Documentation

Mathlib.Algebra.Tropical.Basic

Tropical algebraic structures #

This file defines algebraic structures of the (min-)tropical numbers, up to the tropical semiring. Some basic lemmas about conversion from the base type R to Tropical R are provided, as well as the expected implementations of tropical addition and tropical multiplication.

Main declarations #

Tropical R: The type synonym of the tropical interpretation of R. If [LinearOrder R], then addition on R is via min.
Semiring (Tropical R): A LinearOrderedAddCommMonoidWithTop R induces a Semiring (Tropical R). If one solely has [LinearOrderedAddCommMonoid R], then the "tropicalization of R" would be Tropical (WithTop R).

Implementation notes #

The tropical structure relies on Top and min. For the max-tropical numbers, use OrderDual R.

Inspiration was drawn from the implementation of Additive/Multiplicative/Opposite, where a type synonym is created with some barebones API, and quickly made irreducible.

Algebraic structures are provided with as few typeclass assumptions as possible, even though most references rely on Semiring (Tropical R) for building up the whole theory.

References followed #

https://arxiv.org/pdf/math/0408099.pdf
https://www.mathenjeans.fr/sites/default/files/sujets/tropical_geometry_-_casagrande.pdf

@[irreducible]

def Tropical (R : Type u) :

The tropicalization of a type R.

Equations

Tropical R = R

Instances For

def Tropical.trop {R : Type u} :

R → Tropical R

Reinterpret x : R as an element of Tropical R. See Tropical.tropEquiv for the equivalence.

Equations

Tropical.trop = id

Instances For

def Tropical.untrop {R : Type u} :

Tropical R → R

Reinterpret x : Tropical R as an element of R. See Tropical.tropEquiv for the equivalence.

Equations

Tropical.untrop = id

Instances For

theorem Tropical.trop_injective {R : Type u} :

Function.Injective trop

theorem Tropical.untrop_injective {R : Type u} :

Function.Injective untrop

@[simp]

theorem Tropical.trop_inj_iff {R : Type u} (x y : R) :

trop x = trop y ↔ x = y

@[simp]

theorem Tropical.untrop_inj_iff {R : Type u} (x y : Tropical R) :

untrop x = untrop y ↔ x = y

@[simp]

theorem Tropical.trop_untrop {R : Type u} (x : Tropical R) :

trop (untrop x) = x

@[simp]

theorem Tropical.untrop_trop {R : Type u} (x : R) :

untrop (trop x) = x

theorem Tropical.leftInverse_trop {R : Type u} :

Function.LeftInverse trop untrop

theorem Tropical.rightInverse_trop {R : Type u} :

Function.RightInverse trop untrop

def Tropical.tropEquiv {R : Type u} :

R ≃ Tropical R

Reinterpret x : R as an element of Tropical R. See Tropical.tropOrderIso for the order-preserving equivalence.

Equations

Tropical.tropEquiv = { toFun := Tropical.trop, invFun := Tropical.untrop, left_inv := ⋯, right_inv := ⋯ }

Instances For

@[simp]

theorem Tropical.tropEquiv_coe_fn {R : Type u} :

⇑tropEquiv = trop

@[simp]

theorem Tropical.tropEquiv_symm_coe_fn {R : Type u} :

⇑tropEquiv.symm = untrop

theorem Tropical.trop_eq_iff_eq_untrop {R : Type u} {x : R} {y : Tropical R} :

trop x = y ↔ x = untrop y

theorem Tropical.untrop_eq_iff_eq_trop {R : Type u} {x : Tropical R} {y : R} :

untrop x = y ↔ x = trop y

theorem Tropical.injective_trop {R : Type u} :

Function.Injective trop

theorem Tropical.injective_untrop {R : Type u} :

Function.Injective untrop

theorem Tropical.surjective_trop {R : Type u} :

Function.Surjective trop

theorem Tropical.surjective_untrop {R : Type u} :

Function.Surjective untrop

instance Tropical.instInhabited {R : Type u} [Inhabited R] :

Inhabited (Tropical R)

Equations

Tropical.instInhabited = { default := Tropical.trop default }

def Tropical.tropRec {R : Type u} {F : Tropical R → Sort v} (h : (X : R) → F (trop X)) (X : Tropical R) :

F X

Recursing on an x' : Tropical R is the same as recursing on an x : R reinterpreted as a term of Tropical R via trop x.

