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Proof step 46 from proof validrefs-set-null (previous step)

acc + refs0 = r0 + refs1 ∧ acc + refs0 ≠ [], r0 = st[r -- i '] .refval, 0 < st[r -- _length] .intval, 0 ≤ i, validrefs(str, acc + refs0, st[r -- _type] .type .type, st), is_referencevalue(st[r -- i ']), i < st[r -- _length] .intval, st[r -- _type] .type ≤ ty, is_reftype(ty1), is_obj(str, (r ' + acc) .last, st), mkfs(class, ty1, fieldvar) ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_array(str, r, st), ∀ 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)]) ⊦ refs0 = [], r = jvmref, validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]), dups(acc), r ∈ acc, is_array(str, r, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]) ∧ (st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)][r -- _type]) .type ≤ ty

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 45:
acc + refs0 = st[r -- i '] .refval ' + refs1, is_array(str, r, st), mkfs(class, ty1, fieldvar) ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_obj(str, (r ' + acc) .last, st), is_reftype(ty1), 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 + refs0 = [], r = jvmref, refs0 = [], r ∈ acc, dups(acc), validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]), is_array(str, r, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]) ∧ (st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)][r -- _type]) .type ≤ ty, ¬ st[r -- i '] .refval ∈ acc ∧ ¬ st[r -- i '] .refval ∈ refs0

Rule argument

[emptyarg]

Additional information

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