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Mathlib.Data.Finsupp.Lex

Lexicographic order on finitely supported functions #

This file defines the lexicographic order on Finsupp.

def Finsupp.Lex {α : Type u_1} {N : Type u_2} [Zero N] (r : ααProp) (s : NNProp) (x : α →₀ N) (y : α →₀ N) :

Finsupp.Lex r s is the lexicographic relation on α →₀ N, where α is ordered by r, and N is ordered by s.

The type synonym Lex (α →₀ N) has an order given by Finsupp.Lex (· < ·) (· < ·).

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Instances For
    theorem Pi.lex_eq_finsupp_lex {α : Type u_1} {N : Type u_2} [Zero N] {r : ααProp} {s : NNProp} (a : α →₀ N) (b : α →₀ N) :
    Pi.Lex r (fun {i : α} => s) a b = Finsupp.Lex r s a b
    theorem Finsupp.lex_def {α : Type u_1} {N : Type u_2} [Zero N] {r : ααProp} {s : NNProp} {a : α →₀ N} {b : α →₀ N} :
    Finsupp.Lex r s a b ∃ (j : α), (∀ (d : α), r d ja d = b d) s (a j) (b j)
    theorem Finsupp.lex_eq_invImage_dfinsupp_lex {α : Type u_1} {N : Type u_2} [Zero N] (r : ααProp) (s : NNProp) :
    Finsupp.Lex r s = InvImage (DFinsupp.Lex r fun (x : α) => s) Finsupp.toDFinsupp
    instance Finsupp.instLTLex {α : Type u_1} {N : Type u_2} [Zero N] [LT α] [LT N] :
    LT (Lex (α →₀ N))
    Equations
    • Finsupp.instLTLex = { lt := fun (f g : Lex (α →₀ N)) => Finsupp.Lex (fun (x1 x2 : α) => x1 < x2) (fun (x1 x2 : N) => x1 < x2) (ofLex f) (ofLex g) }
    theorem Finsupp.lex_lt_of_lt_of_preorder {α : Type u_1} {N : Type u_2} [Zero N] [Preorder N] (r : ααProp) [IsStrictOrder α r] {x : α →₀ N} {y : α →₀ N} (hlt : x < y) :
    ∃ (i : α), (∀ (j : α), r j ix j y j y j x j) x i < y i
    theorem Finsupp.lex_lt_of_lt {α : Type u_1} {N : Type u_2} [Zero N] [PartialOrder N] (r : ααProp) [IsStrictOrder α r] {x : α →₀ N} {y : α →₀ N} (hlt : x < y) :
    Pi.Lex r (fun {i : α} (x1 x2 : N) => x1 < x2) x y
    instance Finsupp.Lex.isStrictOrder {α : Type u_1} {N : Type u_2} [Zero N] [LinearOrder α] [PartialOrder N] :
    IsStrictOrder (Lex (α →₀ N)) fun (x1 x2 : Lex (α →₀ N)) => x1 < x2
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    • =
    instance Finsupp.Lex.partialOrder {α : Type u_1} {N : Type u_2} [Zero N] [LinearOrder α] [PartialOrder N] :

    The partial order on Finsupps obtained by the lexicographic ordering. See Finsupp.Lex.linearOrder for a proof that this partial order is in fact linear.

    Equations
    instance Finsupp.Lex.linearOrder {α : Type u_1} {N : Type u_2} [Zero N] [LinearOrder α] [LinearOrder N] :

    The linear order on Finsupps obtained by the lexicographic ordering.

    Equations
    • Finsupp.Lex.linearOrder = LinearOrder.mk LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT
    theorem Finsupp.toLex_monotone {α : Type u_1} {N : Type u_2} [Zero N] [LinearOrder α] [PartialOrder N] :
    Monotone toLex
    theorem Finsupp.lt_of_forall_lt_of_lt {α : Type u_1} {N : Type u_2} [Zero N] [LinearOrder α] [PartialOrder N] (a : Lex (α →₀ N)) (b : Lex (α →₀ N)) (i : α) :
    (∀ j < i, (ofLex a) j = (ofLex b) j)(ofLex a) i < (ofLex b) ia < b

    We are about to sneak in a hypothesis that might appear to be too strong. We assume AddLeftStrictMono (covariant with strict inequality <) also when proving the one with the weak inequality . This is actually necessary: addition on Lex (α →₀ N) may fail to be monotone, when it is "just" monotone on N.

    See Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean for a counterexample.

    Equations
    • =
    instance Finsupp.Lex.addLeftMono {α : Type u_1} {N : Type u_2} [LinearOrder α] [AddMonoid N] [LinearOrder N] [AddLeftStrictMono N] :
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    • =
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    • =
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    • =
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    noncomputable instance Finsupp.Lex.orderedAddCommGroup {α : Type u_1} {N : Type u_2} [LinearOrder α] [OrderedAddCommGroup N] :
    Equations
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    Equations
    • Finsupp.Lex.linearOrderedAddCommGroup = LinearOrderedAddCommGroup.mk LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT