From 748d716efb2f2f75946c8386e441ce1789806a39 Mon Sep 17 00:00:00 2001
From: Enrico Tassi
Date: Wed, 22 May 2019 13:43:08 +0200
Subject: htmldoc regenerated
---
docs/htmldoc/mathcomp.algebra.polyXY.html | 157 +++++++++++++++---------------
1 file changed, 78 insertions(+), 79 deletions(-)
(limited to 'docs/htmldoc/mathcomp.algebra.polyXY.html')
diff --git a/docs/htmldoc/mathcomp.algebra.polyXY.html b/docs/htmldoc/mathcomp.algebra.polyXY.html
index 5506005..53a0eb8 100644
--- a/docs/htmldoc/mathcomp.algebra.polyXY.html
+++ b/docs/htmldoc/mathcomp.algebra.polyXY.html
@@ -21,7 +21,6 @@
@@ -67,9 +66,9 @@
-Notation "'Y" := 'X%:P : ring_scope.
-Notation "p ^:P" := (p ^ polyC) (at level 2, format "p ^:P") : ring_scope.
-Notation "p .[ x , y ]" := (p.[x%:P].[y])
+Notation "'Y" := 'X%:P : ring_scope.
+Notation "p ^:P" := (p ^ polyC) (at level 2, format "p ^:P") : ring_scope.
+Notation "p .[ x , y ]" := (p.[x%:P].[y])
(at level 2, left associativity, format "p .[ x , y ]") : ring_scope.
@@ -77,90 +76,90 @@
Variable R : ringType.
-Implicit Types (u : {poly {poly R}}) (p q : {poly R}) (x : R).
+Implicit Types (u : {poly {poly R}}) (p q : {poly R}) (x : R).
-Fact swapXY_key : unit.
-Definition swapXY_def u : {poly {poly R}} := (u ^ map_poly polyC).['Y].
-Definition swapXY := locked_with swapXY_key swapXY_def.
-Canonical swapXY_unlockable := [unlockable fun swapXY].
+Fact swapXY_key : unit.
+Definition swapXY_def u : {poly {poly R}} := (u ^ map_poly polyC).['Y].
+Definition swapXY := locked_with swapXY_key swapXY_def.
+Canonical swapXY_unlockable := [unlockable fun swapXY].
-Definition sizeY u : nat := \max_(i < size u) (size u`_i).
-Definition poly_XaY p : {poly {poly R}} := p^:P \Po ('X + 'Y).
-Definition poly_XmY p : {poly {poly R}} := p^:P \Po ('X × 'Y).
-Definition sub_annihilant p q := resultant (poly_XaY p) q^:P.
-Definition div_annihilant p q := resultant (poly_XmY p) q^:P.
+Definition sizeY u : nat := \max_(i < size u) (size u`_i).
+Definition poly_XaY p : {poly {poly R}} := p^:P \Po ('X + 'Y).
+Definition poly_XmY p : {poly {poly R}} := p^:P \Po ('X × 'Y).
+Definition sub_annihilant p q := resultant (poly_XaY p) q^:P.
+Definition div_annihilant p q := resultant (poly_XmY p) q^:P.
-Lemma swapXY_polyC p : swapXY p%:P = p^:P.
+Lemma swapXY_polyC p : swapXY p%:P = p^:P.
-Lemma swapXY_X : swapXY 'X = 'Y.
+Lemma swapXY_X : swapXY 'X = 'Y.
-Lemma swapXY_Y : swapXY 'Y = 'X.
+Lemma swapXY_Y : swapXY 'Y = 'X.
Lemma swapXY_is_additive : additive swapXY.
Canonical swapXY_addf := Additive swapXY_is_additive.
-Lemma coef_swapXY u i j : (swapXY u)`_i`_j = u`_j`_i.
+Lemma coef_swapXY u i j : (swapXY u)`_i`_j = u`_j`_i.
-Lemma swapXYK : involutive swapXY.
+Lemma swapXYK : involutive swapXY.
-Lemma swapXY_map_polyC p : swapXY p^:P = p%:P.
+Lemma swapXY_map_polyC p : swapXY p^:P = p%:P.
-Lemma swapXY_eq0 u : (swapXY u == 0) = (u == 0).
+Lemma swapXY_eq0 u : (swapXY u == 0) = (u == 0).
Lemma swapXY_is_multiplicative : multiplicative swapXY.
Canonical swapXY_rmorphism := AddRMorphism swapXY_is_multiplicative.
-Lemma swapXY_is_scalable : scalable_for (map_poly polyC \; *%R) swapXY.
+Lemma swapXY_is_scalable : scalable_for (map_poly polyC \; *%R) swapXY.
Canonical swapXY_linear := AddLinear swapXY_is_scalable.
-Canonical swapXY_lrmorphism := [lrmorphism of swapXY].
+Canonical swapXY_lrmorphism := [lrmorphism of swapXY].
-Lemma swapXY_comp_poly p u : swapXY (p^:P \Po u) = p^:P \Po swapXY u.
