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Proof step 13 from proof rem-preserves-monls (previous step)

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

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [nl1, np] → [nl0, np]

Additional information

Used simplifier rules: monls-append1: ⊦ mon<(⌜nl⌝ + ⌜nl0⌝ ) → (¬ mon<(⌜nl⌝ ) ↔ false);%append-assoc: ⊦ (nσ + nσ0) + nσ1 = nσ + nσ0 + nσ1; from specification stream[natstream]%len: ⊦ #(nl + nl0) = # nl + # nl0; from specification list[natstream]%len-02: ⊦ # m ' = 0 +1; from specification list[natstream]%a: ⊦ (nl + nl0) + nl1 = nl + nl0 + nl1; from specification list[natstream]%s: ⊦ m < n + 1 ↔ ¬ n < m; from specification nat-basic2%ref-1: ⊦ ¬ n < n; from specification nat-basic1%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2 Used terms: nl0, np