Truncated logarithm over the natural numbers#1311
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…t/Mod` ; move `≤-·sk-cancel` and `<-·sk-cancel` from `Fin/Properties` to `Nat/Order`, giving shorter inductive proofs
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This PR introduces the truncated logarithm of
xin baseb, as the unique natural numberlsuch thatb ^ l ≤ x < b ^ (suc l).The base
band argumentxare also natural numbers subject to2 ≤ band1 ≤ x; these assumptions are enforced by implicit arguments of type2 ≤ᵗ band1 ≤ᵗ x, to make computing withlogeasier.However, for stating and proving the properties, it was more straightforward to work with a base of the form
suc (suc m), rather than using an implicit argument.Some auxiliary lemmas were also added in the
Nat.ModandNat.Ordermodules; in particular,≤-·sk-canceland<-·sk-cancelhave been moved fromFin.PropertiestoNat.Order, using shorter inductive proofs.