# Stabilisation of gcf Computations Under Termination #

## Summary #

We show that the continuants and convergents of a gcf stabilise once the gcf terminates.

theorem GeneralizedContinuedFraction.terminated_stable {K : Type u_1} {g : } {n : } {m : } (n_le_m : n m) (terminated_at_n : ) :

If a gcf terminated at position n, it also terminated at m ≥ n.

theorem GeneralizedContinuedFraction.continuantsAux_stable_step_of_terminated {K : Type u_1} {g : } {n : } [] (terminated_at_n : ) :
theorem GeneralizedContinuedFraction.continuantsAux_stable_of_terminated {K : Type u_1} {g : } {n : } {m : } [] (n_lt_m : n < m) (terminated_at_n : ) :
theorem GeneralizedContinuedFraction.convergents'Aux_stable_of_terminated {K : Type u_1} {n : } {m : } [] (n_le_m : n m) (terminated_at_n : ) :
theorem GeneralizedContinuedFraction.continuants_stable_of_terminated {K : Type u_1} {g : } {n : } {m : } [] (n_le_m : n m) (terminated_at_n : ) :
theorem GeneralizedContinuedFraction.numerators_stable_of_terminated {K : Type u_1} {g : } {n : } {m : } [] (n_le_m : n m) (terminated_at_n : ) :
theorem GeneralizedContinuedFraction.denominators_stable_of_terminated {K : Type u_1} {g : } {n : } {m : } [] (n_le_m : n m) (terminated_at_n : ) :
theorem GeneralizedContinuedFraction.convergents_stable_of_terminated {K : Type u_1} {g : } {n : } {m : } [] (n_le_m : n m) (terminated_at_n : ) :
theorem GeneralizedContinuedFraction.convergents'_stable_of_terminated {K : Type u_1} {g : } {n : } {m : } [] (n_le_m : n m) (terminated_at_n : ) :