Research proof index

Self-log fixed point. The two log layers produced by oa.withQueryLog.withQueryLog agree on every support point: simulating the logging oracle over oa.withQueryLog records exactly the queries that the inner withQueryLog already recorded, since withQueryLog does not add new queries to the underlying OracleComp.

Query-log lookup is the OracleSpec specialization of dependent lookup on PolyFun traces.

Universal property (extension) for the Dedekind–MacNeille completion. Given an order embedding f : α ↪o β into a complete lattice β, this theorem produces an order embedding f' : DedekindMacNeilleCompletion α ↪o β such that f = f' ∘ coe'. API note: the constructed f' is defined by a sSup over lower bounds of upper bounds of the image of x. T...

Strict comparison of Padic.mulValuation matches strict comparison of the norm.

For an $L^2$ function on a circle of period $T$, the squared $L^2$ norm of its $N$th partial Fourier sum equals the sum of the squared magnitudes of its Fourier coefficients from $-N$ through $N$:

$\lVert S_Nf\rVert_2^2=\sum_{n=-N}^{N}|\widehat f(n)|^2.$

The expansion of asTensor into complexContrBasis basis of tensor product vectors.