text:
using JuMP, EAGO

                     m = Model()

                     EAGO.register_eago_operators!(m)

                     @variable(m, -1 <= x[i=1:5] <= 1)

                     @variable(m, -53.467800048111016 <= q <= 50.665869089598125)

                     add_NL_constraint(m, :(softplus(-0.14052056007525238 + -0.5593120910347902*softplus(0.7927050633346799 + -0.9855209344073468*softplus(-0.5279345513587295 + 0.26674570975926315*softplus(-0.7911508587829115 + -0.06506911132939441*$(x[1]) + -0.852429148445272*$(x[2]) + 0.9902340539920638*$(x[3]) + -0.5325045469076417*$(x[4]) + 0.16346424911659252*$(x[5])) + 0.6014540451740125*softplus(-0.31894961218704987 + -0.02324224975324851*$(x[1]) + -0.016870940285415248*$(x[2]) + 0.446182370860321*$(x[3]) + -0.9288490609590947*$(x[4]) + 0.29295311736763896*$(x[5])) + -0.7842020578742437*softplus(-0.4972259981592049 + -0.8938534018236086*$(x[1]) + 0.008380370033680418*$(x[2]) + -0.18784778862206242*$(x[3]) + 0.20554715210351482*$(x[4]) + 0.7932560928666454*$(x[5])) + 0.7348995303901047*softplus(-0.6657792028971241 + -0.5409713712241491*$(x[1]) + 0.49245843557230895*$(x[2]) + 0.7906244992303582*$(x[3]) + 0.4705847767166267*$(x[4]) + -0.3481500575812202*$(x[5]))) + -0.8618242155643285*softplus(-0.9736042039183683 + -0.3730380073565449*softplus(-0.7911508587829115 + -0.06506911132939441*$(x[1]) + -0.852429148445272*$(x[2]) + 0.9902340539920638*$(x[3]) + -0.5325045469076417*$(x[4]) + 0.16346424911659252*$(x[5])) + -0.6111131700945078*softplus(-0.31894961218704987 + -0.02324224975324851*$(x[1]) + -0.016870940285415248*$(x[2]) + 0.446182370860321*$(x[3]) + -0.9288490609590947*$(x[4]) + 0.29295311736763896*$(x[5])) + -0.6070668283472709*softplus(-0.4972259981592049 + -0.8938534018236086*$(x[1]) + 0.008380370033680418*$(x[2]) + -0.18784778862206242*$(x[3]) + 0.20554715210351482*$(x[4]) + 0.7932560928666454*$(x[5])) + 0.602250903818204*softplus(-0.6657792028971241 + -0.5409713712241491*$(x[1]) + 0.49245843557230895*$(x[2]) + 0.7906244992303582*$(x[3]) + 0.4705847767166267*$(x[4]) + -0.3481500575812202*$(x[5]))) + 0.1409847014129051*softplus(-0.7008144662498323 + 0.1168473746120049*softplus(-0.7911508587829115 + -0.06506911132939441*$(x[1]) + -0.852429148445272*$(x[2]) + 0.9902340539920638*$(x[3]) + -0.5325045469076417*$(x[4]) + 0.16346424911659252*$(x[5])) + 0.28223204928638745*softplus(-0.31894961218704987 + -0.02324224975324851*$(x[1]) + -0.016870940285415248*$(x[2]) + 0.446182370860321*$(x[3]) + -0.9288490609590947*$(x[4]) + 0.29295311736763896*$(x[5])) + -0.1750913505933096*softplus(-0.4972259981592049 + -0.8938534018236086*$(x[1]) + 0.008380370033680418*$(x[2]) + -0.18784778862206242*$(x[3]) + 0.20554715210351482*$(x[4]) + 0.7932560928666454*$(x[5])) + -0.6765062198397387*softplus(-0.6657792028971241 + -0.5409713712241491*$(x[1]) + 0.49245843557230895*$(x[2]) + 