is_derive_trmx

theories/derive.v

@@ -2341,6 +2341,66 @@ apply/derivable_mxP => i0 j0.
 by have [] := MdM i0 j0.
 Qed.
 
+Lemma derivable_trmx m n (M : V -> 'M[R]_(m, n)) t v :
+  derivable (fun x => (M x)^T) t v = derivable M t v.
+Proof.
+rewrite propeqE; split; rewrite /derivable/=.
+- move=> /cvg_ex[/= l Hl].
+  apply/cvg_ex => /=; exists l^T.
+  apply/cvgrPdist_le => /= e e0.
+  move/cvgrPdist_le : Hl => /(_ _ e0)[/= r r0 re].
+  near=> x.
+  rewrite [leLHS](_ : _ =
+      `|l - x^-1 *: ((M (x *: v + t))^T - (M t)^T)|); last 2 first.
+    rewrite -[RHS]norm_trmx.
+    rewrite [in RHS]linearD/=.
+    rewrite [in RHS]linearN/=.
+    congr (`| _ - _ |).
+    rewrite [RHS]linearZ/=.
+    rewrite [in RHS]linearB.
+    by rewrite /= !trmxK.
+  apply: re => /=.
+    rewrite sub0r normrN.
+    by near: x; exact: dnbhs0_lt.
+  by near: x; exact: nbhs_dnbhs_neq.
+- move=> /cvg_ex[/= l Hl].
+  apply/cvg_ex => /=; exists l^T.
+  apply/cvgrPdist_le => /= e e0.
+  move/cvgrPdist_le : Hl => /(_ _ e0)[/= r r0 re].
+  near=> x.
+  rewrite [leLHS](_ : _ = `|l - x^-1 *: ((M (x *: v + t)) - (M t))|); last 2 first.
+    rewrite -[RHS]norm_trmx.
+    rewrite [in RHS]linearD/=.
+    rewrite [in RHS]linearN/=.
+    congr (`| _ - _ |).
+    rewrite [RHS]linearZ/=.
+    by rewrite [in RHS]linearB.
+  apply: re => /=.
+    rewrite sub0r normrN.
+    by near: x; exact: dnbhs0_lt.
+  by near: x; exact: nbhs_dnbhs_neq.
+Unshelve. all: by end_near. Qed.
+
+Lemma derive_trmx {m n : nat} (M : V -> 'M[R]_(m, n)) t v :
+  derivable M t v -> 'D_v (trmx \o M) t = ('D_v M t)^T.
+Proof.
+move=> Mt1.
+rewrite !derive_mx//=.
+  by rewrite derivable_trmx.
+apply/matrixP => i j; rewrite !mxE.
+by under eq_fun do rewrite mxE.
+Qed.
+
+Global Instance is_derive_trmx {m n : nat}
+  (f : V -> 'M[R]_(m, n)) (f' : 'M[R]_(m, n)) (t : V) w:
+  is_derive t w f f' -> is_derive t w (fun x => (f x)^T) f'^T.
+Proof.
+move=> fD.
+have fDer : derivable f t w by case: fD.
+apply/DeriveDef; last by have [_ <-] := fD; rewrite derive_trmx.
+by rewrite derivable_trmx.
+Qed.
+
 Fact dmx {m n : nat} (M : V -> 'M[R]_(m, n)) (x : V) :
   let g := fun t : V => (\matrix_(i < m, j < n) 'd M x t i j) in
   differentiable M x ->

theories/normedtype_theory/matrix_normedtype.v

@@ -172,6 +172,34 @@ Qed.
 
 End mx_norm.
 
+Section norm_trmx.
+
+Import Order.TTheory GRing.Theory Num.Def Num.Theory.
+Import Order.Def.
+Local Open Scope ring_scope.
+Lemma nng_max0r {K : realFieldType} : left_id ((0:K)%:nng) (@maxr {nonneg K}).
+Proof.
+move=> x.
+rewrite /max; case: ifPn => //.
+rewrite -leNgt => x0.
+apply/eqP; rewrite eq_le; apply/andP; split; last first.
+  exact: x0.
+by have : 0 <= x%:nngnum by []. (* NB: this should be automatic *)
+Qed.
+
+HB.instance Definition _  {K : realFieldType} :=
+  Monoid.isComLaw.Build {nonneg K} 0%:nng max maxA maxC nng_max0r.
+
+Lemma norm_trmx m n {R : realFieldType} (M : 'M[R]_(m, n)) : mx_norm (M^T) = mx_norm M.
+Proof.
+rewrite [LHS]mx_normE/=.
+under eq_bigr do rewrite mxE /=.
+rewrite -(pair_big xpredT xpredT (fun i j => `|M j i|%:nng))/=.
+by rewrite exchange_big//= pair_big.
+Qed.
+
+End norm_trmx.
+
 Lemma mx_normrE (K : realDomainType) (m n : nat) (x : 'M[K]_(m, n)) :
   mx_norm x = \big[maxr/0]_ij `|x ij.1 ij.2|.
 Proof.