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Proof step 12 from proof ex-validrefs-fits-stronger (previous step)

n = m + 1, st[r -- mkfs(class, ty0, fieldvar) '] = refval(r0), # sks = # refs1, is_obj(str, r, st), st[r -- _type] .type ≤ ty, mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), validrefs(str, r0 ' + refs1, ty0, 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]))) ⊦ n = 0, r = jvmref, st[r -- (mkfs(class, ty0, fieldvar) ' ' + firstn(m, sks))] .refval = (r ' + r0 ' + refs1)[n], is_arraytype(st[r -- _type] .type), ∃ sks. fits(r, sks, st) ∧ # sks = 1 + # refs1 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + r0 ' + refs1)[n])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
st[r -- mkfs(class, ty0, fieldvar) '] = refval(r0), # sks = # refs1, is_obj(str, r, st), st[r -- _type] .type ≤ ty, mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), validrefs(str, r0 ' + refs1, ty0, st), fits(r0, sks, st), m ≤ # 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 -- (mkfs(class, ty0, fieldvar) ' ' + firstn(m, sks))] .refval = (r0 ' + refs1)[m], is_arraytype(st[r -- _type] .type), ∃ sks. fits(r, sks, st) ∧ # sks = 1 + # refs1 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + r0 ' + refs1)[n])

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m + n ≤ m + n0 ↔ n ≤ n0], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ (r ' + refs)[n + 1] = refs[n]], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + 1 ≠ 0])]]