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Proof step 3 from proof index-update-at-last-oc (previous step)

# ax = # kx0 + # kx1 + 1 ⊦ ri[kx0 + key ' + kx1, ax][key] = ax[# kx0], key ∈ kx1

Heuristic structural induction applied rule structural induction

And obtained the following new goals

Proof step 4:
# ax = # [] + # kx1 + 1 ⊦ ri[[] + key ' + kx1, ax][key] = ax[# []], key ∈ kx1
Proof step 9:
# ax = #(key0 + kx) + # kx1 + 1, ∀ ax, kx1, ri, key. # ax = # kx + # kx1 + 1 → ri[kx + key ' + kx1, ax][key] = ax[# kx] ∨ key ∈ kx1 ⊦ ri[(key0 + kx) + key ' + kx1, ax][key] = ax[#(key0 + kx)], key ∈ kx1

Rule argument

term: kx0

Additional information

Used simplifier rules: key-list freely generated by [], +;;