Documentation

Mathlib.Algebra.Tropical.Basic

Tropical algebraic structures #

This file defines algebraic structures of the min/max-tropical numbers, up to the tropical semiring. All declarations about MinTropical are translated to MaxTropical using to_dual. Some basic lemmas about conversion from the base type R to MinTropical R/MaxTropical R are provided, as well as the expected implementations of tropical addition and tropical multiplication.

Main declarations #

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

Implementation notes #

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 (MinTropical 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 MinTropical (R : Type u) :

The min-tropicalization of a type R.

Instances For

@[irreducible]

def MaxTropical (R : Type u) :

The max-tropicalization of a type R.

Instances For

@[deprecated MinTropical (since := "2026-07-24")]

def Tropical (R : Type u) :

The min-tropicalization of a type R.

Instances For

def MinTropical.trop {R : Type u} :

R → MinTropical R

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

Instances For

def MaxTropical.trop {R : Type u} :

R → MaxTropical R

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

Instances For

def MinTropical.untrop {R : Type u} :

MinTropical R → R

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

Instances For

def MaxTropical.untrop {R : Type u} :

MaxTropical R → R

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

Instances For

theorem MinTropical.trop_injective {R : Type u} :

Function.Injective trop

theorem MaxTropical.trop_injective {R : Type u} :

Function.Injective trop

theorem MinTropical.untrop_injective {R : Type u} :

Function.Injective untrop

theorem MaxTropical.untrop_injective {R : Type u} :

Function.Injective untrop

@[simp]

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

trop x = trop y ↔ x = y

@[simp]

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

trop x = trop y ↔ x = y

@[simp]

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

untrop x = untrop y ↔ x = y

@[simp]

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

untrop x = untrop y ↔ x = y

@[simp]

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

trop (untrop x) = x

@[simp]

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

trop (untrop x) = x

@[simp]

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

untrop (trop x) = x

@[simp]

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

untrop (trop x) = x

theorem MinTropical.leftInverse_trop {R : Type u} :

Function.LeftInverse trop untrop

theorem MaxTropical.leftInverse_trop {R : Type u} :

Function.LeftInverse trop untrop

theorem MinTropical.rightInverse_trop {R : Type u} :

Function.RightInverse trop untrop

theorem MaxTropical.rightInverse_trop {R : Type u} :

Function.RightInverse trop untrop

def MinTropical.tropEquiv {R : Type u} :

R ≃ MinTropical R

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

Instances For

def MaxTropical.tropEquiv {R : Type u} :

R ≃ MaxTropical R

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

Instances For

@[simp]

theorem MinTropical.tropEquiv_coe_fn {R : Type u} :

⇑tropEquiv = trop

@[simp]

theorem MaxTropical.tropEquiv_coe_fn {R : Type u} :

⇑tropEquiv = trop

@[simp]

theorem MinTropical.tropEquiv_symm_coe_fn {R : Type u} :

⇑tropEquiv.symm = untrop

@[simp]

theorem MaxTropical.tropEquiv_symm_coe_fn {R : Type u} :

⇑tropEquiv.symm = untrop

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

trop x = y ↔ x = untrop y

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

trop x = y ↔ x = untrop y

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

untrop x = y ↔ x = trop y

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

untrop x = y ↔ x = trop y

theorem MinTropical.injective_trop {R : Type u} :

Function.Injective trop

theorem MaxTropical.injective_trop {R : Type u} :

Function.Injective trop

theorem MinTropical.injective_untrop {R : Type u} :

Function.Injective untrop

theorem MaxTropical.injective_untrop {R : Type u} :

Function.Injective untrop

theorem MinTropical.surjective_trop {R : Type u} :

Function.Surjective trop

theorem MaxTropical.surjective_trop {R : Type u} :

Function.Surjective trop

theorem MinTropical.surjective_untrop {R : Type u} :

Function.Surjective untrop

theorem MaxTropical.surjective_untrop {R : Type u} :

Function.Surjective untrop

@[instance_reducible]

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

Inhabited (MinTropical R)

@[instance_reducible]

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

Inhabited (MaxTropical R)

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

F X

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

Instances For

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

F X

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

Instances For

@[instance_reducible]

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

DecidableEq (MinTropical R)

@[instance_reducible]

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

DecidableEq (MaxTropical R)

@[instance_reducible]

instance MinTropical.instLE {R : Type u} [LE R] :

LE (MinTropical R)