0.7906244992303582*$(x[3]) + 0.4705847767166267*$(x[4]) + -0.3481500575812202*$(x[5]))) + -0.8981625305472791*softplus(0.7932740070802415 + 0.6457743599334909*softplus(-0.7911508587829115 + -0.06506911132939441*$(x[1]) + -0.852429148445272*$(x[2]) + 0.9902340539920638*$(x[3]) + -0.5325045469076417*$(x[4]) + 0.16346424911659252*$(x[5])) + 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-0.852429148445272*$(x[2]) + 0.9902340539920638*$(x[3]) + -0.5325045469076417*$(x[4]) + 0.16346424911659252*$(x[5])) + -0.6111131700945078*softplus(-0.31894961218704987 + -0.02324224975324851*$(x[1]) + -0.016870940285415248*$(x[2]) + 0.446182370860321*$(x[3]) + -0.9288490609590947*$(x[4]) + 0.29295311736763896*$(x[5])) + -0.6070668283472709*softplus(-0.4972259981592049 + -0.8938534018236086*$(x[1]) + 0.008380370033680418*$(x[2]) + -0.18784778862206242*$(x[3]) + 0.20554715210351482*$(x[4]) + 0.7932560928666454*$(x[5])) + 0.602250903818204*softplus(-0.6657792028971241 + -0.5409713712241491*$(x[1]) + 0.49245843557230895*$(x[2]) + 0.7906244992303582*$(x[3]) + 0.4705847767166267*$(x[4]) + -0.3481500575812202*$(x[5]))) + -0.8696160844053109*softplus(-0.7008144662498323 + 0.1168473746120049*softplus(-0.7911508587829115 + -0.06506911132939441*$(x[1]) + -0.852429148445272*$(x[2]) + 0.9902340539920638*$(x[3]) + -0.5325045469076417*$(x[4]) + 0.16346424911659252*$(x[5])) + 0.28223204928638745*softplus(-0.31894961218704987 + -0.02324224975324851*$(x[1]) + -0.016870940285415248*$(x[2]) + 0.446182370860321*$(x[3]) + -0.9288490609590947*$(x[4]) + 0.29295311736763896*$(x[5])) + -0.1750913505933096*softplus(-0.4972259981592049 + -0.8938534018236086*$(x[1]) + 0.008380370033680418*$(x[2]) + -0.18784778862206242*$(x[3]) + 0.20554715210351482*$(x[4]) + 0.7932560928666454*$(x[5])) + -0.6765062198397387*softplus(-0.6657792028971241 + -0.5409713712241491*$(x[1]) + 0.49245843557230895*$(x[2]) + 0.7906244992303582*$(x[3]) + 0.4705847767166267*$(x[4]) + -0.3481500575812202*$(x[5]))) + -0.357347274050821*softplus(0.7932740070802415 + 0.6457743599334909*softplus(-0.7911508587829115 + -0.06506911132939441*$(x[1]) + -0.852429148445272*$(x[2]) + 0.9902340539920638*$(x[3]) + -0.5325045469076417*$(x[4]) + 0.16346424911659252*$(x[5])) + 0.13450970941280982*softplus(-0.31894961218704987 + -0.02324224975324851*$(x[1]) + -0.016870940285415248*$(x[2]) + 0.446182370860321*$(x[3]) + -0.9288490609590947*$(x[4]) + 0.29295311736763896*$(x[5])) + -0.4617332497282156*softplus(-0.4972259981592049 + -0.8938534018236086*$(x[1]) + 0.008380370033680418*$(x[2]) + -0.18784778862206242*$(x[3]) + 0.20554715210351482*$(x[4]) + 0.7932560928666454*$(x[5])) + -0.2655397829444155*softplus(-0.6657792028971241 + -0.5409713712241491*$(x[1]) + 0.49245843557230895*$(x[2]) + 0.7906244992303582*$(x[3]) + 0.4705847767166267*$(x[4]) + -0.3481500575812202*$(x[5]))))) - $q <= 0.0))

