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Proof step 16 from proof new-inode-exists (previous step)

∃ ks0. ks = ks0 ++ key ∧ ¬ key ∈ ks0, ∀ ks0. # ks0 < # ks → true → (∃ ino. ∀ ino0. inodekey(ino0) ∈ ks0 → ino0 < ino) ⊦ # ks = 0, ∃ ino. ∀ ino0. inodekey(ino0) ∈ ks → ino0 < ino

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
∀ ks1. # ks1 < #(ks0 ++ key) → true → (∃ ino. ∀ ino0. inodekey(ino0) ∈ ks1 → ino0 < ino) ⊦ key ∈ ks0, ∃ ino. ∀ ino0. inodekey(ino0) = key ∨ inodekey(ino0) ∈ ks0 → ino0 < ino

Rule argument

Additional information

Used simplifier rules: ⊦ key ∈ ks ++ key0 ↔ key = key0 ∨ key ∈ ks; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ ¬ key ∈ ks → #(ks ++ key) = # ks + 1; (s)