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Proof step 14 from proof rem-preserves-monls (previous step)

mon<(rem(⌜nl0⌝ , np)), mon<(⌜nl0⌝ + ⌜m '⌝ ), ∀ nl1, np. # nl1 < #(nl0 + m ') → mon<(⌜nl1⌝ ) → mon<(rem(⌜nl1⌝ , np)) ⊦ mon<(rem(⌜nl0⌝ , np) + ⌜m '⌝ ), np(# nl0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ mon<(nσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # nσ → nσ[n] < nσ[n + 1])
Proof step 16:
mon<(rem(⌜nl0⌝ , np)), mon<(⌜nl0⌝ + ⌜m '⌝ ), ∀ nl1, np. # nl1 < #(nl0 + m ') → mon<(⌜nl1⌝ ) → mon<(rem(⌜nl1⌝ , np)) ⊦ ∀ n. ⌜n⌝ + ⌜1⌝ < #(rem(⌜nl0⌝ , np) + ⌜m '⌝ ) → (rem(⌜nl0⌝ , np) + ⌜m '⌝ )[n] < (rem(⌜nl0⌝ , np) + ⌜m '⌝ )[n + 1], np(# nl0)

Rule argument

spec: inst: name: monls-def seq: ⊦ mon<(nσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # nσ → nσ[n] < nσ[n + 1]) subst: [nσ] → [rem(⌜nl0⌝ , np) + ⌜m '⌝ ] flags: rewrite, current

Additional information

Used terms: rem(⌜nl0⌝ , np) + ⌜m '⌝