                     @objective(m, Min, q)

                     return m

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text:
module Cartpole

# The convention for packages is to explicitly import all required functions; for a quick
# project this isn't necessary and we can just do `using OrdinaryDiffEq` and pull in
# everything

using OrdinaryDiffEq: ODEProblem, solve, Tsit5  # ODE integrator
using StaticArrays: SVector  # Fast statically-sized arrays
using MatrixEquations: arec  # Solve a Riccati equation
using Plots: plot  # Basic plotting
using ForwardDiff: jacobian  # Automatic differentiation
using LinearAlgebra: diagm, I  # Basic linear algebra

export cartpole, cartpole_defaults, cartpole_example

"""
    cartpole(u, p, t)

Implement the cart-pole equations of motion where `u = [x, x′, θ, θ′]` with `M` as the cart
mass, `m` as the pendulum mass, `L` as the length of the pendulum, and `g` as gravity.

```math
(M + m)x″ + mLsin(θ)θ′² - mLcos(θ)θ″ = f(u, p, t)
```

```math
mL²θ″ - mLcos(θ)x″ - mgLsin(θ) = 0
```

`f(u, p, t)` is an arbitrary control force.
"""
function cartpole(u, p, t)
    # Unpack the state
    (x, x′, θ, θ′) = u
    # Force to apply (a function)
    force = p.f(u, p, t)
    # Right-hand side
    return SVector(
        x′,
        (force + p.m * p.L * sin(θ) * θ′^2 + p.m * p.g * cos(θ) * sin(θ)) /
        (p.M + p.m - p.m * cos(θ)^2),
        θ′,
        (force * cos(θ) - (p.M + p.m) * p.g * sin(θ) + p.m * p.L * cos(θ) * sin(θ) * x′^2) /
        (p.m * p.L * cos(θ)^2 - (p.M + p.m) * p.L),
    )
end

"""
    cartpole_defaults()

Return the default parameter values for the cart-pole example.
"""
function cartpole_defaults()
    return (M=5, m=1, L=2, g=10, f=(u, p, t) -> false)
end

function cartpole_integrate(u0, p, tend)
    # This assume Float64 as the underlying floating point type - could write this
    # generically which might be better for GPU implementations
    prob = ODEProblem(cartpole, SVector{4}(u0), (0.0, tend), p)
    return solve(prob, Tsit5())
end

"""
    cartpole_optimal_control(u, p, Q, R)

Calculate the optimal control gains (LQR) around a specific operating point.
"""
function cartpole_optimal_control(u, p, Q, R)
    A = jacobian(uu -> cartpole(uu, p, false), u)
    B = jacobian(pp -> cartpole(u, merge(p, (f=(args...) -> only(pp),)), false), [0.0])  # this is slightly cumbersome since I've made f as a function
    # Solve the Riccati equation
    P, _ = arec(A, B, R, Q)  # returns a tuple; the first element is the one of interest
    # Calculate optimal gains
    K = R \ (B' * P)
    return K
end

"""
    cartpole_example(; plotting)

Calculate the solution of the cart-pole example with optimal (LQR) control. Plotting is
optional, use `cartpole_example(plotting=true)` to display.
"""
function cartpole_example(; plotting=false)
    u0 = [0, 0, 0.1, 0]
    p0 = cartpole_defaults()
    tend = 10.0
    R = Matrix(I, 1, 1)
    Q = diagm([500.0, 250, 1, 1])
    K = cartpole_optimal_control(u0, p0, Q, R)  # calculate control gains
    p = merge(p0, (f=(u, p, t) -> only(-p.K * u), K=K))  # add a linear controller
    prob = ODEProblem(cartpole, SVector{4}(u0), (0.0, tend), p)
    sol = solve(prob, Tsit5())
    plotting && display(plot(sol))
    return sol
end

end  # module

ext: jl


text:
@testset "674.longest-continuous-increasing-subsequence.jl" begin
    @test find_length_of_lcis([1,3,5,4,7]) == 3
    @test find_length_of_lcis([2, 2, 2, 2, 2]) == 1
    @test find_length_of_lcis(Int[]) == 0
end


text:
module Tprint

import Term: highlight, theme, unescape_brackets
import ..renderables: AbstractRenderable
import ..style: apply_style
import ..layout: hstack

export tprint, tprintln
"""
tprint

Similar to standard lib's `print` function but with added
styling functionality
"""
function tprint end
tprint(x) = tprint(stdout, x)
tprint(io::IO, x) = tprint(io, string(x))
tprint(io::IO, ::MIME"text/html", x) = tprint(io, x)

"""
tprint(x::AbstractString)

Apply style to a string and print it to a new line
"""
tprint(io::IO, x::AbstractString) = print(io, (unescape_brackets ∘  apply_style)(x))

"""
tprint(x::AbstractRenderable)

Print an `AbstractRenderable`.

Equivalent to `println(x)`
"""
tprint(io::IO, x::AbstractRenderable) =  print(io, x)

"""
tprint(x::Symbol)

Print highlighted as a Symbol
"""
tprint(io::IO, x::Symbol) = tprint(io, highlight(":" * string(x), :symbol))

"""
tprint(x::Number)

Print highlighted as a Number
"""
tprint(io::IO, x::Number) = tprint(io, highlight(string(x), :number))

"""
tprint(x::Function)

Print highlighted as a Function
"""
tprint(io::IO, x::Function) = tprint(io, highlight(string(x), :func))

"""
tprint(x::DataType)

Print highlighted as a DataType
"""
tprint(io::IO, x::DataType) = tprint(io, highlight(string(x), :type))

function tprint(io::IO, args...)
    for (n, arg) in enumerate(args)
        tprint(io, arg)

        if n < length(args)
            args[n+1] isa AbstractRenderable || print(io, " ")
        end
    end
    return nothing
end

function tprint(args...)
    for (n, arg) in enumerate(args)
        tprint(arg)

        if n < length(args)
            args[n+1] isa AbstractRenderable || print(" ")
        end
    end
    return nothing
end

