Documentation

Mathlib.Algebra.Order.Group.PiLex

Lexicographic product of algebraic order structures #

This file proves that the lexicographic order on pi types is compatible with the pointwise algebraic operations.

source
instance Pi.Lex.orderedCancelCommMonoid {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → OrderedCancelCommMonoid (α i)] :
OrderedCancelCommMonoid (Lex ((i : ι) → α i))
Equations
source
theorem Pi.Lex.orderedAddCancelCommMonoid.proof_2 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → OrderedCancelAddCommMonoid (α i)] :
∀ (x x_1 x_2 : Lex ((i : ι) → α i)), x + x_1 ≤ x + x_2 → x_1 ≤ x_2
source
theorem Pi.Lex.orderedAddCancelCommMonoid.proof_1 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → OrderedCancelAddCommMonoid (α i)] :
∀ (x x_1 : Lex ((i : ι) → α i)), x ≤ x_1 → ∀ (z : Lex ((i : ι) → α i)), z + x ≤ z + x_1
source
instance Pi.Lex.orderedAddCancelCommMonoid {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → OrderedCancelAddCommMonoid (α i)] :
OrderedCancelAddCommMonoid (Lex ((i : ι) → α i))
Equations
source
instance Pi.Lex.orderedCommGroup {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → OrderedCommGroup (α i)] :
OrderedCommGroup (Lex ((i : ι) → α i))
Equations
source
instance Pi.Lex.orderedAddCommGroup {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → OrderedAddCommGroup (α i)] :
OrderedAddCommGroup (Lex ((i : ι) → α i))
Equations
source
theorem Pi.Lex.orderedAddCommGroup.proof_1 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → OrderedAddCommGroup (α i)] :
∀ (x x_1 : Lex ((i : ι) → α i)), x ≤ x_1 → ∀ (a : Lex ((i : ι) → α i)), a + x ≤ a + x_1
source
noncomputable instance Pi.Lex.linearOrderedCancelCommMonoid {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedCancelCommMonoid (α i)] :
LinearOrderedCancelCommMonoid (Lex ((i : ι) → α i))
Equations
source
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_6 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedCancelAddCommMonoid (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
compare a b = compareOfLessAndEq a b
source
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_1 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → LinearOrderedCancelAddCommMonoid (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
a ≤ b → ∀ (c : Lex ((i : ι) → α i)), c + a ≤ c + b
source
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_5 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedCancelAddCommMonoid (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
max a b = if a ≤ b then b else a
source
noncomputable instance Pi.Lex.linearOrderedAddCancelCommMonoid {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedCancelAddCommMonoid (α i)] :
LinearOrderedCancelAddCommMonoid (Lex ((i : ι) → α i))
Equations
source
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_3 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedCancelAddCommMonoid (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
a ≤ b ∨ b ≤ a
source
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_4 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedCancelAddCommMonoid (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
min a b = if a ≤ b then a else b
source
theorem Pi.Lex.linearOrderedAddCancelCommMonoid.proof_2 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [(i : ι) → LinearOrderedCancelAddCommMonoid (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) (c : Lex ((i : ι) → α i)) :
a + b ≤ a + c → b ≤ c
source
noncomputable instance Pi.Lex.linearOrderedCommGroup {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedCommGroup (α i)] :
LinearOrderedCommGroup (Lex ((i : ι) → α i))
Equations
  • Pi.Lex.linearOrderedCommGroup = LinearOrderedCommGroup.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯
source
theorem Pi.Lex.linearOrderedAddCommGroup.proof_5 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedAddCommGroup (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
compare a b = compareOfLessAndEq a b
source
theorem Pi.Lex.linearOrderedAddCommGroup.proof_1 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedAddCommGroup (α i)] :
∀ (x x_1 : Lex ((i : ι) → α i)), x ≤ x_1 → ∀ (a : Lex ((i : ι) → α i)), a + x ≤ a + x_1
source
noncomputable instance Pi.Lex.linearOrderedAddCommGroup {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedAddCommGroup (α i)] :
LinearOrderedAddCommGroup (Lex ((i : ι) → α i))
Equations
  • Pi.Lex.linearOrderedAddCommGroup = LinearOrderedAddCommGroup.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯
source
theorem Pi.Lex.linearOrderedAddCommGroup.proof_2 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedAddCommGroup (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
a ≤ b ∨ b ≤ a
source
theorem Pi.Lex.linearOrderedAddCommGroup.proof_3 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedAddCommGroup (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
min a b = if a ≤ b then a else b
source
theorem Pi.Lex.linearOrderedAddCommGroup.proof_4 {ι : Type u_1} {α : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → LinearOrderedAddCommGroup (α i)] (a : Lex ((i : ι) → α i)) (b : Lex ((i : ι) → α i)) :
max a b = if a ≤ b then b else a