+Lemma swapXY_comp_poly p u : swapXY (p^:P \Po u) = p^:P \Po swapXY u.
-Lemma max_size_coefXY u i : size u`_i ≤ sizeY u.
+Lemma max_size_coefXY u i : size u`_i ≤ sizeY u.
-Lemma max_size_lead_coefXY u : size (lead_coef u) ≤ sizeY u.
+Lemma max_size_lead_coefXY u : size (lead_coef u) ≤ sizeY u.
-Lemma max_size_evalX u : size u.['X] ≤ sizeY u + (size u).-1.
+Lemma max_size_evalX u : size u.['X] ≤ sizeY u + (size u).-1.
-Lemma max_size_evalC u x : size u.[x%:P] ≤ sizeY u.
+Lemma max_size_evalC u x : size u.[x%:P] ≤ sizeY u.
-Lemma sizeYE u : sizeY u = size (swapXY u).
+Lemma sizeYE u : sizeY u = size (swapXY u).
-Lemma sizeY_eq0 u : (sizeY u == 0%N) = (u == 0).
+Lemma sizeY_eq0 u : (sizeY u == 0%N) = (u == 0).
-Lemma sizeY_mulX u : sizeY (u × 'X) = sizeY u.
+Lemma sizeY_mulX u : sizeY (u × 'X) = sizeY u.
-Lemma swapXY_poly_XaY p : swapXY (poly_XaY p) = poly_XaY p.
+Lemma swapXY_poly_XaY p : swapXY (poly_XaY p) = poly_XaY p.
-Lemma swapXY_poly_XmY p : swapXY (poly_XmY p) = poly_XmY p.
+Lemma swapXY_poly_XmY p : swapXY (poly_XmY p) = poly_XmY p.
-Lemma poly_XaY0 : poly_XaY 0 = 0.
+Lemma poly_XaY0 : poly_XaY 0 = 0.
-Lemma poly_XmY0 : poly_XmY 0 = 0.
+Lemma poly_XmY0 : poly_XmY 0 = 0.
End PolyXY_Ring.
@@ -168,30 +167,30 @@
-Lemma swapXY_map (R S : ringType) (f : {additive R → S}) u :
- swapXY (u ^ map_poly f) = swapXY u ^ map_poly f.
+Lemma swapXY_map (R S : ringType) (f : {additive R → S}) u :
+ swapXY (u ^ map_poly f) = swapXY u ^ map_poly f.
Section PolyXY_ComRing.
Variable R : comRingType.
-Implicit Types (u : {poly {poly R}}) (p : {poly R}) (x y : R).
+Implicit Types (u : {poly {poly R}}) (p : {poly R}) (x y : R).
-Lemma horner_swapXY u x : (swapXY u).[x%:P] = u ^ eval x.
+Lemma horner_swapXY u x : (swapXY u).[x%:P] = u ^ eval x.
-Lemma horner_polyC u x : u.[x%:P] = swapXY u ^ eval x.
+Lemma horner_polyC u x : u.[x%:P] = swapXY u ^ eval x.
-Lemma horner2_swapXY u x y : (swapXY u).[x, y] = u.[y, x].
+Lemma horner2_swapXY u x y : (swapXY u).[x, y] = u.[y, x].
-Lemma horner_poly_XaY p v : (poly_XaY p).[v] = p \Po (v + 'X).
+Lemma horner_poly_XaY p v : (poly_XaY p).[v] = p \Po (v + 'X).
-Lemma horner_poly_XmY p v : (poly_XmY p).[v] = p \Po (v × 'X).
+Lemma horner_poly_XmY p v : (poly_XmY p).[v] = p \Po (v × 'X).
End PolyXY_ComRing.
@@ -201,49 +200,49 @@
Variable R : idomainType.
-Implicit Types (p q : {poly R}) (x y : R).
+Implicit Types (p q : {poly R}) (x y : R).
-Lemma size_poly_XaY p : size (poly_XaY p) = size p.
+Lemma size_poly_XaY p : size (poly_XaY p) = size p.
-Lemma poly_XaY_eq0 p : (poly_XaY p == 0) = (p == 0).
+Lemma poly_XaY_eq0 p : (poly_XaY p == 0) = (p == 0).
-Lemma size_poly_XmY p : size (poly_XmY p) = size p.
+Lemma size_poly_XmY p : size (poly_XmY p) = size p.
-Lemma poly_XmY_eq0 p : (poly_XmY p == 0) = (p == 0).
+Lemma poly_XmY_eq0 p : (poly_XmY p == 0) = (p == 0).
-Lemma lead_coef_poly_XaY p : lead_coef (poly_XaY p) = (lead_coef p)%:P.
+Lemma lead_coef_poly_XaY p : lead_coef (poly_XaY p) = (lead_coef p)%:P.
