# 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 #

@[irreducible]
def Tropical (R : Type u) :

The tropicalization of a type R.

Equations
Instances For
def Tropical.trop {R : Type u} :
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} :
R

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

Equations
• Tropical.untrop = id
Instances For
@[simp]
theorem Tropical.trop_inj_iff {R : Type u} (x : R) (y : R) :
x = y
@[simp]
theorem Tropical.untrop_inj_iff {R : Type u} (x : ) (y : ) :
x.untrop = y.untrop x = y
@[simp]
theorem Tropical.trop_untrop {R : Type u} (x : ) :
Tropical.trop x.untrop = x
@[simp]
theorem Tropical.untrop_trop {R : Type u} (x : R) :
().untrop = x
theorem Tropical.leftInverse_trop {R : Type u} :
Function.LeftInverse Tropical.trop Tropical.untrop
theorem Tropical.rightInverse_trop {R : Type u} :
Function.RightInverse Tropical.trop Tropical.untrop
def Tropical.tropEquiv {R : Type u} :
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} :
Tropical.tropEquiv = Tropical.trop
@[simp]
theorem Tropical.tropEquiv_symm_coe_fn {R : Type u} :
Tropical.tropEquiv.symm = Tropical.untrop
theorem Tropical.trop_eq_iff_eq_untrop {R : Type u} {x : R} {y : } :
x = y.untrop
theorem Tropical.untrop_eq_iff_eq_trop {R : Type u} {x : } {y : R} :
x.untrop = y
instance Tropical.instInhabited {R : Type u} [] :
Equations
def Tropical.tropRec {R : Type u} {F : Sort v} (h : (X : R) → F ()) (X : ) :
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
• = h X.untrop
Instances For
instance Tropical.instDecidableEq {R : Type u} [] :
Equations
instance Tropical.instLETropical {R : Type u} [LE R] :
LE ()
Equations
• Tropical.instLETropical = { le := fun (x y : ) => x.untrop y.untrop }
@[simp]
theorem Tropical.untrop_le_iff {R : Type u} [LE R] {x : } {y : } :
x.untrop y.untrop x y
instance Tropical.decidableLE {R : Type u} [LE R] [DecidableRel fun (x x_1 : R) => x x_1] :
DecidableRel fun (x x_1 : ) => x x_1
Equations
• x.decidableLE y = inst x.untrop y.untrop
instance Tropical.instLTTropical {R : Type u} [LT R] :
LT ()
Equations
• Tropical.instLTTropical = { lt := fun (x y : ) => x.untrop < y.untrop }
@[simp]
theorem Tropical.untrop_lt_iff {R : Type u} [LT R] {x : } {y : } :
x.untrop < y.untrop x < y
instance Tropical.decidableLT {R : Type u} [LT R] [DecidableRel fun (x x_1 : R) => x < x_1] :
DecidableRel fun (x x_1 : ) => x < x_1
Equations
• x.decidableLT y = inst x.untrop y.untrop
Equations
• Tropical.instPreorderTropical = let __src := Tropical.instLETropical; let __src_1 := Tropical.instLTTropical; Preorder.mk
def Tropical.tropOrderIso {R : Type u} [] :
R ≃o

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

Equations
• Tropical.tropOrderIso = let __src := Tropical.tropEquiv; { toEquiv := __src, map_rel_iff' := }
Instances For
@[simp]
theorem Tropical.tropOrderIso_coe_fn {R : Type u} [] :
Tropical.tropOrderIso = Tropical.trop
@[simp]
theorem Tropical.tropOrderIso_symm_coe_fn {R : Type u} [] :
Tropical.tropOrderIso.symm = Tropical.untrop
theorem Tropical.trop_monotone {R : Type u} [] :
Monotone Tropical.trop
theorem Tropical.untrop_monotone {R : Type u} [] :
Monotone Tropical.untrop
Equations
• Tropical.instPartialOrderTropical = let __src := Tropical.instPreorderTropical;
instance Tropical.instZeroTropical {R : Type u} [Top R] :
Zero ()
Equations
• Tropical.instZeroTropical = { zero := }
instance Tropical.instTopTropical {R : Type u} [Top R] :
Top ()
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) :
0
@[simp]
theorem Tropical.zero_ne_trop_coe {R : Type u} (x : R) :
0
@[simp]
theorem Tropical.le_zero {R : Type u} [LE R] [] (x : ) :
x 0
instance Tropical.instOrderTop {R : Type u} [LE R] [] :
Equations
• Tropical.instOrderTop = let __src := Tropical.instTopTropical;
instance Tropical.instAdd {R : Type u} [] :

