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Proof step 4 from proof index-update-dom (previous step)

# kx0 + 1 = # ax, ax = adr + ax0, ∀ ax, ri. # kx0 = # ax → dom ri[kx0, ax] = dom ri ∪ kx0.toSet ⊦ dom ri[key ' + kx0, ax] = (kx0.toSet ∪ dom ri) ++ key

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
# kx0 = # ax0, ∀ ax, ri. # ax0 = # ax → dom ri[kx0, ax] = dom ri ∪ kx0.toSet ⊦ dom ri[key, adr][kx0, ax0] = (kx0.toSet ∪ dom ri) ++ key

Rule argument

Additional information

Used simplifier rules: index-update-key-rec: ⊦ ri[key ' + kx, adr ' + ax] = ri[key, adr][kx, ax];%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ m +1 = m + 1; (s)% ⊦ #(adr ' + ax) = (# ax)+1; (s)% ⊦ adr + ax = adr ' + ax; (s)