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Proof step 42 from proof validref-putrkvs (previous step)

sk .fs = mkfs(class, ty0, fieldvar), okval(refval(r1), st[r0 -- _type] .type .type), validref(str, r, ty, st), validvals(str, rkvs0, st), is_indexkey(sk), validref(str, r1, st[r0 -- _type] .type .type, st), ∀ str, r, ty, st. validref(str, r, ty, st) ∧ validvals(str, rkvs0, st) → validref(str, r, ty, st[rkvs0]) ⊦ validref(str, r, ty, st[r0 -- sk, refval(r1)]), validref(str, r, ty, st[r0 -- sk, refval(r1)][rkvs0]), r0 ≠ jvmref ∧ is_arraytype(st[r0 -- _type] .type) ∧ (r1 = jvmref → is_reftype(st[r0 -- _type] .type .type))

User applied rule cut formula

And obtained the following new goals

Proof step 40:
r0 ≠ jvmref, sk .fs = mkfs(class, ty0, fieldvar), okval(refval(r1), st[r0 -- _type] .type .type), validref(str, r, ty, st), validvals(str, rkvs0, st), is_indexkey(sk), validref(str, r1, st[r0 -- _type] .type .type, st), ∀ str, r, ty, st. validref(str, r, ty, st) ∧ validvals(str, rkvs0, st) → validref(str, r, ty, st[rkvs0]) ⊦ validref(str, r, ty, st[r0 -- sk, refval(r1)]), validref(str, r, ty, st[r0 -- sk, refval(r1)][rkvs0]), r0 ≠ jvmref ∧ is_arraytype(st[r0 -- _type] .type) ∧ (r1 = jvmref → is_reftype(st[r0 -- _type] .type .type))
Proof step 41:
⊦ validref(str, r, ty1, st[jvmref -- sk, val]) ↔ validref(str, r, ty1, st)

Rule argument

[fma-arg: r0 ≠ jvmref]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRAFMAS : (r0 ≠ jvmref)]]