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derive row_mx #1866

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2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -65,6 +65,8 @@

- in `derive.v`:
+ lemmas `compact_EVT_max`, `compact_EVT_min`, `EVT_max_rV`, `EVT_min_rV`
- in `derive.v`:
+ lemmas `derivable_row_mx`, `derive_row_mx`

### Changed
- in `set_interval.v`
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61 changes: 61 additions & 0 deletions theories/derive.v
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Expand Up @@ -1023,6 +1023,67 @@ Proof. by move=> /differentiableP df; rewrite diff_val. Qed.

End DifferentialR3_numFieldType.

Lemma derivable_row_mx {R : realFieldType} {n1 n2 : nat}
(f : R -> 'rV[R]_n1) (g : R -> 'rV[R]_n2) t v :
(forall x, derivable f x v) -> (forall x, derivable g x v) ->
derivable (fun x : R => row_mx (f x) (g x)) t v.
Proof.
move=> /= fv gv; apply/derivable_mxP => i j.
rewrite (ord1 i)/=.
have /cvg_ex[/= l Hl]:= fv t.
have /cvg_ex[/= k Hk]:= gv t.
apply/cvg_ex => /=; exists (row_mx l k)``_j.
apply/cvgrPdist_le => /= e e0.
move/cvgrPdist_le : Hl => /(_ _ e0) Hl.
move/cvgrPdist_le : Hk => /(_ _ e0) Hk.
move: Hl Hk; apply: filterS2 => x Hl Hk.
rewrite !mxE.
case: fintype.splitP => j1 jj1.
apply: le_trans Hl.
rewrite [in leRHS]/Num.Def.normr/= mx_normrE.
apply: le_trans; last first.
exact: (le_bigmax _ _ (ord0, j1)).
by rewrite !mxE/=.
apply: le_trans Hk.
rewrite [in leRHS]/Num.Def.normr/= mx_normrE.
apply: le_trans; last first.
exact: (le_bigmax _ _ (ord0, j1)).
by rewrite !mxE/=.
Qed.

Lemma derive_row_mx {R : realFieldType} {n1 n2 : nat}
(f : R -> 'rV[R]_n1) (g : R -> 'rV[R]_n2) t v :
(forall x : R, derivable f x v) ->
(forall x : R, derivable g x v) ->
'D_v (fun x => row_mx (f x) (g x)) t = row_mx ('D_v f t) ('D_v g t).
Proof.
move=> fv gv.
apply/matrixP => i j.
rewrite [in LHS]derive_mx ?mxE//=; last first.
by apply: derivable_row_mx; [exact: fv|exact: gv].
do 2 rewrite derive_mx ?mxE//=.
case: fintype.split_ordP => /= j1 jj1; rewrite !mxE; congr ('D_v _ t).
apply/funext => x; rewrite !mxE.
case: fintype.split_ordP => k jE.
congr (f x i _).
move: jE.
by rewrite jj1 => /(congr1 val) => /= /val_inj.
move: jE.
rewrite jj1 => /(congr1 val)/=.
have /[swap] -> := ltn_ord j1.
by rewrite ltnNge/= leq_addr.
apply/funext => x; rewrite !mxE.
case: fintype.split_ordP => k jE.
move: jE.
rewrite jj1 => /(congr1 val)/=.
have /[swap] <- := ltn_ord k.
by rewrite ltnNge/= leq_addr.
congr (g x i _).
move: jE.
rewrite jj1 => /(congr1 val) => /= /eqP.
by rewrite eqn_add2l => /eqP /val_inj.
Qed.

Section DeriveRU.
Variables (R : numFieldType) (U : normedModType R).
Implicit Types f : R -> U.
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