Equations

Tropical.tropRec h X = h (Tropical.untrop X)

Instances For

instance Tropical.instDecidableEq {R : Type u} [DecidableEq R] :

DecidableEq (Tropical R)

Equations

x✝¹.instDecidableEq x✝ = decidable_of_iff (Tropical.untrop x✝¹ = Tropical.untrop x✝) ⋯

instance Tropical.instLETropical {R : Type u} [LE R] :

LE (Tropical R)

Equations

Tropical.instLETropical = { le := fun (x y : Tropical R) => Tropical.untrop x ≤ Tropical.untrop y }

@[simp]

theorem Tropical.untrop_le_iff {R : Type u} [LE R] {x y : Tropical R} :

untrop x ≤ untrop y ↔ x ≤ y

instance Tropical.decidableLE {R : Type u} [LE R] [DecidableLE R] :

DecidableLE (Tropical R)

Equations

x.decidableLE y = inst✝ (Tropical.untrop x) (Tropical.untrop y)

instance Tropical.instLTTropical {R : Type u} [LT R] :

LT (Tropical R)

Equations

Tropical.instLTTropical = { lt := fun (x y : Tropical R) => Tropical.untrop x < Tropical.untrop y }

@[simp]

theorem Tropical.untrop_lt_iff {R : Type u} [LT R] {x y : Tropical R} :

untrop x < untrop y ↔ x < y

instance Tropical.decidableLT {R : Type u} [LT R] [DecidableLT R] :

DecidableLT (Tropical R)

Equations

x.decidableLT y = inst✝ (Tropical.untrop x) (Tropical.untrop y)

instance Tropical.instPreorderTropical {R : Type u} [Preorder R] :

Preorder (Tropical R)

Equations

Tropical.instPreorderTropical = { toLE := Tropical.instLETropical, toLT := Tropical.instLTTropical, le_refl := ⋯, le_trans := ⋯, lt_iff_le_not_le := ⋯ }

def Tropical.tropOrderIso {R : Type u} [Preorder R] :

R ≃o Tropical R

Reinterpret x : R as an element of Tropical R, preserving the order.

Equations

Tropical.tropOrderIso = { toEquiv := Tropical.tropEquiv, map_rel_iff' := ⋯ }

Instances For

@[simp]

theorem Tropical.tropOrderIso_coe_fn {R : Type u} [Preorder R] :

⇑tropOrderIso = trop

@[simp]

theorem Tropical.tropOrderIso_symm_coe_fn {R : Type u} [Preorder R] :

⇑tropOrderIso.symm = untrop

theorem Tropical.trop_monotone {R : Type u} [Preorder R] :

theorem Tropical.untrop_monotone {R : Type u} [Preorder R] :

Monotone untrop

instance Tropical.instPartialOrderTropical {R : Type u} [PartialOrder R] :

PartialOrder (Tropical R)

Equations

Tropical.instPartialOrderTropical = { toPreorder := Tropical.instPreorderTropical, le_antisymm := ⋯ }

instance Tropical.instZeroTropical {R : Type u} [Top R] :

Zero (Tropical R)

Equations

Tropical.instZeroTropical = { zero := Tropical.trop ⊤ }

instance Tropical.instTopTropical {R : Type u} [Top R] :

Top (Tropical R)

Equations

Tropical.instTopTropical = { top := 0 }

@[simp]

theorem Tropical.untrop_zero {R : Type u} [Top R] :

@[simp]

theorem Tropical.trop_top {R : Type u} [Top R] :

