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Proof step 1 from proof index-update-dom (conclusion)

# kx = # ax ⊦ dom ri[kx, ax] = dom ri ∪ kx.toSet

User applied rule structural induction

And closed the goal!

Rule argument

indvar: kx

Additional information

Used simplifier rules: key-list freely generated by [], +;;% ⊦ key + kx = key ' + kx; (s)% ⊦ #(key ' + kx) = (# kx)+1; (s)% ⊦ m +1 = m + 1; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]%# kx = # ax ⊦ dom ri[kx, ax] = dom ri ∪ kx.toSet; (s)% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (s)% ⊦ key '.toSet = ∅ ++ key; (s)% ⊦ (ks ++ key) ∪ ks0 = (ks ∪ ks0) ++ key; (s)% ⊦ ks ∪ ∅ = ks; (s)%una: ⊦ (ks ∪ ks0) ∪ ks1 = ks ∪ ks0 ∪ ks1; from specification set-union[node-index]%unc: ⊦ ks ∪ ks0 = ks0 ∪ ks; from specification set-union[node-index]% ⊦ # [] = 0; (s)% ⊦ # ax = 0 ↔ ax = []; (s)% ⊦ # [] = 0; (s)% ⊦ [].toSet = ∅; (s)