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Proof step 4 from proof keys-length (conclusion)

⊦ # fis .keys = # fis

User applied rule structural induction

And obtained the following new goals

Proof step 1:
⊦ [] .keys = []
Proof step 2:
⊦ (fis + fis0) .keys = fis .keys + fis0 .keys
Proof step 3:
⊦ fv ' .keys = fv .1 '

Rule argument

[VARWITHVARSEQSARG : fis, ()]

Additional information

[[CONCRETESIMPRULES : ([CSIMPGEN : fieldinits generated by [], +;], [CSIMPNAMEDSEQ : list-data, fieldinit-assoclista, size-01, ⊦ # [] = 0], [CSIMPNAMEDSEQ : list-data, storekeys, size-01, ⊦ # [] = 0], [CSIMPNAMEDSEQ : list, fieldinit-assoclista, c, ⊦ fv + fis = fv ' + fis], [CSIMPNAMEDSEQ : list, fieldinit-assoclista, a, ⊦ (fis + fis0) + fis1 = fis + fis0 + fis1], [CSIMPNAMEDSEQ : list, storekeys, len, ⊦ # (sks + sks0) = # sks + # sks0], [CSIMPNAMEDSEQ : list, storekeys, len-02, ⊦ # sk ' = 0 +1], [CSIMPNAMEDSEQ : list, storekeys, a, ⊦ (sks + sks0) + sks1 = sks + sks0 + sks1], [CSIMPNAMEDSEQ : list, fieldinit-assoclista, len, ⊦ # (fis + fis0) = # fis + # fis0], [CSIMPNAMEDSEQ : list, fieldinit-assoclista, len-02, ⊦ # fv ' = 0 +1], [CSIMPNAMEDSEQ : nat-basic2, , eq-01, ⊦ m + n = m + n0 ↔ n = n0], [CSIMPNAMEDSEQ : nat-basic2, , a, ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPNAMEDSEQ : nat-basic2, , c, ⊦ m + n = n + m])]]