@[simp]

theorem Tropical.trop_coe_ne_zero {R : Type u} (x : R) :

trop ↑x ≠ 0

@[simp]

theorem Tropical.zero_ne_trop_coe {R : Type u} (x : R) :

0 ≠ trop ↑x

@[simp]

theorem Tropical.le_zero {R : Type u} [LE R] [OrderTop R] (x : Tropical R) :

x ≤ 0

instance Tropical.instOrderTop {R : Type u} [LE R] [OrderTop R] :

OrderTop (Tropical R)

Equations

Tropical.instOrderTop = { toTop := Tropical.instTopTropical, le_top := ⋯ }

instance Tropical.instAdd {R : Type u} [LinearOrder R] :

Add (Tropical R)

Tropical addition is the minimum of two underlying elements of R.

Equations

Tropical.instAdd = { add := fun (x y : Tropical R) => Tropical.trop (min (Tropical.untrop x) (Tropical.untrop y)) }

instance Tropical.instAddCommSemigroupTropical {R : Type u} [LinearOrder R] :

AddCommSemigroup (Tropical R)

Equations

Tropical.instAddCommSemigroupTropical = { add := fun (x1 x2 : Tropical R) => x1 + x2, add_assoc := ⋯, add_comm := ⋯ }

@[simp]

theorem Tropical.untrop_add {R : Type u} [LinearOrder R] (x y : Tropical R) :

untrop (x + y) = min (untrop x) (untrop y)

@[simp]

theorem Tropical.trop_min {R : Type u} [LinearOrder R] (x y : R) :

trop (min x y) = trop x + trop y

@[simp]

theorem Tropical.trop_inf {R : Type u} [LinearOrder R] (x y : R) :

trop (min x y) = trop x + trop y

theorem Tropical.trop_add_def {R : Type u} [LinearOrder R] (x y : Tropical R) :

x + y = trop (min (untrop x) (untrop y))

instance Tropical.instLinearOrderTropical {R : Type u} [LinearOrder R] :

LinearOrder (Tropical R)

Equations

One or more equations did not get rendered due to their size.

@[simp]

theorem Tropical.untrop_sup {R : Type u} [LinearOrder R] (x y : Tropical R) :

untrop (max x y) = max (untrop x) (untrop y)

@[simp]

theorem Tropical.untrop_max {R : Type u} [LinearOrder R] (x y : Tropical R) :

untrop (max x y) = max (untrop x) (untrop y)

@[simp]

theorem Tropical.min_eq_add {R : Type u} [LinearOrder R] :

min = fun (x1 x2 : Tropical R) => x1 + x2

@[simp]

theorem Tropical.inf_eq_add {R : Type u} [LinearOrder R] :

(fun (x1 x2 : Tropical R) => min x1 x2) = fun (x1 x2 : Tropical R) => x1 + x2

theorem Tropical.trop_max_def {R : Type u} [LinearOrder R] (x y : Tropical R) :

max x y = trop (max (untrop x) (untrop y))

theorem Tropical.trop_sup_def {R : Type u} [LinearOrder R] (x y : Tropical R) :

max x y = trop (max (untrop x) (untrop y))

@[simp]

theorem Tropical.add_eq_left {R : Type u} [LinearOrder R] ⦃x y : Tropical R⦄ (h : x ≤ y) :

x + y = x

@[simp]

theorem Tropical.add_eq_right {R : Type u} [LinearOrder R] ⦃x y : Tropical R⦄ (h : y ≤ x) :

x + y = y

theorem Tropical.add_eq_left_iff {R : Type u} [LinearOrder R] {x y : Tropical R} :

x + y = x ↔ x ≤ y

theorem Tropical.add_eq_right_iff {R : Type u} [LinearOrder R] {x y : Tropical R} :

x + y = y ↔ y ≤ x

theorem Tropical.add_self {R : Type u} [LinearOrder R] (x : Tropical R) :

x + x = x

theorem Tropical.add_eq_iff {R : Type u} [LinearOrder R] {x y z : Tropical R} :

x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x

@[simp]

theorem Tropical.add_eq_zero_iff {R : Type u} [LinearOrder R] {a b : Tropical (WithTop R)} :

a + b = 0 ↔ a = 0 ∧ b = 0

instance Tropical.instAddCommMonoidTropical {R : Type u} [LinearOrder R] [OrderTop R] :