@[instance_reducible]

instance MaxTropical.instLE {R : Type u} [LE R] :

LE (MaxTropical R)

@[simp]

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

untrop x ≤ untrop y ↔ x ≤ y

@[simp]

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

untrop y ≤ untrop x ↔ y ≤ x

@[instance_reducible]

instance MinTropical.instDecidableLE {R : Type u} [LE R] [DecidableLE R] :

DecidableLE (MinTropical R)

@[instance_reducible]

instance MaxTropical.instDecidableLE {R : Type u} [LE R] [DecidableLE R] :

DecidableLE (MaxTropical R)

@[instance_reducible]

instance MinTropical.instLT {R : Type u} [LT R] :

LT (MinTropical R)

@[instance_reducible]

instance MaxTropical.instLT {R : Type u} [LT R] :

LT (MaxTropical R)

@[simp]

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

untrop x < untrop y ↔ x < y

@[simp]

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

untrop y < untrop x ↔ y < x

@[instance_reducible]

instance MinTropical.instDecidableLT {R : Type u} [LT R] [DecidableLT R] :

DecidableLT (MinTropical R)

@[instance_reducible]

instance MaxTropical.instDecidableLT {R : Type u} [LT R] [DecidableLT R] :

DecidableLT (MaxTropical R)

@[instance_reducible]

instance MinTropical.instPreorder {R : Type u} [Preorder R] :

Preorder (MinTropical R)

@[instance_reducible]

instance MaxTropical.instPreorder {R : Type u} [Preorder R] :

Preorder (MaxTropical R)

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

R ≃o MinTropical R

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

Instances For

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

R ≃o MaxTropical R

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

Instances For

@[simp]

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

⇑tropOrderIso = trop

@[simp]

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

⇑tropOrderIso = trop

@[simp]

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

⇑tropOrderIso.symm = untrop

@[simp]

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

⇑tropOrderIso.symm = untrop

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

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

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

Monotone untrop

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

Monotone untrop

@[instance_reducible]

instance MinTropical.instPartialOrder {R : Type u} [PartialOrder R] :

PartialOrder (MinTropical R)

@[instance_reducible]

instance MaxTropical.instPartialOrder {R : Type u} [PartialOrder R] :

PartialOrder (MaxTropical R)

@[instance_reducible]

instance MinTropical.instZeroOfTop {R : Type u} [Top R] :

Zero (MinTropical R)

@[instance_reducible]

instance MaxTropical.instZeroOfBot {R : Type u} [Bot R] :

Zero (MaxTropical R)

@[instance_reducible]

instance MinTropical.instTop {R : Type u} [Top R] :

Top (MinTropical R)

@[instance_reducible]

instance MaxTropical.instBot {R : Type u} [Bot R] :

Bot (MaxTropical R)

@[simp]

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

@[simp]

theorem MaxTropical.untrop_zero {R : Type u} [Bot R] :

@[simp]

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

@[simp]

theorem MaxTropical.trop_bot {R : Type u} [Bot R] :

@[simp]

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

trop ↑x ≠ 0

@[simp]

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

trop ↑x ≠ 0

@[simp]

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

0 ≠ trop ↑x

@[simp]

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

0 ≠ trop ↑x

@[simp]

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

x ≤ 0

@[simp]

theorem MaxTropical.le_zero {R : Type u} [LE R] [OrderBot R] (x : MaxTropical R) :

0 ≤ x

@[instance_reducible]

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

OrderTop (MinTropical R)

@[instance_reducible]

instance MaxTropical.instOrderBot {R : Type u} [LE R] [OrderBot R] :

OrderBot (MaxTropical R)

@[instance_reducible]

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

Add (MinTropical R)

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

@[instance_reducible]

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

Add (MaxTropical R)

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

@[instance_reducible]

instance MinTropical.instAddCommSemigroup {R : Type u} [LinearOrder R] :

AddCommSemigroup (MinTropical R)

@[instance_reducible]

instance MaxTropical.instAddCommSemigroup {R : Type u} [LinearOrder R] :

AddCommSemigroup (MaxTropical R)

@[simp]

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

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

@[simp]

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

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

@[simp]

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

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

@[simp]

theorem MaxTropical.trop_max {R : Type u} [LinearOrder R] (x y : R) :

trop (max x y) = trop x + trop y

@[simp]

