# Documentation

Mathlib.Data.Pi.Lex

# Lexicographic order on Pi types #

This file defines the lexicographic order for Pi types. a is less than b if a i = b i for all i up to some point k, and a k < b k.

## Notation #

• Πₗ i, α i: Pi type equipped with the lexicographic order. Type synonym of Π i, α i.

Related files are:

• Data.Finset.Colex: Colexicographic order on finite sets.
• Data.List.Lex: Lexicographic order on lists.
• Data.Sigma.Order: Lexicographic order on Σₗ i, α i.
• Data.PSigma.Order: Lexicographic order on Σₗ' i, α i.
• Data.Prod.Lex: Lexicographic order on α × β.
def Pi.Lex {ι : Type u_1} {β : ιType u_2} (r : ιιProp) (s : {i : ι} → β iβ iProp) (x : (i : ι) → β i) (y : (i : ι) → β i) :

The lexicographic relation on Π i : ι, β i, where ι is ordered by r, and each β i is ordered by s.

Instances For

The notation Πₗ i, α i refers to a pi type equipped with the lexicographic order.

Instances For
@[simp]
theorem Pi.toLex_apply {ι : Type u_1} {β : ιType u_2} (x : (i : ι) → β i) (i : ι) :
toLex x i = x i
@[simp]
theorem Pi.ofLex_apply {ι : Type u_1} {β : ιType u_2} (x : Lex ((i : ι) → β i)) (i : ι) :
ofLex x i = x i
theorem Pi.lex_lt_of_lt_of_preorder {ι : Type u_1} {β : ιType u_2} [(i : ι) → Preorder (β i)] {r : ιιProp} (hwf : ) {x : (i : ι) → β i} {y : (i : ι) → β i} (hlt : x < y) :
i, (∀ (j : ι), r j ix j y j y j x j) x i < y i
theorem Pi.lex_lt_of_lt {ι : Type u_1} {β : ιType u_2} [(i : ι) → PartialOrder (β i)] {r : ιιProp} (hwf : ) {x : (i : ι) → β i} {y : (i : ι) → β i} (hlt : x < y) :
Pi.Lex r (fun i x x_1 => x < x_1) x y
theorem Pi.isTrichotomous_lex {ι : Type u_1} {β : ιType u_2} (r : ιιProp) (s : {i : ι} → β iβ iProp) [∀ (i : ι), IsTrichotomous (β i) (s i)] (wf : ) :
IsTrichotomous ((i : ι) → β i) (Pi.Lex r s)
instance Pi.instLTLexForAll {ι : Type u_1} {β : ιType u_2} [LT ι] [(a : ι) → LT (β a)] :
LT (Lex ((i : ι) → β i))
instance Pi.Lex.isStrictOrder {ι : Type u_1} {β : ιType u_2} [] [(a : ι) → PartialOrder (β a)] :
IsStrictOrder (Lex ((i : ι) → β i)) fun x x_1 => x < x_1
instance Pi.instPartialOrderLexForAll {ι : Type u_1} {β : ιType u_2} [] [(a : ι) → PartialOrder (β a)] :
PartialOrder (Lex ((i : ι) → β i))
noncomputable instance Pi.instLinearOrderLexForAll {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(a : ι) → LinearOrder (β a)] :
LinearOrder (Lex ((i : ι) → β i))

Πₗ i, α i is a linear order if the original order is well-founded.

