Pointwise order on finitely supported functions #

This file lifts order structures on α to ι →₀ α.

Main declarations #

• Finsupp.orderEmbeddingToFun: The order embedding from finitely supported functions to functions.

Order structures #

instance Finsupp.instLEFinsupp {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] :
LE (ι →₀ α)
Equations
• Finsupp.instLEFinsupp = { le := fun (f g : ι →₀ α) => ∀ (i : ι), f i g i }
theorem Finsupp.le_def {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] {f : ι →₀ α} {g : ι →₀ α} :
f g ∀ (i : ι), f i g i
@[simp]
theorem Finsupp.coe_le_coe {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] {f : ι →₀ α} {g : ι →₀ α} :
f g f g
def Finsupp.orderEmbeddingToFun {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] :
(ι →₀ α) ↪o (ια)

The order on Finsupps over a partial order embeds into the order on functions

Equations
• Finsupp.orderEmbeddingToFun = { toFun := fun (f : ι →₀ α) => f, inj' := , map_rel_iff' := }
Instances For
@[simp]
theorem Finsupp.orderEmbeddingToFun_apply {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] {f : ι →₀ α} {i : ι} :
Finsupp.orderEmbeddingToFun f i = f i
instance Finsupp.preorder {ι : Type u_1} {α : Type u_2} [Zero α] [] :
Equations
• Finsupp.preorder = let __src := Finsupp.instLEFinsupp; Preorder.mk
theorem Finsupp.lt_def {ι : Type u_1} {α : Type u_2} [Zero α] [] {f : ι →₀ α} {g : ι →₀ α} :
f < g f g ∃ (i : ι), f i < g i
@[simp]
theorem Finsupp.coe_lt_coe {ι : Type u_1} {α : Type u_2} [Zero α] [] {f : ι →₀ α} {g : ι →₀ α} :
f < g f < g
theorem Finsupp.coe_mono {ι : Type u_1} {α : Type u_2} [Zero α] [] :
Monotone Finsupp.toFun
theorem Finsupp.coe_strictMono {ι : Type u_1} {α : Type u_2} [Zero α] [] :
Monotone Finsupp.toFun
instance Finsupp.partialorder {ι : Type u_1} {α : Type u_2} [Zero α] [] :
Equations
• Finsupp.partialorder = let __src := Finsupp.preorder;
instance Finsupp.semilatticeInf {ι : Type u_1} {α : Type u_2} [Zero α] [] :
Equations
• Finsupp.semilatticeInf = let __src := Finsupp.partialorder;
@[simp]
theorem Finsupp.inf_apply {ι : Type u_1} {α : Type u_2} [Zero α] [] {i : ι} {f : ι →₀ α} {g : ι →₀ α} :
(f g) i = f i g i
instance Finsupp.semilatticeSup {ι : Type u_1} {α : Type u_2} [Zero α] [] :
Equations
• Finsupp.semilatticeSup = let __src := Finsupp.partialorder;
@[simp]
theorem Finsupp.sup_apply {ι : Type u_1} {α : Type u_2} [Zero α] [] {i : ι} {f : ι →₀ α} {g : ι →₀ α} :
(f g) i = f i g i
instance Finsupp.lattice {ι : Type u_1} {α : Type u_2} [Zero α] [] :
Lattice (ι →₀ α)
Equations
• Finsupp.lattice = let __src := Finsupp.semilatticeInf; let __src_1 := Finsupp.semilatticeSup; Lattice.mk
theorem Finsupp.support_inf_union_support_sup {ι : Type u_1} {α : Type u_2} [Zero α] [] [] (f : ι →₀ α) (g : ι →₀ α) :
(f g).support (f g).support = f.support g.support
theorem Finsupp.support_sup_union_support_inf {ι : Type u_1} {α : Type u_2} [Zero α] [] [] (f : ι →₀ α) (g : ι →₀ α) :
(f g).support (f g).support = f.support g.support

