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Proof step 12 from proof prepost-replay-log (previous step)

valid-indices(replay-log(lpt1, log), log), valid-indices(replay-log(lpt1, log), log0), valid-indices(lpt1, log), valid-indices(lpt1, log0), # lpt1 ≤ log0.head ⊦ log0 = [], dups(log + log0), dups(log), dups(log0)

Heuristic elimination applied rule insert elim lemma

And closed the goal!

Rule argument

spec: list-data, inst: nat-list-set name: elim-list-c seq: nx ≠ [] ⊦ n = nx.head ∧ nx0 = nx.tail ↔ nx = n + nx0 subst: [nx, n, nx0] → [log0, n, nx]

Additional information

Used simplifier rules: ⊦ n + nx = n ' + nx; (s)% ⊦ n ' + nx ≠ []; (s)%a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]%⊦ valid-indices(lpt, nx0 + nx1) ↔ valid-indices(lpt, nx0) ∧ valid-indices(lpt, nx1); (ws)% ⊦ valid-indices(lpt, n ') ↔ n < # lpt; (s)% ⊦ # replay-log(lpt, nx) = # lpt; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%nx ≠ [] ⊦ n = nx.head ∧ nx0 = nx.tail ↔ nx = n + nx0; (elim) Used terms: log0, n, nx