Lemma sub_annihilant_in_ideal p q :
- 1 < size p → 1 < size q →
- {uv : {poly {poly R}} × {poly {poly R}}
- | size uv.1 < size q ∧ size uv.2 < size p
- & ∀ x y,
- (sub_annihilant p q).[y] = uv.1.[x, y] × p.[x + y] + uv.2.[x, y] × q.[x]}.
+ 1 < size p → 1 < size q →
+ {uv : {poly {poly R}} × {poly {poly R}}
+ | size uv.1 < size q ∧ size uv.2 < size p
+ & ∀ x y,
+ (sub_annihilant p q).[y] = uv.1.[x, y] × p.[x + y] + uv.2.[x, y] × q.[x]}.
Lemma sub_annihilantP p q x y :
- p != 0 → q != 0 → p.[x] = 0 → q.[y] = 0 →
- (sub_annihilant p q).[x - y] = 0.
+ p != 0 → q != 0 → p.[x] = 0 → q.[y] = 0 →
+ (sub_annihilant p q).[x - y] = 0.
-Lemma sub_annihilant_neq0 p q : p != 0 → q != 0 → sub_annihilant p q != 0.
+Lemma sub_annihilant_neq0 p q : p != 0 → q != 0 → sub_annihilant p q != 0.
Lemma div_annihilant_in_ideal p q :
- 1 < size p → 1 < size q →
- {uv : {poly {poly R}} × {poly {poly R}}
- | size uv.1 < size q ∧ size uv.2 < size p
- & ∀ x y,
- (div_annihilant p q).[y] = uv.1.[x, y] × p.[x × y] + uv.2.[x, y] × q.[x]}.
+ 1 < size p → 1 < size q →
+ {uv : {poly {poly R}} × {poly {poly R}}
+ | size uv.1 < size q ∧ size uv.2 < size p
+ & ∀ x y,
+ (div_annihilant p q).[y] = uv.1.[x, y] × p.[x × y] + uv.2.[x, y] × q.[x]}.
-Lemma div_annihilant_neq0 p q : p != 0 → q.[0] != 0 → div_annihilant p q != 0.
+Lemma div_annihilant_neq0 p q : p != 0 → q.[0] != 0 → div_annihilant p q != 0.
End PolyXY_Idomain.
@@ -252,35 +251,35 @@
Section PolyXY_Field.
-Variables (F E : fieldType) (FtoE : {rmorphism F → E}).
+Variables (F E : fieldType) (FtoE : {rmorphism F → E}).
-Lemma div_annihilantP (p q : {poly E}) (x y : E) :
- p != 0 → q != 0 → y != 0 → p.[x] = 0 → q.[y] = 0 →
- (div_annihilant p q).[x / y] = 0.
+Lemma div_annihilantP (p q : {poly E}) (x y : E) :
+ p != 0 → q != 0 → y != 0 → p.[x] = 0 → q.[y] = 0 →
+ (div_annihilant p q).[x / y] = 0.
-Lemma map_sub_annihilantP (p q : {poly F}) (x y : E) :
- p != 0 → q != 0 →(p ^ FtoE).[x] = 0 → (q ^ FtoE).[y] = 0 →
- (sub_annihilant p q ^ FtoE).[x - y] = 0.
+Lemma map_sub_annihilantP (p q : {poly F}) (x y : E) :
+ p != 0 → q != 0 →(p ^ FtoE).[x] = 0 → (q ^ FtoE).[y] = 0 →
+ (sub_annihilant p q ^ FtoE).[x - y] = 0.
-Lemma map_div_annihilantP (p q : {poly F}) (x y : E) :
- p != 0 → q != 0 → y != 0 → (p ^ FtoE).[x] = 0 → (q ^ FtoE).[y] = 0 →
- (div_annihilant p q ^ FtoE).[x / y] = 0.
+Lemma map_div_annihilantP (p q : {poly F}) (x y : E) :
+ p != 0 → q != 0 → y != 0 → (p ^ FtoE).[x] = 0 → (q ^ FtoE).[y] = 0 →
+ (div_annihilant p q ^ FtoE).[x / y] = 0.
-Lemma root_annihilant x p (pEx := (p ^ pFtoE).[x%:P]) :
- pEx != 0 → algebraicOver FtoE x →
- exists2 r : {poly F}, r != 0 & ∀ y, root pEx y → root (r ^ FtoE) y.
+Lemma root_annihilant x p (pEx := (p ^ pFtoE).[x%:P]) :
+ pEx != 0 → algebraicOver FtoE x →
+ exists2 r : {poly F}, r != 0 & ∀ y, root pEx y → root (r ^ FtoE) y.
Lemma algebraic_root_polyXY x y :
- (let pEx p := (p ^ map_poly FtoE).[x%:P] in
- exists2 p, pEx p != 0 & root (pEx p) y) →
- algebraicOver FtoE x → algebraicOver FtoE y.
+ (let pEx p := (p ^ map_poly FtoE).[x%:P] in
+ exists2 p, pEx p != 0 & root (pEx p) y) →
+ algebraicOver FtoE x → algebraicOver FtoE y.
End PolyXY_Field.
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