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Proof step 9 from proof at-write-at (previous step)

#(adr0 + ax) + # ax0 + 1 = # ndx, ∀ ax0, ndx, fs, adr. # ax + # ax0 + 1 = # ndx → fs[ax + adr ' + ax0, ndx][adr] = ndx[# ax] ∨ dups(ax + ax0) ∨ dups(ax) ∨ dups(ax0) ∨ adr ∈ ax ∨ adr ∈ ax0 ⊦ fs[(adr0 + ax) + adr ' + ax0, ndx][adr] = ndx[#(adr0 + ax)], dups(ax0), dups(adr0 + ax), dups((adr0 + ax) + ax0), adr ∈ ax0, adr ∈ adr0 + ax

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
2 + # ax + # ax0 = # ndx, ∀ ax1, ndx0, fs0, adr1. # ax + # ax1 + 1 = # ndx0 → fs0[ax + adr1 ' + ax1, ndx0][adr1] = ndx0[# ax] ∨ dups(ax + ax1) ∨ dups(ax1) ∨ adr1 ∈ ax ∨ adr1 ∈ ax1 ⊦ fs[(adr0 ' + ax) + adr ' + ax0, ndx][adr] = ndx[# ax + 1], adr = adr0, dups(ax + ax0), dups(ax), dups(ax0), adr ∈ ax, adr ∈ ax0, adr0 ∈ ax, adr0 ∈ ax0

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(ax + ax0) → ¬ dups(ax) ∧ ¬ dups(ax0); (forward) from specification list-dup[address-list-set]% ⊦ ¬ adr ∈ ax → (dups(adr ' + ax) ↔ dups(ax)); (s)% ⊦ adr ∈ ax + ax0 ↔ adr ∈ ax ∨ adr ∈ ax0; (s)% ⊦ dups(adr ' + ax) ↔ dups(ax) ∨ adr ∈ ax; (ws)% ⊦ adr ∈ adr0 ' + ax ↔ ¬(adr ≠ adr0 ∧ ¬ adr ∈ ax); (ss)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%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)