < st0[r -- _length] .intval, is_array(str, r, st), st[r -- _type] .type ≤ ty, st ⊆ st0, st0[r -- _type] .type ≤ ty, i < st0[r -- _length] .intval, is_referencevalue(st0[r -- i ']), validrefs(str, st0[r -- i '] .refval ' + refs1, st0[r -- _type] .type .type, st0), ∀ st, st1, ty, r1, str, refs0. # refs0 < # (st0[r -- i '] .refval + refs1) → (validrefs(str, r1 ' + refs0, ty, st1) ∧ st ⊆ st1 ∧ validref(str, r1, st[r1 -- _type] .type, st) → validrefs(str, r1 ' + refs0, ty, st)) ⊦ r = jvmref, junkref(str, r, st), ∃ i0. ¬ i0 < 0 ∧ ¬ st[r -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r -- i0 ']) ∧ st0[r -- i '] .refval = st[r -- i0 '] .refval ∧ validrefs(str, st0[r -- i '] .refval ' + refs1, st[r -- _type] .type .type, st)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ r ' + refs ≠ []], [CSIMPSEQ : ⊦ r + refs = r ' + refs])]]