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Proof step 6 from proof index-update-adr-in (previous step)

# kx0 + 1 = # ax, ax = adr1 + ax1, ∀ ax0, adr0, ri0. # kx0 = # ax0 → (adr0 ∈ ri0[kx0, ax0] ↔ adr0 ∈ ax0 ∨ adr0 ∈ ri0 \ kx0.toSet) ⊦ adr ∈ ri[key ' + kx0, ax] ↔ adr ∈ ax ∨ adr ∈ ri \ (kx0.toSet ++ key), dups(kx0), key ∈ kx0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
# kx0 = # ax1, ∀ ax0, adr0, ri0. # ax1 = # ax0 → (adr0 ∈ ri0[kx0, ax0] ↔ adr0 ∈ ax0 ∨ adr0 ∈ ri0 \ kx0.toSet) ⊦ adr ∈ ri[key, adr1][kx0, ax1] ↔ adr = adr1 ∨ adr ∈ ax1 ∨ adr ∈ ri \ (kx0.toSet ++ key), dups(kx0), key ∈ kx0

Rule argument

Additional information

Used simplifier rules: ⊦ adr ∈ adr0 ' + ax ↔ adr = adr0 ∨ adr ∈ ax; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ ri[key ' + kx, adr ' + ax] = ri[key, adr][kx, ax]; (s)%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)