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Proof step 4 from proof validref-put-any-01 (previous step)

sk .fs = mkfs(class, ty0, fieldvar), validref(str, r, ty, st), ¬ is_indexkey(sk) → (is_referencevalue(val) → validref(str, val .refval, ty0, st) ∧ is_reftype(ty0)) ∧ (¬ is_referencevalue(val) → okval(val, ty0) ∧ ty0 ≠ void_type ∧ sk ≠ _length), ¬ is_fskey(sk) → ( is_referencevalue(val) → validref(str, val .refval, st[r0 -- _type] .type .type, st) ∧ r0 ≠ jvmref ∧ is_arraytype(st[r0 -- _type] .type) ∧ (val .refval = jvmref → is_reftype(st[r0 -- _type] .type .type))) ∧ (¬ is_referencevalue(val) → okval(val, st[r0 -- _type] .type .type)) ⊦ validref(str, r, ty, st[r0 -- sk, val])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 1:
⊦ validref(str, r, ty, st) ∧ ( is_fskey(sk) → (is_referencevalue(val) → validref(str, val .refval, sk .fs .type, st) ∧ is_reftype(sk .fs .type)) ∧ (¬ is_referencevalue(val) → is_fskey(sk) ∧ okval(val, sk .fs .type) ∧ sk .fs .type ≠ void_type ∧ sk ≠ _length)) ∧ ( is_indexkey(sk) → ( is_referencevalue(val) → validref(str, val .refval, st[r0 -- _type] .type .type, st) ∧ r0 ≠ jvmref ∧ is_arraytype(st[r0 -- _type] .type) ∧ (val .refval = jvmref → is_reftype(st[r0 -- _type] .type .type))) ∧ (¬ is_referencevalue(val) → okval(val, st[r0 -- _type] .type .type))) → validref(str, r, ty, st[r0 -- sk, val])
Proof step 3:
¬ ( validref(str, r, ty, st) ∧ ( is_fskey(sk) → (is_referencevalue(val) → validref(str, val .refval, sk .fs .type, st) ∧ is_reftype(sk .fs .type)) ∧ (¬ is_referencevalue(val) → is_fskey(sk) ∧ okval(val, sk .fs .type) ∧ sk .fs .type ≠ void_type ∧ sk ≠ _length)) ∧ ( is_indexkey(sk) → ( is_referencevalue(val) → validref(str, val .refval, st[r0 -- _type] .type .type, st) ∧ r0 ≠ jvmref ∧ is_arraytype(st[r0 -- _type] .type) ∧ (val .refval = jvmref → is_reftype(st[r0 -- _type] .type .type))) ∧ (¬ is_referencevalue(val) → okval(val, st[r0 -- _type] .type .type)))), sk .fs = mkfs(class, ty0, fieldvar), validref(str, r, ty, st), ¬ is_indexkey(sk) → (is_referencevalue(val) → validref(str, val .refval, ty0, st) ∧ is_reftype(ty0)) ∧ (¬ is_referencevalue(val) → okval(val, ty0) ∧ ty0 ≠ void_type ∧ sk ≠ _length), ¬ is_fskey(sk) → ( is_referencevalue(val) → validref(str, val .refval, st[r0 -- _type] .type .type, st) ∧ r0 ≠ jvmref ∧ is_arraytype(st[r0 -- _type] .type) ∧ (val .refval = jvmref → is_reftype(st[r0 -- _type] .type .type))) ∧ (¬ is_referencevalue(val) → okval(val, st[r0 -- _type] .type .type)) ⊦ validref(str, r, ty, st[r0 -- sk, val])

Rule argument

[crewrite-arg: validref-put-any , F , [val, sk, r0, st, ty, r, str] → [val, sk, r0, st, ty, r, str] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (val, sk, r0, st, ty, r, str)]]