AddCommMonoid (Tropical R)

Equations

One or more equations did not get rendered due to their size.

instance Tropical.instMulOfAdd {R : Type u} [Add R] :

Mul (Tropical R)

Tropical multiplication is the addition in the underlying R.

Equations

Tropical.instMulOfAdd = { mul := fun (x y : Tropical R) => Tropical.trop (Tropical.untrop x + Tropical.untrop y) }

@[simp]

theorem Tropical.trop_add {R : Type u} [Add R] (x y : R) :

trop (x + y) = trop x * trop y

@[simp]

theorem Tropical.untrop_mul {R : Type u} [Add R] (x y : Tropical R) :

untrop (x * y) = untrop x + untrop y

theorem Tropical.trop_mul_def {R : Type u} [Add R] (x y : Tropical R) :

x * y = trop (untrop x + untrop y)

instance Tropical.instOneTropical {R : Type u} [Zero R] :

One (Tropical R)

Equations

Tropical.instOneTropical = { one := Tropical.trop 0 }

@[simp]

theorem Tropical.trop_zero {R : Type u} [Zero R] :

trop 0 = 1

@[simp]

theorem Tropical.untrop_one {R : Type u} [Zero R] :

instance Tropical.instAddMonoidWithOneTropical {R : Type u} [LinearOrder R] [OrderTop R] [Zero R] :

AddMonoidWithOne (Tropical R)

Equations

One or more equations did not get rendered due to their size.

instance Tropical.instNontrivialWithTopOfZero {R : Type u} [Zero R] :

Nontrivial (Tropical (WithTop R))

instance Tropical.instInvOfNeg {R : Type u} [Neg R] :

Inv (Tropical R)

Equations

Tropical.instInvOfNeg = { inv := fun (x : Tropical R) => Tropical.trop (-Tropical.untrop x) }

@[simp]

theorem Tropical.untrop_inv {R : Type u} [Neg R] (x : Tropical R) :

untrop x⁻¹ = -untrop x

instance Tropical.instDivOfSub {R : Type u} [Sub R] :

Div (Tropical R)

Equations

Tropical.instDivOfSub = { div := fun (x y : Tropical R) => Tropical.trop (Tropical.untrop x - Tropical.untrop y) }

@[simp]

theorem Tropical.untrop_div {R : Type u} [Sub R] (x y : Tropical R) :

untrop (x / y) = untrop x - untrop y

instance Tropical.instSemigroupTropical {R : Type u} [AddSemigroup R] :

Semigroup (Tropical R)

Equations

Tropical.instSemigroupTropical = { mul := fun (x1 x2 : Tropical R) => x1 * x2, mul_assoc := ⋯ }

instance Tropical.instCommSemigroupTropical {R : Type u} [AddCommSemigroup R] :

CommSemigroup (Tropical R)

Equations

Tropical.instCommSemigroupTropical = { toSemigroup := Tropical.instSemigroupTropical, mul_comm := ⋯ }

instance Tropical.instPowOfSMul {R : Type u} {α : Type u_1} [SMul α R] :

Pow (Tropical R) α

Equations

Tropical.instPowOfSMul = { pow := fun (x : Tropical R) (n : α) => Tropical.trop (n • Tropical.untrop x) }

@[simp]

theorem Tropical.untrop_pow {R : Type u} {α : Type u_1} [SMul α R] (x : Tropical R) (n : α) :

untrop (x ^ n) = n • untrop x

@[simp]

theorem Tropical.trop_smul {R : Type u} {α : Type u_1} [SMul α R] (x : R) (n : α) :

trop (n • x) = trop x ^ n

instance Tropical.instMulOneClassTropical {R : Type u} [AddZeroClass R] :

MulOneClass (Tropical R)

Equations

Tropical.instMulOneClassTropical = { one := 1, mul := fun (x1 x2 : Tropical R) => x1 * x2, one_mul := ⋯, mul_one := ⋯ }

instance Tropical.instMonoidTropical {R : Type u} [AddMonoid R] :

Monoid (Tropical R)

Equations

One or more equations did not get rendered due to their size.