theorem Pi.toLex_monotone {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] :
Monotone toLex
theorem Pi.toLex_strictMono {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] :
StrictMono toLex
@[simp]
theorem Pi.lt_toLex_update_self_iff {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} :
toLex x < toLex () x i < a
@[simp]
theorem Pi.toLex_update_lt_self_iff {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} :
toLex () < toLex x a < x i
@[simp]
theorem Pi.le_toLex_update_self_iff {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} :
toLex x toLex () x i a
@[simp]
theorem Pi.toLex_update_le_self_iff {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} :
toLex () toLex x a x i
instance Pi.instOrderBotLexForAllToLEToPreorderInstPartialOrderLexForAll {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(a : ι) → PartialOrder (β a)] [(a : ι) → OrderBot (β a)] :
OrderBot (Lex ((a : ι) → β a))
instance Pi.instOrderTopLexForAllToLEToPreorderInstPartialOrderLexForAll {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(a : ι) → PartialOrder (β a)] [(a : ι) → OrderTop (β a)] :
OrderTop (Lex ((a : ι) → β a))
instance Pi.instBoundedOrderLexForAllToLEToPreorderInstPartialOrderLexForAll {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(a : ι) → PartialOrder (β a)] [(a : ι) → BoundedOrder (β a)] :
BoundedOrder (Lex ((a : ι) → β a))
instance Pi.instDenselyOrderedLexForAllInstLTLexForAllToLT {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → LT (β i)] [∀ (i : ι), DenselyOrdered (β i)] :
DenselyOrdered (Lex ((i : ι) → β i))
theorem Pi.Lex.noMaxOrder' {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → LT (β i)] (i : ι) [NoMaxOrder (β i)] :
NoMaxOrder (Lex ((i : ι) → β i))
instance Pi.instNoMaxOrderLexForAllInstLTLexForAllToLTToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLTToPreorder {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [] [(i : ι) → PartialOrder (β i)] [∀ (i : ι), NoMaxOrder (β i)] :
NoMaxOrder (Lex ((i : ι) → β i))
instance Pi.instNoMinOrderLexForAllInstLTLexForAllToLTToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLTToPreorder {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [] [(i : ι) → PartialOrder (β i)] [∀ (i : ι), NoMinOrder (β i)] :
NoMinOrder (Lex ((i : ι) → β i))
instance Pi.Lex.orderedAddCancelCommMonoid {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] :
OrderedCancelAddCommMonoid (Lex ((i : ι) → β i))
abbrev Pi.Lex.orderedAddCancelCommMonoid.match_1 {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] :
(x x_1 : Lex ((i : ι) → β i)) → (motive : x < x_1Prop) → ∀ (x_2 : x < x_1), (∀ (i : ι) (hi : (∀ (j : ι), j < ix j = x_1 j) (fun x x_3 x_4 => x_3 < x_4) i (x i) (x_1 i)), motive (_ : i, (∀ (j : ι), j < ix j = x_1 j) (fun x x_3 x_4 => x_3 < x_4) i (x i) (x_1 i))) → motive x_2
Instances For
theorem Pi.Lex.orderedAddCancelCommMonoid.proof_2 {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] :
∀ (x x_1 x_2 : Lex ((i : ι) → β i)), x + x_1 x + x_2x_1 x_2
abbrev Pi.Lex.orderedAddCancelCommMonoid.match_2 {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] :
(x x_1 x_2 : Lex ((i : ι) → β i)) → (motive : x + x_1 < x + x_2Prop) → ∀ (x_3 : x + x_1 < x + x_2), (∀ (i : ι) (hi : (∀ (j : ι), j < i(Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_1 j = (Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_2 j) (fun x x_4 x_5 => x_4 < x_5) i ((Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_1 i) ((Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_2 i)), motive (_ : i, (∀ (j : ι), j < i(Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_1 j = (Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_2 j) (fun x x_4 x_5 => x_4 < x_5) i ((Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_1 i) ((Lex ((i : ι) → β i) + Lex ((i : ι) → β i)) (Lex ((i : ι) → β i)) instHAdd x x_2 i))) → motive x_3
Instances For
theorem Pi.Lex.orderedAddCancelCommMonoid.proof_1 {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] :
∀ (x x_1 : Lex ((i : ι) → β i)), x x_1∀ (z : Lex ((i : ι) → β i)), z + x z + x_1
instance Pi.Lex.orderedCancelCommMonoid {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] :
OrderedCancelCommMonoid (Lex ((i : ι) → β i))
theorem Pi.Lex.orderedAddCommGroup.proof_1 {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → OrderedAddCommGroup (β i)] :
∀ (x x_1 : Lex ((i : ι) → β i)), x x_1∀ (a : Lex ((i : ι) → β i)), a + x a + x_1
instance Pi.Lex.orderedAddCommGroup {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → OrderedAddCommGroup (β i)] :
OrderedAddCommGroup (Lex ((i : ι) → β i))
instance Pi.Lex.orderedCommGroup {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → OrderedCommGroup (β i)] :
OrderedCommGroup (Lex ((i : ι) → β i))
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_5 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
max a b = if a b then b else a
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_4 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
min a b = if a b then a else b
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_6 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
compare a b =
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_2 {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) (c : Lex ((i : ι) → β i)) :
a + b a + cb c
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_1 {ι : Type u_1} {β : ιType u_2} [] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
a b∀ (c : Lex ((i : ι) → β i)), c + a c + b
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_3 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
a b b a
noncomputable instance Pi.Lex.linearOrderedAddCancelCommMonoid {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] :
noncomputable instance Pi.Lex.linearOrderedCancelCommMonoid {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] :
LinearOrderedCancelCommMonoid (Lex ((i : ι) → β i))
theorem Pi.Lex.linearOrderedAddCommGroup.proof_1 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] :
∀ (x x_1 : Lex ((i : ι) → β i)), x x_1∀ (a : Lex ((i : ι) → β i)), a + x a + x_1
theorem Pi.Lex.linearOrderedAddCommGroup.proof_3 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
min a b = if a b then a else b
theorem Pi.Lex.linearOrderedAddCommGroup.proof_4 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
max a b = if a b then b else a
noncomputable instance Pi.Lex.linearOrderedAddCommGroup {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] :
LinearOrderedAddCommGroup (Lex ((i : ι) → β i))
theorem Pi.Lex.linearOrderedAddCommGroup.proof_5 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
compare a b =
theorem Pi.Lex.linearOrderedAddCommGroup.proof_2 {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] (a : Lex ((i : ι) → β i)) (b : Lex ((i : ι) → β i)) :
a b b a
noncomputable instance Pi.Lex.linearOrderedCommGroup {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → ] :
LinearOrderedCommGroup (Lex ((i : ι) → β i))
theorem Pi.lex_desc {ι : Type u_1} {α : Type u_3} [] [] [] {f : ια} {i : ι} {j : ι} (h₁ : i j) (h₂ : f j < f i) :
toLex (f ↑()) < toLex f

If we swap two strictly decreasing values in a function, then the result is lexicographically smaller than the original function.