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Proof step 19 from proof rem-preserves-monls (previous step)

∀ n. ⌜n⌝ + ⌜1⌝ < # rem(⌜nl0⌝ , np) → rem(⌜nl0⌝ , np)[n] < rem(⌜nl0⌝ , np)[n + 1], mon<(⌜nl0⌝ + ⌜m '⌝ ), ⌜n⌝ < # rem(⌜nl0⌝ , np), (rem(⌜nl0⌝ , np) + ⌜m '⌝ )[n + 1] ≤ rem(⌜nl0⌝ , np)[n], ∀ nl1, np. # nl1 < #(nl0 + m ') → mon<(⌜nl1⌝ ) → mon<(rem(⌜nl1⌝ , np)) ⊦ np(# nl0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ ⌜m⌝ < # nσ → (nσ + nσ0)[m] = nσ[m]
Proof step 21:
¬ ⌜n + 1⌝ < # rem(⌜nl0⌝ , np), ∀ n. ⌜n⌝ + ⌜1⌝ < # rem(⌜nl0⌝ , np) → rem(⌜nl0⌝ , np)[n] < rem(⌜nl0⌝ , np)[n + 1], mon<(⌜nl0⌝ + ⌜m '⌝ ), ⌜n⌝ < # rem(⌜nl0⌝ , np), (rem(⌜nl0⌝ , np) + ⌜m '⌝ )[n + 1] ≤ rem(⌜nl0⌝ , np)[n], ∀ nl1, np. # nl1 < #(nl0 + m ') → mon<(⌜nl1⌝ ) → mon<(rem(⌜nl1⌝ , np)) ⊦ np(# nl0)
Proof step 27:
⌜n + 1⌝ < # rem(⌜nl0⌝ , np), ∀ n. ⌜n⌝ + ⌜1⌝ < # rem(⌜nl0⌝ , np) → rem(⌜nl0⌝ , np)[n] < rem(⌜nl0⌝ , np)[n + 1], mon<(⌜nl0⌝ + ⌜m '⌝ ), ⌜n⌝ < # rem(⌜nl0⌝ , np), rem(⌜nl0⌝ , np)[n + 1] ≤ rem(⌜nl0⌝ , np)[n], ∀ nl1, np. # nl1 < #(nl0 + m ') → mon<(⌜nl1⌝ ) → mon<(rem(⌜nl1⌝ , np)) ⊦ np(# nl0)

Rule argument

spec: stream inst: natstream name: get-append-ls seq: ⊦ ⌜m⌝ < # nσ → (nσ + nσ0)[m] = nσ[m] subst: [m, nσ0, nσ] → [n + 1, ⌜m '⌝ , rem(⌜nl0⌝ , np)] flags: rewrite

Additional information

Used terms: n + 1, ⌜m '⌝ , rem(⌜nl0⌝ , np)