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

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

@[simp]

theorem MaxTropical.trop_sup {R : Type u} [LinearOrder R] (x y : R) :

trop (max x y) = trop x + trop y

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

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

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

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

@[instance_reducible]

instance MinTropical.instLinearOrder {R : Type u} [LinearOrder R] :

LinearOrder (MinTropical R)

@[instance_reducible]

instance MaxTropical.instLinearOrder {R : Type u} [LinearOrder R] :

LinearOrder (MaxTropical R)

@[simp]

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

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

@[simp]

theorem MaxTropical.untrop_inf {R : Type u} [LinearOrder R] (x y : MaxTropical R) :

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

@[simp]

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

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

@[simp]

theorem MaxTropical.untrop_min {R : Type u} [LinearOrder R] (x y : MaxTropical R) :

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

@[simp]

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

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

@[simp]

theorem MaxTropical.max_eq_add {R : Type u} [LinearOrder R] :

max = fun (x1 x2 : MaxTropical R) => x1 + x2

@[simp]

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

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

@[simp]

theorem MaxTropical.sup_eq_add {R : Type u} [LinearOrder R] :

(fun (x1 x2 : MaxTropical R) => max x1 x2) = fun (x1 x2 : MaxTropical R) => x1 + x2

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

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

theorem MaxTropical.trop_min_def {R : Type u} [LinearOrder R] (x y : MaxTropical R) :

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

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

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

theorem MaxTropical.trop_inf_def {R : Type u} [LinearOrder R] (x y : MaxTropical R) :

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

@[simp]

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

x + y = x

@[simp]

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

x + y = x

@[simp]

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

x + y = y

@[simp]

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

x + y = y

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

x + y = x ↔ x ≤ y

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

x + y = x ↔ y ≤ x

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

x + y = y ↔ y ≤ x

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

x + y = y ↔ x ≤ y

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

x + x = x

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

x + x = x

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

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

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

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

@[simp]

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

a + b = 0 ↔ a = 0 ∧ b = 0

@[simp]

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

a + b = 0 ↔ a = 0 ∧ b = 0

@[instance_reducible]

instance MinTropical.instAddCommMonoidOfOrderTop {R : Type u} [LinearOrder R] [OrderTop R] :

AddCommMonoid (MinTropical R)

@[instance_reducible]

instance MaxTropical.instAddCommMonoidOfOrderBot {R : Type u} [LinearOrder R] [OrderBot R] :

AddCommMonoid (MaxTropical R)

@[instance_reducible]

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

Mul (MinTropical R)

Tropical multiplication is the addition in the underlying R.

@[instance_reducible]

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

Mul (MaxTropical R)

Tropical multiplication is the addition in the underlying R.

@[simp]

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

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

@[simp]

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

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

@[simp]

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

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

@[simp]

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

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

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

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

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

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

def MinTropical.equivMaxTropical {R : Type u} [LinearOrder R] [Add R] :

MinTropical R ≃+* MaxTropical Rᵒᵈ

The ring structure on MinTropical R is the same as on MaxTropical Rᵒᵈ.

Instances For

def MaxTropical.equivMinTropical {R : Type u} [LinearOrder R] [Add R] :

MaxTropical R ≃+* MinTropical Rᵒᵈ

The ring structure on MaxTropical R is the same as on MinTropical Rᵒᵈ.

Instances For

@[instance_reducible]

instance MinTropical.instOneOfZero {R : Type u} [Zero R] :

One (MinTropical R)

@[instance_reducible]

instance MaxTropical.instOneOfZero {R : Type u} [Zero R] :

One (MaxTropical R)

@[simp]

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

trop 0 = 1

@[simp]

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

trop 0 = 1

@[simp]

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

@[simp]

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

@[instance_reducible]

instance MinTropical.instAddMonoidWithOneOfOrderTopOfZero {R : Type u} [LinearOrder R] [OrderTop R] [Zero R] :

AddMonoidWithOne (MinTropical R)

@[instance_reducible]

instance MaxTropical.instAddMonoidWithOneOfOrderBotOfZero {R : Type u} [LinearOrder R] [OrderBot R] [Zero R] :

AddMonoidWithOne (MaxTropical R)

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

Nontrivial (MinTropical (WithTop R))

instance MaxTropical.instNontrivialWithBotOfZero {R : Type u} [Zero R] :

Nontrivial (MaxTropical (WithBot R))

@[instance_reducible]

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

Inv (MinTropical R)

@[instance_reducible]

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

Inv (MaxTropical R)

@[simp]

