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Proof step 57 from proof validrefs-set-null (previous step)

(acc + refs0) .first = st[r -- i '] .refval, is_array(str, r, st[(r ' + acc) .last -- fs ', refval(jvmref)]), (st[(r ' + acc) .last -- fs ', refval(jvmref)][r -- _type]) .type ≤ ty, is_array(str, r, st), fs ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_obj(str, (r ' + acc) .last, st), is_reftype(fs .type), st[r -- _type] .type ≤ ty, i < st[r -- _length] .intval, is_referencevalue(st[r -- i ']), validrefs(str, acc + refs0, st[r -- _type] .type .type, st), 0 ≤ i, 0 < st[r -- _length] .intval, ∀ str0, refs, ty0, st0, fs0. validrefs(str0, acc + refs, ty0, st0) ∧ acc ≠ [] ∧ is_obj(str0, acc .last, st0) ∧ fs0 ' ∈ instfields(st0[acc .last -- _type] .type .class) ∧ is_reftype(fs0 .type) ∧ refs ≠ [] → validrefs(str0, acc + jvmref ', ty0, st0[acc .last -- fs0 ', refval(jvmref)]) ⊦ r = jvmref, refs0 = [], r ∈ acc, dups(acc), ∃ i0. ¬ i0 < 0 ∧ ¬ (st[(r ' + acc) .last -- fs ', refval(jvmref)][r -- _length]) .intval ≤ i0 ∧ is_referencevalue(st[r -- i0 ']) ∧ (acc + jvmref ') .first = st[r -- i0 '] .refval ∧ validrefs (str, acc + jvmref ', (st[(r ' + acc) .last -- fs ', refval(jvmref)][r -- _type]) .type .type, st[(r ' + acc) .last -- fs ', refval(jvmref)])

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 56:
(acc + refs0) .first = st[r -- i '] .refval, is_array(str, r, st[(r ' + acc) .last -- fs ', refval(jvmref)]), (st[(r ' + acc) .last -- fs ', refval(jvmref)][r -- _type]) .type ≤ ty, is_array(str, r, st), fs ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_obj(str, (r ' + acc) .last, st), is_reftype(fs .type), st[r -- _type] .type ≤ ty, i < st[r -- _length] .intval, is_referencevalue(st[r -- i ']), validrefs(str, acc + refs0, st[r -- _type] .type .type, st), 0 ≤ i, 0 < st[r -- _length] .intval, ∀ str0, refs, ty0, st0, fs0. validrefs(str0, acc + refs, ty0, st0) ∧ acc ≠ [] ∧ is_obj(str0, acc .last, st0) ∧ fs0 ' ∈ instfields(st0[acc .last -- _type] .type .class) ∧ is_reftype(fs0 .type) ∧ refs ≠ [] → validrefs(str0, acc + jvmref ', ty0, st0[acc .last -- fs0 ', refval(jvmref)]) ⊦ acc = [], r = jvmref, refs0 = [], r ∈ acc, dups(acc), ∃ i0. ¬ i0 < 0 ∧ ¬ (st[(r ' + acc) .last -- fs ', refval(jvmref)][r -- _length]) .intval ≤ i0 ∧ is_referencevalue(st[r -- i0 ']) ∧ (acc + jvmref ') .first = st[r -- i0 '] .refval ∧ validrefs (str, acc + jvmref ', (st[(r ' + acc) .last -- fs ', refval(jvmref)][r -- _type]) .type .type, st[(r ' + acc) .last -- fs ', refval(jvmref)])

Rule argument

[fma-arg: acc = []]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ [] + refs = refs], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ refs + [] = refs], [CSIMPSEQ : ⊦ r ' .last = r], [CSIMPSEQ : ⊦ is_array(str, r, st) → ¬ is_obj(str, r, st)])], [EXTRAFMAS : (acc = [])]]