"""
    tprintln

Similar to standard lib's `println` function but with added
styling functionality.
"""
tprintln(x) = tprint(x, "\n")
tprintln(args...) = tprint(args..., "\n")

tprintln(io::IO, x) = tprint(io, x, "\n")
tprintln(io::IO, args...) = tprint(io, args..., "\n")

end


text:
module HeapQueueModule

import Base.popfirst!

export Max,
       Min,
       MinHeapQueue,
       MaxHeapQueue,
       IndexedMinHQ,
       IndexedMaxHQ,
       push!,
       decreasekey!,
       heapify,
       heapify!,
       heapsort!
       popfirst!

abstract type HeapType end
struct MinHeap <: HeapType end
struct MaxHeap <: HeapType end
const Min = MinHeap()
const Max = MaxHeap()

abstract type AbstractHeapQueue end

mutable struct HeapQueue{H<:HeapType, T} <: AbstractHeapQueue
    arr::Vector{T}
    n::Int

    function HeapQueue{H, T}() where {T,H<:HeapType}
        new{H, T}(Vector{T}[], 0)
    end
end

MinHeapQueue{T} = HeapQueue{MinHeap, T}
MaxHeapQueue{T} = HeapQueue{MaxHeap, T}

Base.getindex(hq::HeapQueue, i::Int) = hq.arr[i]
Base.setindex!(hq::HeapQueue, v, i::Int) = hq.arr[i] = v
Base.length(hq::AbstractHeapQueue) = hq.n
Base.isempty(hq::AbstractHeapQueue) = hq.n == 0

"""
Test the heap dominance between i and j.
"""
testheapdom(hq::MinHeapQueue, i, j) = hq[i] < hq[j]
testheapdom(hq::MaxHeapQueue, i, j) = hq[i] > hq[j]

function Base.push!(hq::HeapQueue, transaction)
    push!(hq.arr, transaction)
    hq.n += 1
    swim!(hq, hq.n)
end

function Base.push!(hq::HeapQueue, transactions...)
    for transaction in transactions
        push!(hq, transaction)
    end
end

function popfirst!(hq::HeapQueue)
    last_transaction = hq.arr[1]
    swap!(hq, 1, hq.n)
    hq.n -= 1
    sink!(hq, 1)
    Base.pop!(hq.arr)
    last_transaction
end

function swap!(hq::HeapQueue, i, j)
    hq[i], hq[j] = hq[j], hq[i]
    nothing
end

"""
Move node from bottom to top of the tree.
"""
function swim!(hq::AbstractHeapQueue, k::Int)
    # if k-th node dominates its parent,
    # swap with its the parent node
    while k > 1 && testheapdom(hq, k, div(k,2))
        swap!(hq, div(k,2), k)
        k = div(k,2)
    end
end

function sink!(hq::AbstractHeapQueue, k::Int, n::Int)
    while 2*k <= n
        # child node to swap with, start with the left child
        j = 2*k
        # test whether the right child dominates he left child
        # if so, swap with the right child node
        if j < n && testheapdom(hq, j+1, j)
            j += 1
        end
        # now j-th node is the most dominant node in child-level
        # if parent node dominates j-th node, break
        # if not, then swap with j-th node, in this way we still
        # maintain the heap invariance
        if testheapdom(hq, k, j)
            break
        else
            swap!(hq, k, j)
            # move k to next child level
            k = j
        end
    end
end

sink!(hq::AbstractHeapQueue, k::Int) = sink!(hq, k, hq.n)

function heapify(v::Vector{T}, ::H=Min) where {T,H<:HeapType}
    hq = HeapQueue{H, T}()
    for elem in v
        push!(hq, elem)
    end
    hq
end

function heapify!(v::Vector{T}, ::H=Min) where{T,H<:HeapType}
    n = length(v)
    hq = HeapQueue{H, T}()
    hq.arr = v
    hq.n = n
    # we start from middle because the right half index will be
    # the last level and there will be no where to sink
    for i in div(n, 2):-1:1
        sink!(hq, i, n)
    end
    hq
end

"""
In-place heapsort. It is similar to `popfirst!` and the only difference is
we don't pop the array but manipulate the index.
"""
function heapsort!(v::Vector{T}) where {T}
    hq = heapify!(v, Max)
    n = hq.n
    while n > 1
        swap!(hq, 1, n)
        n -= 1
        sink!(hq, 1, n)
    end
    v
    nothing
end

