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Proof step 16 from proof index-update-range (previous step)

# kx0 = # ax0, range ri ∩ ax0.toSet = ∅, ri-inj(ri), key ∈ ri, kx0.toSet ⊆ dom ri, ∀ kx1, ax, ri. # kx1 < #(kx0 + key ') → # kx1 = # ax ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ kx1.toSet ⊆ dom ri → range ri[kx1, ax] = (range ri \ ri[kx1].toSet) ∪ ax.toSet ∨ dups(kx1) ∨ dups(ax) ⊦ (range ri[kx0, ax0] -- ri[key]) ++ adr = (ax0.toSet ∪ ((range ri \ ri[kx0].toSet) -- ri[key])) ++ adr, dups(kx0), dups(ax0), adr ∈ ri, key ∈ kx0, adr ∈ ax0

User applied rule apply induction

And obtained the following new goals

Proof step 17:
# kx0 = # ax0, range ri ∩ ax0.toSet = ∅, ri-inj(ri), key ∈ ri, kx0.toSet ⊆ dom ri, range ri[kx0, ax0] = (range ri \ ri[kx0].toSet) ∪ ax0.toSet ∨ dups(kx0) ∨ dups(ax0), ∀ kx1, ax, ri. # kx1 < #(kx0 + key ') → # kx1 = # ax ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ kx1.toSet ⊆ dom ri → range ri[kx1, ax] = (range ri \ ri[kx1].toSet) ∪ ax.toSet ∨ dups(kx1) ∨ dups(ax) ⊦ (range ri[kx0, ax0] -- ri[key]) ++ adr = (ax0.toSet ∪ ((range ri \ ri[kx0].toSet) -- ri[key])) ++ adr, dups(ax0), dups(kx0), adr ∈ ax0, key ∈ kx0, adr ∈ ri

Rule argument

subst: [kx1, ax, ri] → [kx0, ax0, ri]

Additional information

Used simplifier rules: int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ #(kx + kx0) = # kx + # kx0; (s)% ⊦ # key ' = 0 +1; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: kx0, ax0, ri