@[simp]

theorem Tropical.trop_nsmul {R : Type u} [AddMonoid R] (x : R) (n : ℕ) :

trop (n • x) = trop x ^ n

instance Tropical.instCommMonoidTropical {R : Type u} [AddCommMonoid R] :

CommMonoid (Tropical R)

Equations

Tropical.instCommMonoidTropical = { toMonoid := Tropical.instMonoidTropical, mul_comm := ⋯ }

instance Tropical.instGroupTropical {R : Type u} [AddGroup R] :

Group (Tropical R)

Equations

One or more equations did not get rendered due to their size.

instance Tropical.instCommGroupOfAddCommGroup {R : Type u} [AddCommGroup R] :

CommGroup (Tropical R)

Equations

Tropical.instCommGroupOfAddCommGroup = { toGroup := Tropical.instGroupTropical, mul_comm := ⋯ }

@[simp]

theorem Tropical.untrop_zpow {R : Type u} [AddGroup R] (x : Tropical R) (n : ℤ) :

untrop (x ^ n) = n • untrop x

@[simp]

theorem Tropical.trop_zsmul {R : Type u} [AddGroup R] (x : R) (n : ℤ) :

trop (n • x) = trop x ^ n

instance Tropical.mulLeftMono {R : Type u} [LE R] [Add R] [AddLeftMono R] :

MulLeftMono (Tropical R)

instance Tropical.mulRightMono {R : Type u} [LE R] [Add R] [AddRightMono R] :

MulRightMono (Tropical R)

instance Tropical.addLeftMono {R : Type u} [LinearOrder R] :

AddLeftMono (Tropical R)

instance Tropical.mulLeftStrictMono {R : Type u} [LT R] [Add R] [AddLeftStrictMono R] :

MulLeftStrictMono (Tropical R)

instance Tropical.mulRightStrictMono {R : Type u} [Preorder R] [Add R] [AddRightStrictMono R] :

MulRightStrictMono (Tropical R)

instance Tropical.instDistribTropical {R : Type u} [LinearOrder R] [Add R] [AddLeftMono R] [AddRightMono R] :

Distrib (Tropical R)

Equations

Tropical.instDistribTropical = { mul := fun (x1 x2 : Tropical R) => x1 * x2, add := fun (x1 x2 : Tropical R) => x1 + x2, left_distrib := ⋯, right_distrib := ⋯ }

@[simp]

theorem Tropical.add_pow {R : Type u} [LinearOrder R] [AddMonoid R] [AddLeftMono R] [AddRightMono R] (x y : Tropical R) (n : ℕ) :

(x + y) ^ n = x ^ n + y ^ n

instance Tropical.instCommSemiring {R : Type u} [LinearOrderedAddCommMonoidWithTop R] :

CommSemiring (Tropical R)

Equations

One or more equations did not get rendered due to their size.

@[simp]

theorem Tropical.succ_nsmul {R : Type u_1} [LinearOrder R] [OrderTop R] (x : Tropical R) (n : ℕ) :

(n + 1) • x = x

theorem Tropical.mul_eq_zero_iff {R : Type u_1} [AddCommMonoid R] {a b : Tropical (WithTop R)} :

a * b = 0 ↔ a = 0 ∨ b = 0

instance Tropical.instNoZeroDivisorsWithTop {R : Type u_1} [AddCommMonoid R] :

NoZeroDivisors (Tropical (WithTop R))