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Proof step 44 from proof ex-validrefs-fits-stronger (previous step)

# sks = # refs1, st[r -- i '] = refval(r0), fits(r0, sks, st), validrefs(str, r0 ' + refs1, st[r -- _type] .type .type, st), i < st[r -- _length] .intval, st[r -- _type] .type ≤ ty, is_array(str, r, st), 0 < st[r -- _length] .intval, 0 ≤ i, ∀ n. n ≠ 0 ∧ ¬ # refs1 < n → st[r0 -- firstn(n, sks)] .refval = (r0 ' + refs1)[n], ∀ st0, ty, r1, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, r1 ' + refs, ty, st0) → (∃ sks. fits(r1, sks, st0) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st0[r1 -- firstn(n, sks)] .refval = (r1 ' + refs)[n]))) ⊦ r = jvmref, r ≠ jvmref ∧ is_referencevalue(st[r -- i ']) ∧ (is_classtype(st[r -- _type] .type) → i ' ∈ instfields(st[r -- _type] .type .class)) ∧ (is_arraytype(st[r -- _type] .type) → is_indexkey(i ') ∧ 0 ≤ i ' .index ∧ i ' .index < st[r -- _length] .intval) ∧ fits(st[r -- i '] .refval, sks, st) ∧ (n ≠ 0 ∧ n ≤ 1 + # refs1 → st[r -- firstn(n, i ' ' + sks)] .refval = (r ' + r0 ' + refs1)[n])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 43:
st[r -- i '] = refval(r0), # sks = # refs1, 0 ≤ i, 0 < st[r -- _length] .intval, is_array(str, r, st), st[r -- _type] .type ≤ ty, i < st[r -- _length] .intval, validrefs(str, r0 ' + refs1, st[r -- _type] .type .type, st), fits(r0, sks, st), n ≤ 1 + # refs1, ∀ n. n ≠ 0 ∧ ¬ # refs1 < n → st[r0 -- firstn(n, sks)] .refval = (r0 ' + refs1)[n], ∀ st0, ty, r1, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, r1 ' + refs, ty, st0) → (∃ sks. fits(r1, sks, st0) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st0[r1 -- firstn(n, sks)] .refval = (r1 ' + refs)[n]))) ⊦ r = jvmref, st[r -- firstn(n, i ' ' + sks)] .refval = (r ' + r0 ' + refs1)[n], n = 0, is_classtype(st[r -- _type] .type)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ is_referencevalue(refval(r))], [CSIMPSEQ : ⊦ refval(r) .refval = r], [CSIMPSEQ : ⊦ is_classtype (st[r -- _type] .type) → ¬ is_array(str, r, st)], [CSIMPSEQ : ⊦ i < j → (¬ ¬ j ≤ i ↔ false)], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i ' .index = i], [CSIMPSEQ : ⊦ is_indexkey(i ')], [CSIMPSEQ : ⊦ ¬ i ' ∈ instfields(class)])]]