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Proof step 34 from proof ac-shift-03 (previous step)

is_indexkey(sk), ∀ class, st, ty, fieldvar, val. (addclass(class, fis0, st)[jvmref -- mkfs(class, ty, fieldvar) ', val]) = addclass(class, fis0[mkfs(class, ty, fieldvar) ', val], st) ⊦ (addclass(class, fis0, st)[jvmref -- mkfs(class, ty, fieldvar) ', val]) = addclass(class, (sk × val0) ' + fis0[mkfs(class, ty, fieldvar) ', val], st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ is_indexkey(fv .1) → addclass(class, fv ' + fis, st) = addclass(class, fis, st)
Proof step 33:
is_indexkey((sk × val0) .1), is_indexkey(sk), ∀ class, st, ty, fieldvar, val. (addclass(class, fis0, st)[jvmref -- mkfs(class, ty, fieldvar) ', val]) = addclass(class, fis0[mkfs(class, ty, fieldvar) ', val], st) ⊦ (addclass(class, fis0, st)[jvmref -- mkfs(class, ty, fieldvar) ', val]) = addclass(class, fis0[mkfs(class, ty, fieldvar) ', val], st)

Rule argument

[crewrite-arg: ac-no2 , F , [st, fis, fv, class] → [st, fis0[mkfs(class, ty, fieldvar) ', val], sk × val0, class] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (sk × val) .1 = sk], [CSIMPSEQ : ⊦ ¬ is_indexkey(sk) ↔ is_fskey(sk)], [CSIMPSEQ : ⊦ is_indexkey(sk) → ¬ is_fskey(sk)])], [EXTRATERMS : (st, fis0[mkfs(class, ty, fieldvar) ', val], sk × val0, class)]]