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Proof step 44 from proof unchNorec-validrefs (previous step)

refs0 = r1 + refs, r1 = st[r -- i '] .refval, st0[r -- _type] .type = mkarraytype(ty0), st[r -- _type] = st0[r -- _type], st[r -- _length] = st0[r -- _length], 0 < st0[r -- _length] .intval, 0 ≤ i, validrefs(str, refs0, ty0, st), is_referencevalue(st[r -- i ']), i < st0[r -- _length] .intval, is_array(str, r, st), unchangedNorec(r, st, st0), mkarraytype(ty0) ≤ ty, is_array(str, r, st0), ∀ str, ty, st, st0. validrefs(str, refs0, ty, st) ∧ (∀ r. r ∈ refs0 → unchangedNorec(r, st, st0)) → validrefs(str, refs0, ty, st0), ∀ r0. r0 ∈ refs0 → unchangedNorec(r0, st, st0) ⊦ refs0 = [], r = jvmref, ∃ i0. ¬ i0 < 0 ∧ ¬ st0[r -- _length] .intval ≤ i0 ∧ is_referencevalue(st0[r -- i0 ']) ∧ r1 = st0[r -- i0 '] .refval ∧ validrefs(str, refs0, ty0, st0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 43:
st0[r -- _type] .type = mkarraytype(ty0), st[r -- _type] = st0[r -- _type], st[r -- _length] = st0[r -- _length], is_referencevalue(st[r -- i ']), i < st0[r -- _length] .intval, is_array(str, r, st), unchangedNorec(r, st, st0), mkarraytype(ty0) ≤ ty, is_array(str, r, st0), 0 < st0[r -- _length] .intval, 0 ≤ i, validrefs(str, st[r -- i '] .refval ' + refs, ty0, st), ∀ str0, ty1, st1, st2. validrefs(str0, st[r -- i '] .refval ' + refs, ty1, st1) ∧ (∀ r0. r0 = st[r -- i '] .refval ∨ r0 ∈ refs → unchangedNorec(r0, st1, st2)) → validrefs(str0, st[r -- i '] .refval ' + refs, ty1, st2), ∀ r0. r0 = st[r -- i '] .refval ∨ r0 ∈ refs → unchangedNorec(r0, st, st0) ⊦ r = jvmref, ∃ i0. ¬ i0 < 0 ∧ ¬ st0[r -- _length] .intval ≤ i0 ∧ is_referencevalue(st0[r -- i0 ']) ∧ st[r -- i '] .refval = st0[r -- i0 '] .refval ∧ validrefs(str, st[r -- i '] .refval ' + refs, ty0, st0)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ r ∈ r0 ' + refs ↔ r ≠ r0 ∧ ¬ r ∈ refs], [CSIMPSEQ : ⊦ r ∈ r0 ' ↔ r = r0], [CSIMPSEQ : ⊦ r ∈ refs + refs0 ↔ ¬ ¬ (r ∈ refs ∨ r ∈ refs0)], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ r ' + refs ≠ []], [CSIMPSEQ : ⊦ r + refs = r ' + refs])]]