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Proof step 69 from proof validrefs-subset (previous step)

st0[r0 -- i '] = refval(r), st[r0 -- _type] = st0[r0 -- _type], r0 -- _type ∈ st, validrefs(str, r ' + refs, st0[r0 -- _type] .type .type, st0), i < st0[r0 -- _length] .intval, is_valid_ref(str, r0, st0), is_array(str, r0, st), st0[r0 -- _type] .type ≤ ty, 0 < st0[r0 -- _length] .intval, 0 ≤ i, r0 -- _type ∈ st0, r0 ∈ st, r0 ∈ st0, r0 -- _length ∈ st → r0 -- _length ∈ st0 ∧ st[r0 -- _length] = st0[r0 -- _length], ∀ rk. rk ∈ st → rk ∈ st0 ∧ st[rk] = st0[rk], ∀ str0, ty0, st1, st2. validrefs(str0, r ' + refs, ty0, st1) ∧ st2 ⊆ st1 ∧ ¬ junkref(str0, r, st2) → validrefs(str0, r ' + refs, ty0, st2) ⊦ r0 = jvmref, ¬ st[r0 -- _length] .intval ≤ i ∧ is_referencevalue(st[r0 -- i ']) ∧ st[r0 -- i '] .refval ≠ jvmref ∧ (¬ st[st[r0 -- i '] .refval -- _type] .type ≤ st0[r0 -- _type] .type .type ∨ junkref(str, st[r0 -- i '] .refval, st)), ¬ st[r0 -- _length] .intval ≤ i ∧ is_referencevalue(st[r0 -- i ']) ∧ r = st[r0 -- i '] .refval ∧ validrefs(str, st[r0 -- i '] .refval ' + refs, st0[r0 -- _type] .type .type, st)

User applied rule cut formula

And obtained the following new goals

Proof step 68:
r0 -- _length ∈ st, st0[r0 -- i '] = refval(r), st[r0 -- _type] = st0[r0 -- _type], r0 -- _type ∈ st, validrefs(str, r ' + refs, st0[r0 -- _type] .type .type, st0), i < st0[r0 -- _length] .intval, is_valid_ref(str, r0, st0), is_array(str, r0, st), st0[r0 -- _type] .type ≤ ty, 0 < st0[r0 -- _length] .intval, 0 ≤ i, r0 -- _type ∈ st0, r0 ∈ st, r0 ∈ st0, r0 -- _length ∈ st → r0 -- _length ∈ st0 ∧ st[r0 -- _length] = st0[r0 -- _length], ∀ rk. rk ∈ st → rk ∈ st0 ∧ st[rk] = st0[rk], ∀ str0, ty0, st1, st2. validrefs(str0, r ' + refs, ty0, st1) ∧ st2 ⊆ st1 ∧ ¬ junkref(str0, r, st2) → validrefs(str0, r ' + refs, ty0, st2) ⊦ r0 = jvmref, ¬ st[r0 -- _length] .intval ≤ i ∧ is_referencevalue(st[r0 -- i ']) ∧ st[r0 -- i '] .refval ≠ jvmref ∧ (¬ st[st[r0 -- i '] .refval -- _type] .type ≤ st0[r0 -- _type] .type .type ∨ junkref(str, st[r0 -- i '] .refval, st)), ¬ st[r0 -- _length] .intval ≤ i ∧ is_referencevalue(st[r0 -- i ']) ∧ r = st[r0 -- i '] .refval ∧ validrefs(str, st[r0 -- i '] .refval ' + refs, st0[r0 -- _type] .type .type, st)

Rule argument

[fma-arg: r0 -- _length ∈ st]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ r -- _length ∈ st → ¬ is_array(str, r, st)])], [EXTRAFMAS : (r0 -- _length ∈ st)]]