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

untrop x⁻¹ = -untrop x

@[simp]

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

untrop x⁻¹ = -untrop x

@[instance_reducible]

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

Div (MinTropical R)

@[instance_reducible]

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

Div (MaxTropical R)

@[simp]

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

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

@[simp]

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

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

@[instance_reducible]

instance MinTropical.instSemigroupOfAddSemigroup {R : Type u} [AddSemigroup R] :

Semigroup (MinTropical R)

@[instance_reducible]

instance MaxTropical.instSemigroupOfAddSemigroup {R : Type u} [AddSemigroup R] :

Semigroup (MaxTropical R)

@[instance_reducible]

instance MinTropical.instCommSemigroupOfAddCommSemigroup {R : Type u} [AddCommSemigroup R] :

CommSemigroup (MinTropical R)

@[instance_reducible]

instance MaxTropical.instCommSemigroupOfAddCommSemigroup {R : Type u} [AddCommSemigroup R] :

CommSemigroup (MaxTropical R)

@[instance_reducible]

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

Pow (MinTropical R) α

@[instance_reducible]

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

Pow (MaxTropical R) α

@[simp]

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

untrop (x ^ n) = n • untrop x

@[simp]

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

untrop (x ^ n) = n • untrop x

@[simp]

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

trop (n • x) = trop x ^ n

@[simp]

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

trop (n • x) = trop x ^ n

@[instance_reducible]

instance MinTropical.instMulOneClassOfAddZeroClass {R : Type u} [AddZeroClass R] :

MulOneClass (MinTropical R)

@[instance_reducible]

instance MaxTropical.instMulOneClassOfAddZeroClass {R : Type u} [AddZeroClass R] :

MulOneClass (MaxTropical R)

@[instance_reducible]

instance MinTropical.instMonoidOfAddMonoid {R : Type u} [AddMonoid R] :

Monoid (MinTropical R)

@[instance_reducible]

instance MaxTropical.instMonoidOfAddMonoid {R : Type u} [AddMonoid R] :

Monoid (MaxTropical R)

@[simp]

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

trop (n • x) = trop x ^ n

@[simp]

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

trop (n • x) = trop x ^ n

@[instance_reducible]

instance MinTropical.instCommMonoidOfAddCommMonoid {R : Type u} [AddCommMonoid R] :

CommMonoid (MinTropical R)

@[instance_reducible]

instance MaxTropical.instCommMonoidOfAddCommMonoid {R : Type u} [AddCommMonoid R] :

CommMonoid (MaxTropical R)

@[instance_reducible]

instance MinTropical.instGroupOfAddGroup {R : Type u} [AddGroup R] :

Group (MinTropical R)

@[instance_reducible]

instance MaxTropical.instGroupOfAddGroup {R : Type u} [AddGroup R] :

Group (MaxTropical R)

@[instance_reducible]

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

CommGroup (MinTropical R)

@[instance_reducible]

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

CommGroup (MaxTropical R)

@[simp]

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

untrop (x ^ n) = n • untrop x

@[simp]

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

untrop (x ^ n) = n • untrop x

@[simp]

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

trop (n • x) = trop x ^ n

@[simp]

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

trop (n • x) = trop x ^ n

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

MulLeftMono (MinTropical R)

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

MulRightMono (MinTropical R)

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

AddLeftMono (MinTropical R)

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

MulLeftStrictMono (MinTropical R)

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

MulRightStrictMono (MinTropical R)

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

MulLeftMono (MaxTropical R)

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

MulRightMono (MaxTropical R)

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

AddLeftMono (MaxTropical R)

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

MulLeftStrictMono (MaxTropical R)

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

MulRightStrictMono (MaxTropical R)

@[instance_reducible]

instance MinTropical.instDistribOfAddLeftMonoOfAddRightMono {R : Type u} [LinearOrder R] [Add R] [AddLeftMono R] [AddRightMono R] :

Distrib (MinTropical R)

@[instance_reducible]

instance MaxTropical.instDistribOfAddLeftMonoOfAddRightMono {R : Type u} [LinearOrder R] [Add R] [AddLeftMono R] [AddRightMono R] :

Distrib (MaxTropical R)

@[simp]

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

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

@[simp]

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

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

@[instance_reducible]

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

CommSemiring (MinTropical R)

@[simp]

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

(n + 1) • x = x

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

a * b = 0 ↔ a = 0 ∨ b = 0

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

NoZeroDivisors (MinTropical (WithTop R))