mathlib3 documentation

data.sigma.lex

Lexicographic order on a sigma type #

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This defines the lexicographical order of two arbitrary relations on a sigma type and proves some lemmas about psigma.lex, which is defined in core Lean.

Given a relation in the index type and a relation on each summand, the lexicographical order on the sigma type relates a and b if their summands are related or they are in the same summand and related by the summand's relation.

See also #

Related files are:

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inductive sigma.lex {ι : Type u_1} {α : ι → Type u_2} (r : ι → ι → Prop) (s : Π (i : ι), α i → α i → Prop) (a b : Σ (i : ι), α i) :
Prop

The lexicographical order on a sigma type. It takes in a relation on the index type and a relation for each summand. a is related to b iff their summands are related or they are in the same summand and are related through the summand's relation.

Instances for sigma.lex
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theorem sigma.lex_iff {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} {a b : Σ (i : ι), α i} :
sigma.lex r s a b ↔ r a.fst b.fst ∨ ∃ (h : a.fst = b.fst), s b.fst (eq.rec a.snd h) b.snd
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@[protected, instance]
def sigma.lex.decidable {ι : Type u_1} {α : ι → Type u_2} (r : ι → ι → Prop) (s : Π (i : ι), α i → α i → Prop) [decidable_eq ι] [decidable_rel r] [Π (i : ι), decidable_rel (s i)] :
decidable_rel (sigma.lex r s)
Equations
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theorem sigma.lex.mono {ι : Type u_1} {α : ι → Type u_2} {r₁ r₂ : ι → ι → Prop} {s₁ s₂ : Π (i : ι), α i → α i → Prop} (hr : ∀ (a b : ι), r₁ a b → r₂ a b) (hs : ∀ (i : ι) (a b : α i), s₁ i a b → s₂ i a b) {a b : Σ (i : ι), α i} (h : sigma.lex r₁ s₁ a b) :
sigma.lex r₂ s₂ a b
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theorem sigma.lex.mono_left {ι : Type u_1} {α : ι → Type u_2} {r₁ r₂ : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} (hr : ∀ (a b : ι), r₁ a b → r₂ a b) {a b : Σ (i : ι), α i} (h : sigma.lex r₁ s a b) :
sigma.lex r₂ s a b
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theorem sigma.lex.mono_right {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s₁ s₂ : Π (i : ι), α i → α i → Prop} (hs : ∀ (i : ι) (a b : α i), s₁ i a b → s₂ i a b) {a b : Σ (i : ι), α i} (h : sigma.lex r s₁ a b) :
sigma.lex r s₂ a b
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theorem sigma.lex_swap {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} {a b : Σ (i : ι), α i} :
sigma.lex (function.swap r) s a b ↔ sigma.lex r (λ (i : ι), function.swap (s i)) b a
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@[protected, instance]
def sigma.lex.is_refl {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} [∀ (i : ι), is_refl (α i) (s i)] :
is_refl (Σ (i : ι), α i) (sigma.lex r s)
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@[protected, instance]
def sigma.lex.is_irrefl {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} [is_irrefl ι r] [∀ (i : ι), is_irrefl (α i) (s i)] :
is_irrefl (Σ (i : ι), α i) (sigma.lex r s)
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@[protected, instance]
def sigma.lex.is_trans {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} [is_trans ι r] [∀ (i : ι), is_trans (α i) (s i)] :
is_trans (Σ (i : ι), α i) (sigma.lex r s)
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@[protected, instance]
def sigma.lex.is_symm {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} [is_symm ι r] [∀ (i : ι), is_symm (α i) (s i)] :
is_symm (Σ (i : ι), α i) (sigma.lex r s)
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@[protected, instance]
def sigma.lex.is_antisymm {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} [is_asymm ι r] [∀ (i : ι), is_antisymm (α i) (s i)] :
is_antisymm (Σ (i : ι), α i) (sigma.lex r s)
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@[protected, instance]
def sigma.lex.is_total {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} [is_trichotomous ι r] [∀ (i : ι), is_total (α i) (s i)] :
is_total (Σ (i : ι), α i) (sigma.lex r s)
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@[protected, instance]
def sigma.lex.is_trichotomous {ι : Type u_1} {α : ι → Type u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} [is_trichotomous ι r] [∀ (i : ι), is_trichotomous (α i) (s i)] :
is_trichotomous (Σ (i : ι), α i) (sigma.lex r s)

psigma #

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theorem psigma.lex_iff {ι : Sort u_1} {α : ι → Sort u_2} {r : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} {a b : Σ' (i : ι), α i} :
psigma.lex r s a b ↔ r a.fst b.fst ∨ ∃ (h : a.fst = b.fst), s b.fst (eq.rec a.snd h) b.snd
source
@[protected, instance]
def psigma.lex.decidable {ι : Sort u_1} {α : ι → Sort u_2} (r : ι → ι → Prop) (s : Π (i : ι), α i → α i → Prop) [decidable_eq ι] [decidable_rel r] [Π (i : ι), decidable_rel (s i)] :
decidable_rel (psigma.lex r s)
Equations
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theorem psigma.lex.mono {ι : Sort u_1} {α : ι → Sort u_2} {r₁ r₂ : ι → ι → Prop} {s₁ s₂ : Π (i : ι), α i → α i → Prop} (hr : ∀ (a b : ι), r₁ a b → r₂ a b) (hs : ∀ (i : ι) (a b : α i), s₁ i a b → s₂ i a b) {a b : Σ' (i : ι), α i} (h : psigma.lex r₁ s₁ a b) :
psigma.lex r₂ s₂ a b
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theorem psigma.lex.mono_left {ι : Sort u_1} {α : ι → Sort u_2} {r₁ r₂ : ι → ι → Prop} {s : Π (i : ι), α i → α i → Prop} (hr : ∀ (a b : ι), r₁ a b → r₂ a b) {a b : Σ' (i : ι), α i} (h : psigma.lex r₁ s a b) :
psigma.lex r₂ s a b
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theorem psigma.lex.mono_right {ι : Sort u_1} {α : ι → Sort u_2} {r : ι → ι → Prop} {s₁ s₂ : Π (i : ι), α i → α i → Prop} (hs : ∀ (i : ι) (a b : α i), s₁ i a b → s₂ i a b) {a b : Σ' (i : ι), α i} (h : psigma.lex r s₁ a b) :
psigma.lex r s₂ a b