Algebraic order structures #

instance Finsupp.orderedAddCommMonoid {ι : Type u_1} {α : Type u_2} :
Equations
instance Finsupp.orderedCancelAddCommMonoid {ι : Type u_1} {α : Type u_2} :
Equations
instance Finsupp.contravariantClass {ι : Type u_1} {α : Type u_2} [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] :
ContravariantClass (ι →₀ α) (ι →₀ α) (fun (x x_1 : ι →₀ α) => x + x_1) fun (x x_1 : ι →₀ α) => x x_1
Equations
• =
instance Finsupp.instPosSMulMono {ι : Type u_1} {α : Type u_2} {β : Type u_3} [Zero α] [] [Zero β] [] [] [] :
PosSMulMono α (ι →₀ β)
Equations
• =
instance Finsupp.instSMulPosMono {ι : Type u_1} {α : Type u_2} {β : Type u_3} [] [Zero β] [] [] [] :
SMulPosMono α (ι →₀ β)
Equations
• =
instance Finsupp.instPosSMulReflectLE {ι : Type u_1} {α : Type u_2} {β : Type u_3} [Zero α] [] [Zero β] [] [] [] :
Equations
• =
instance Finsupp.instSMulPosReflectLE {ι : Type u_1} {α : Type u_2} {β : Type u_3} [] [Zero β] [] [] [] :
Equations
• =
instance Finsupp.instPosSMulStrictMono {ι : Type u_1} {α : Type u_2} {β : Type u_3} [Zero α] [] [Zero β] [] [] [] :
Equations
• =
instance Finsupp.instSMulPosStrictMono {ι : Type u_1} {α : Type u_2} {β : Type u_3} [Zero α] [] [Zero β] [] [] [] :
Equations
• =
instance Finsupp.instSMulPosReflectLT {ι : Type u_1} {α : Type u_2} {β : Type u_3} [Zero α] [] [Zero β] [] [] [] :
Equations
• =
instance Finsupp.orderBot {ι : Type u_1} {α : Type u_2} :
Equations
• Finsupp.orderBot =
theorem Finsupp.bot_eq_zero {ι : Type u_1} {α : Type u_2} :
= 0
@[simp]
theorem Finsupp.add_eq_zero_iff {ι : Type u_1} {α : Type u_2} (f : ι →₀ α) (g : ι →₀ α) :
f + g = 0 f = 0 g = 0
theorem Finsupp.le_iff' {ι : Type u_1} {α : Type u_2} (f : ι →₀ α) (g : ι →₀ α) {s : } (hf : f.support s) :
f g is, f i g i
theorem Finsupp.le_iff {ι : Type u_1} {α : Type u_2} (f : ι →₀ α) (g : ι →₀ α) :
f g if.support, f i g i
theorem Finsupp.support_monotone {ι : Type u_1} {α : Type u_2} :
Monotone Finsupp.support
theorem Finsupp.support_mono {ι : Type u_1} {α : Type u_2} {f : ι →₀ α} {g : ι →₀ α} (hfg : f g) :
f.support g.support
instance Finsupp.decidableLE {ι : Type u_1} {α : Type u_2} [DecidableRel LE.le] :
Equations
instance Finsupp.decidableLT {ι : Type u_1} {α : Type u_2} [DecidableRel LE.le] :
Equations
• Finsupp.decidableLT = decidableLTOfDecidableLE
@[simp]
theorem Finsupp.single_le_iff {ι : Type u_1} {α : Type u_2} {i : ι} {x : α} {f : ι →₀ α} :
f x f i
instance Finsupp.tsub {ι : Type u_1} {α : Type u_2} [Sub α] [] :
Sub (ι →₀ α)

This is called tsub for truncated subtraction, to distinguish it with subtraction in an additive group.

Equations
instance Finsupp.orderedSub {ι : Type u_1} {α : Type u_2} [Sub α] [] :
Equations
• =
instance Finsupp.instCanonicallyOrderedAddCommMonoid {ι : Type u_1} {α : Type u_2} [Sub α] [] :
Equations
@[simp]
theorem Finsupp.coe_tsub {ι : Type u_1} {α : Type u_2} [Sub α] [] (f : ι →₀ α) (g : ι →₀ α) :
(f - g) = f - g
theorem Finsupp.tsub_apply {ι : Type u_1} {α : Type u_2} [Sub α] [] (f : ι →₀ α) (g : ι →₀ α) (a : ι) :
(f - g) a = f a - g a
@[simp]
theorem Finsupp.single_tsub {ι : Type u_1} {α : Type u_2} [Sub α] [] {i : ι} {a : α} {b : α} :
Finsupp.single i (a - b) = -
theorem Finsupp.support_tsub {ι : Type u_1} {α : Type u_2} [Sub α] [] {f1 : ι →₀ α} {f2 : ι →₀ α} :
(f1 - f2).support f1.support
theorem Finsupp.subset_support_tsub {ι : Type u_1} {α : Type u_2} [Sub α] [] [] {f1 : ι →₀ α} {f2 : ι →₀ α} :
f1.support \ f2.support (f1 - f2).support
@[simp]
theorem Finsupp.support_inf {ι : Type u_1} {α : Type u_2} [] (f : ι →₀ α) (g : ι →₀ α) :
(f g).support = f.support g.support
@[simp]
theorem Finsupp.support_sup {ι : Type u_1} {α : Type u_2} [] (f : ι →₀ α) (g : ι →₀ α) :
(f g).support = f.support g.support
theorem Finsupp.disjoint_iff {ι : Type u_1} {α : Type u_2} {f : ι →₀ α} {g : ι →₀ α} :
Disjoint f g Disjoint f.support g.support

Some lemmas about ℕ#

theorem Finsupp.sub_single_one_add {ι : Type u_1} {a : ι} {u : ι →₀ } {u' : ι →₀ } (h : u a 0) :
u - + u' = u + u' -
theorem Finsupp.add_sub_single_one {ι : Type u_1} {a : ι} {u : ι →₀ } {u' : ι →₀ } (h : u' a 0) :
u + (u' - ) = u + u' -