diff --git a/Nevanlinna/partialDeriv.lean b/Nevanlinna/partialDeriv.lean index cd3f922..47d6596 100644 --- a/Nevanlinna/partialDeriv.lean +++ b/Nevanlinna/partialDeriv.lean @@ -93,6 +93,25 @@ lemma partialDeriv_fderiv {f : E โ†’ F} (hf : ContDiff ๐•œ 2 f) (z a b : E) : ยท simp +section restrictScalars + +variable (๐•œ : Type*) [NontriviallyNormedField ๐•œ] +variable {๐•œ' : Type*} [NontriviallyNormedField ๐•œ'] [NormedAlgebra ๐•œ ๐•œ'] +variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ๐•œ E] [NormedSpace ๐•œ' E] +variable [IsScalarTower ๐•œ ๐•œ' E] +variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ๐•œ F] [NormedSpace ๐•œ' F] +variable [IsScalarTower ๐•œ ๐•œ' F] +--variable {f : E โ†’ F} + +theorem partialDeriv_restrictScalars {f : E โ†’ F} {v : E} : + Differentiable ๐•œ' f โ†’ partialDeriv ๐•œ v f = partialDeriv ๐•œ' v f := by + intro hf + unfold partialDeriv + funext x + rw [(hf x).fderiv_restrictScalars ๐•œ] + simp + + theorem partialDeriv_comm {E : Type*} [NormedAddCommGroup E] [NormedSpace โ„ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace โ„ F]