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

Equations
Equations
@[simp]
theorem Tropical.untrop_add {R : Type u} [] (x : ) (y : ) :
(x + y).untrop = min x.untrop y.untrop
@[simp]
theorem Tropical.trop_min {R : Type u} [] (x : R) (y : R) :
@[simp]
theorem Tropical.trop_inf {R : Type u} [] (x : R) (y : R) :
theorem Tropical.trop_add_def {R : Type u} [] (x : ) (y : ) :
x + y = Tropical.trop (min x.untrop y.untrop)
Equations
• Tropical.instLinearOrderTropical = let __src := Tropical.instPartialOrderTropical; LinearOrder.mk Tropical.decidableLE decidableEqOfDecidableLE decidableLTOfDecidableLE
@[simp]
theorem Tropical.untrop_sup {R : Type u} [] (x : ) (y : ) :
(x y).untrop = x.untrop y.untrop
@[simp]
theorem Tropical.untrop_max {R : Type u} [] (x : ) (y : ) :
(max x y).untrop = max x.untrop y.untrop
@[simp]
theorem Tropical.min_eq_add {R : Type u} [] :
min = fun (x x_1 : ) => x + x_1
@[simp]
theorem Tropical.inf_eq_add {R : Type u} [] :
(fun (x x_1 : ) => x x_1) = fun (x x_1 : ) => x + x_1
theorem Tropical.trop_max_def {R : Type u} [] (x : ) (y : ) :
max x y = Tropical.trop (max x.untrop y.untrop)
theorem Tropical.trop_sup_def {R : Type u} [] (x : ) (y : ) :
x y = Tropical.trop (x.untrop y.untrop)
@[simp]
theorem Tropical.add_eq_left {R : Type u} [] ⦃x : ⦃y : (h : x y) :
x + y = x
@[simp]
theorem Tropical.add_eq_right {R : Type u} [] ⦃x : ⦃y : (h : y x) :
x + y = y
theorem Tropical.add_eq_left_iff {R : Type u} [] {x : } {y : } :
x + y = x x y
theorem Tropical.add_eq_right_iff {R : Type u} [] {x : } {y : } :
x + y = y y x
theorem Tropical.add_self {R : Type u} [] (x : ) :
x + x = x
@[simp]
theorem Tropical.bit0 {R : Type u} [] (x : ) :
bit0 x = x
theorem Tropical.add_eq_iff {R : Type u} [] {x : } {y : } {z : } :
x + y = z x = z x y y = z y x
@[simp]
theorem Tropical.add_eq_zero_iff {R : Type u} [] {a : Tropical ()} {b : Tropical ()} :
a + b = 0 a = 0 b = 0
instance Tropical.instAddCommMonoidTropical {R : Type u} [] [] :
Equations
Mul ()

Tropical multiplication is the addition in the underlying R.