# function naive_heapsort!(v::Vector{T}) where {T}
#     hq = heapify!(v)
#     result = T[]
#     while !isempty(hq)
#         push!(result, pop!(hq))
#     end
#     result
# end

# Indexed heap queue
mutable struct IndexedHeapQueue{H<:HeapType, T} <: AbstractHeapQueue
    keys::Vector{Union{T, Nothing}}
    pq::Vector{Union{Int,Nothing}}
    qp::Vector{Union{Int,Nothing}}
    n::Int
    capacity::Int

    IndexedHeapQueue{H, T}(capacity::Int) where {H<:HeapType,T} = new(
        Vector{Union{T, Nothing}}(nothing, capacity),
        Vector{Union{Int, Nothing}}(nothing, capacity),
        Vector{Union{Int, Nothing}}(nothing, capacity),
        0, capacity
    )
end

IndexedMinHQ{T} = IndexedHeapQueue{MinHeap, T}
IndexedMaxHQ{T} = IndexedHeapQueue{MaxHeap, T}

Base.getindex(hq::IndexedHeapQueue, keyindex::Int) = hq.keys[keyindex]
pqtokey(hq::IndexedHeapQueue, i) = hq.keys[hq.pq[i]]
keytopq(hq::IndexedHeapQueue, keyindex) = hq.qp[keyindex]

testheapdom(hq::IndexedMinHQ, i, j) = pqtokey(hq, i) < pqtokey(hq, j)
testheapdom(hq::IndexedMaxHQ, i, j) = pqtokey(hq, i) > pqtokey(hq, j)

function swap!(hq::IndexedHeapQueue, i, j)
    hq.pq[i], hq.pq[j] = hq.pq[j], hq.pq[i]
    # key indices in i and j has been swapped in pq
    # Since pq[i] ≡ pq[j] ≡ keyindex_j,
    # so we need to assign keyindex_j to i in `qp`
    hq.qp[hq.pq[i]], hq.qp[hq.pq[j]] = i, j
    nothing
end

contains(hq::IndexedHeapQueue, keyindex) = !isnothing(hq.qp[keyindex])

struct NoSuchElementException <: Exception
    msg::String
end

"""
    popfirst!(hq)

Remove the most dominant item in the queue and returns the associated
index in `hq.keys`.
"""
function popfirst!(hq::IndexedHeapQueue)
    hq.n == 0 && throw(NoSuchElementException("priority queue underflow"))

    # pop the minimum
    firstkeyindex = hq.pq[1]
    swap!(hq, 1, hq.n)
    hq.n -= 1
    sink!(hq, 1)
    @assert firstkeyindex == hq.pq[hq.n+1]

    # clean up
    hq.pq[hq.n+1] = nothing
    hq.qp[firstkeyindex] = nothing
    firstkey = hq.keys[firstkeyindex]
    hq.keys[firstkeyindex] = nothing

    firstkeyindex, firstkey
end

"""
    insert!(hq, i, key)

Associate key with index `keyindex` in the heap queue.
"""
function insert!(hq::IndexedHeapQueue, keyindex::Int, key)
    contains(hq, keyindex) && throw(ArgumentError("index $keyindex is already in the heap queue"))
    hq.n += 1
    hq.qp[keyindex] = hq.n
    hq.keys[keyindex] = key
    hq.pq[hq.n] = keyindex
    swim!(hq, hq.n)

    nothing
end

"""
Change the key value at `keyindex`. If it is a MinHeapQueue, then
make sure the updated key is smaller then existing one.
"""
function changekey!(hq::IndexedHeapQueue, keyindex::Int, key)
    contains(hq, keyindex) || throw(NoSuchElementException("index $keyindex is not in the heap queue"))
    checkkey(hq, keyindex, key)
    hq.keys[keyindex] = key
    # a more dominant value updated, should swim! to maintain
    # heap invariance
    swim!(hq, keytopq(hq, keyindex))
    nothing
end

decreasekey!(hq::IndexedMinHQ, keyindex::Int, key) = changekey!(hq, keyindex, key)

struct InvalidKeyChangeError <: Exception
    msg::String
end
function checkkey(hq::IndexedMinHQ, keyindex, key)
    hq.keys[keyindex] <= key && throwerror(hq)
end
throwerror(hq::IndexedMinHQ) = throw(InvalidKeyChangeError("new key should be smaller than current key"))
throwerror(hq::IndexedMaxHQ) = throw(InvalidKeyChangeError("new key should be greater than current key"))

end # module

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