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Proof step 6 from proof update-index-data-notin (previous step)

# kx0 + 1 = # ax, ax = adr0 + ax0, adr ∈ ri[key ' + kx0, ax], ∀ ax, adr, ri. # kx0 = # ax ∧ adr ∈ ri[kx0, ax] → adr ∈ ax ∨ adr ∈ ri ⊦ adr ∈ ri, adr ∈ ax

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
# kx0 = # ax0, adr ∈ ri[key, adr0][kx0, ax0], ∀ ax, adr, ri. # ax0 = # ax ∧ adr ∈ ri[kx0, ax] → adr ∈ ax ∨ adr ∈ ri ⊦ adr = adr0, adr ∈ ri, adr ∈ ax0

Rule argument

Additional information

Used simplifier rules: index-update-key-rec: ⊦ ri[key ' + kx, adr ' + ax] = ri[key, adr][kx, ax];% ⊦ adr ∈ adr0 ' + ax ↔ ¬(adr ≠ adr0 ∧ ¬ adr ∈ ax); (ss)%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)