luxbock

Set partitions in J

https://code.jsoftware.com/wiki/Essays/Set_Partitions

   nextpart=: (#~ ,"1 0 ;@:(i.&.>)@]) >:@#@~."1
   setpart=: nextpart^:(]`(''"_))

The power gerund in setpart is just a way of passing arguments to the power

luxbock

namespace hidden

open classical
open decidable
local attribute [instance] prop_decidable

def divides (m n : ℕ) : Prop := ∃ k, m * k = n

instance : has_dvd ℕ := ⟨divides⟩

luxbock

open classical

variables (α : Type) (p q : α → Prop)
variable r : Prop
variable a : α

example : (∃ x, p x → r) ↔ (∀ x, p x) → r :=
iff.intro
  (assume ⟨a, pair⟩,
   assume h : ∀ x, p x,

luxbock

example : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) :=
iff.intro
  (assume h : p ∧ (q ∨ r),
    have hp : p, from h.left,
    or.elim h.right
      (assume hq, or.inl ⟨hp, hq⟩)
      (assume hr, or.inr ⟨hp, hr⟩))
  (assume h : (p ∧ q) ∨ (p ∧ r),
    or.elim h
      (assume hpq,

luxbock

example : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) := 
iff.intro
  (assume h, _)
  /-
  propositions_as_types.lean:432:13: error 
  don't know how to synthesize placeholder
  context:
  p q r : Prop,
  h : p ∨ q → r
  ⊢ (p → r) ∧ (q → r)

luxbock

/// Santa is delivering presents to an infinite two-dimensional grid of houses.
///
/// He begins by delivering a present to the house at his starting location, and then an elf at the North Pole calls him via radio and tells him where to move next. Moves are always exactly one house to the north (^), south (v), east (>), or west (<). After each move, he delivers another present to the house at his new location.
///
/// However, the elf back at the north pole has had a little too much eggnog, and so his directions are a little off, and Santa ends up visiting some houses more than once. How many houses receive at least one present?
///
/// For example:
///
/// > delivers presents to 2 houses: one at the starting location, and one to the east.
/// ^>v< delivers presents to 4 houses in a square, including twice to the house at his starting/ending location.

luxbock

(defnav every-nth [n]
  (select* [this structure next-fn]
    (next-fn (take-nth n structure)))
  (transform* [this structure next-fn]
    (let [xform (comp
                  (map-indexed
                    (fn [i x]
                      (if (zero? (mod i n))
                        (next-fn x)
                        x)))

luxbock

(defnav drop-nav [n]
  (select* [this structure next-fn]
    (next-fn
      (if (vector? structure)
        (subvec structure (min n (count structure)))
        (drop n structure))))
  (transform* [this structure next-fn]
    (if (vector? structure)
      (let [mi (min n (count structure))]
        (into (subvec structure 0 mi)

luxbock

(defnav take-nth [n]
  (select* [this structure next-fn]
    (next-fn (take-nth n structure)))
  (transform* [this structure next-fn]
    (if (vector? structure)
      (let [c  (count structure)
            is (into []
                 (take-while #(< % c))
                 (iterate #(+ % n) 0))]

luxbock

(defn- get-fspec-attr
  [sym attr]
  (->> (s/form sym)
       (rest)
       (partition 2)
       (keep (fn [[a b]] (when (= a attr) b)))
       (first)))

(s/fdef conform-fn-call
  :args (s/cat :fn-call (s/spec (s/cat :f symbol? :rs (s/* any?))))