Equations
• Tropical.instMulOfAdd = { mul := fun (x y : ) => Tropical.trop (x.untrop + y.untrop) }
@[simp]
theorem Tropical.trop_add {R : Type u} [Add R] (x : R) (y : R) :
@[simp]
theorem Tropical.untrop_mul {R : Type u} [Add R] (x : ) (y : ) :
(x * y).untrop = x.untrop + y.untrop
theorem Tropical.trop_mul_def {R : Type u} [Add R] (x : ) (y : ) :
x * y = Tropical.trop (x.untrop + y.untrop)
instance Tropical.instOneTropical {R : Type u} [Zero R] :
One ()
Equations
• Tropical.instOneTropical = { one := }
@[simp]
theorem Tropical.trop_zero {R : Type u} [Zero R] :
@[simp]
theorem Tropical.untrop_one {R : Type u} [Zero R] :
instance Tropical.instAddMonoidWithOneTropical {R : Type u} [] [] [Zero R] :
Equations
Equations
• =
instance Tropical.instInvOfNeg {R : Type u} [Neg R] :
Inv ()
Equations
@[simp]
theorem Tropical.untrop_inv {R : Type u} [Neg R] (x : ) :
x⁻¹.untrop = -x.untrop
instance Tropical.instDivOfSub {R : Type u} [Sub R] :
Div ()
Equations
• Tropical.instDivOfSub = { div := fun (x y : ) => Tropical.trop (x.untrop - y.untrop) }
@[simp]
theorem Tropical.untrop_div {R : Type u} [Sub R] (x : ) (y : ) :
(x / y).untrop = x.untrop - y.untrop
instance Tropical.instSemigroupTropical {R : Type u} [] :
Equations
• Tropical.instSemigroupTropical =
Equations
• Tropical.instCommSemigroupTropical = let __src := Tropical.instSemigroupTropical;
instance Tropical.instPowOfSMul {R : Type u} {α : Type u_1} [SMul α R] :
Pow () α
Equations
• Tropical.instPowOfSMul = { pow := fun (x : ) (n : α) => Tropical.trop (n x.untrop) }
@[simp]
theorem Tropical.untrop_pow {R : Type u} {α : Type u_1} [SMul α R] (x : ) (n : α) :
(x ^ n).untrop = n x.untrop
@[simp]
theorem Tropical.trop_smul {R : Type u} {α : Type u_1} [SMul α R] (x : R) (n : α) :
Equations
• Tropical.instMulOneClassTropical =
instance Tropical.instMonoidTropical {R : Type u} [] :
Equations
• Tropical.instMonoidTropical = let __src := Tropical.instMulOneClassTropical; let __src_1 := Tropical.instSemigroupTropical; Monoid.mk (fun (n : ) (x : ) => x ^ n)
@[simp]
theorem Tropical.trop_nsmul {R : Type u} [] (x : R) (n : ) :
Equations
• Tropical.instCommMonoidTropical = let __src := Tropical.instMonoidTropical; let __src_1 := Tropical.instCommSemigroupTropical;
instance Tropical.instGroupTropical {R : Type u} [] :
Equations
• Tropical.instGroupTropical = let __src := Tropical.instMonoidTropical;
Equations
• Tropical.instCommGroupOfAddCommGroup = let __src := Tropical.instGroupTropical;
@[simp]
theorem Tropical.untrop_zpow {R : Type u} [] (x : ) (n : ) :
(x ^ n).untrop = n x.untrop
@[simp]
theorem Tropical.trop_zsmul {R : Type u} [] (x : R) (n : ) :
instance Tropical.covariant_mul {R : Type u} [LE R] [Add R] [CovariantClass R R (fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x x_1] :
CovariantClass () () (fun (x x_1 : ) => x * x_1) fun (x x_1 : ) => x x_1
Equations
• =
instance Tropical.covariant_swap_mul {R : Type u} [LE R] [Add R] [CovariantClass R R (Function.swap fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x x_1] :
CovariantClass () () (Function.swap fun (x x_1 : ) => x * x_1) fun (x x_1 : ) => x x_1
Equations
• =
instance Tropical.covariant_add {R : Type u} [] :
CovariantClass () () (fun (x x_1 : ) => x + x_1) fun (x x_1 : ) => x x_1
Equations
• =
instance Tropical.covariant_mul_lt {R : Type u} [LT R] [Add R] [CovariantClass R R (fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x < x_1] :
CovariantClass () () (fun (x x_1 : ) => x * x_1) fun (x x_1 : ) => x < x_1
Equations
• =
instance Tropical.covariant_swap_mul_lt {R : Type u} [] [Add R] [CovariantClass R R (Function.swap fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x < x_1] :
CovariantClass () () (Function.swap fun (x x_1 : ) => x * x_1) fun (x x_1 : ) => x < x_1
Equations
• =
instance Tropical.instDistribTropical {R : Type u} [] [Add R] [CovariantClass R R (fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x x_1] [CovariantClass R R (Function.swap fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x x_1] :
Equations
• Tropical.instDistribTropical =
@[simp]
theorem Tropical.add_pow {R : Type u} [] [] [CovariantClass R R (fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x x_1] [CovariantClass R R (Function.swap fun (x x_1 : R) => x + x_1) fun (x x_1 : R) => x x_1] (x : ) (y : ) (n : ) :
(x + y) ^ n = x ^ n + y ^ n
instance Tropical.instCommSemiring {R : Type u} :
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Tropical.succ_nsmul {R : Type u_1} [] [] (x : ) (n : ) :
(n + 1) x = x
theorem Tropical.mul_eq_zero_iff {R : Type u_1} {a : Tropical ()} {b : Tropical ()} :
a * b = 0 a = 0 b = 0
Equations
• =