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Proof step 1 from proof keys-size (conclusion)

⊦ # keys(fs, ax) = # ax

Heuristic structural induction applied rule structural induction

And closed the goal!

Rule argument

term: ax

Additional information

Used simplifier rules: address-list freely generated by [], +;;% ⊦ adr + ax = adr ' + ax; (s)% ⊦ keys(fs, adr ' + ax) = fs[adr].key + keys(fs, ax); (s)% ⊦ key + kx = key ' + kx; (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ #(key ' + kx) = (# kx)+1; (s)% ⊦ # keys(fs, ax) = # ax; (s)% ⊦ m +1 = m + 1; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ #(adr ' + ax) = (# ax)+1; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ keys(fs, []) = []; (s)% ⊦ # [] = 0; (s)% ⊦ # [] = 0; (s)