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Experimental implementation of Cubical Type Theory
Home Page: https://arxiv.org/abs/1611.02108
License: MIT License
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An idea to simplify terms: flow
Here is something we can do to simplify terms and thus moving closer to computing the Brunerie's number, as an example I will do it by hand for the term test0To3 below.
open import Cubical.HITs.Everything public
p0 : (i j k : I) → join S¹ S¹ -- normal form of test0To3
p0 i j k = hcomp (λ l → λ { (i = i0) → hcomp (λ m → λ { (j = i0) → push base base (~ m ∧ ~ l)
; (j = i1) → push base base (~ m ∧ ~ l)
; (l = i0) → push (loop j) base (~ m)
; (l = i1) → inl base
})
(push base base (~ l))
; (i = i1) → push base (loop k) (~ l)
; (j = i0) → push base (loop k) (i ∧ ~ l)
; (j = i1) → push base (loop k) (i ∧ ~ l)
; (k = i0) → hcomp (λ m → λ { (i = i1) → push base base (~ l)
; (j = i0) → push base base ((i ∨ ~ m) ∧ (~ l))
; (j = i1) → push base base ((i ∨ ~ m) ∧ (~ l))
; (l = i0) → push (loop j) base (i ∨ ~ m)
; (l = i1) → inl base})
(push base base (~ l))
; (k = i1) → hcomp (λ m → λ { (i = i1) → push base base (~ l)
; (j = i0) → push base base ((i ∨ ~ m) ∧ (~ l))
; (j = i1) → push base base ((i ∨ ~ m) ∧ (~ l))
; (l = i0) → push (loop j) base (i ∨ ~ m)
; (l = i1) → inl base})
(push base base (~ l))
}) (push (loop j) (loop k) i)Above is the normal form of test0To3 in Agda.
Now let's add a variable z and write down the following term:
p : (i j k z : I) → join S¹ S¹
p i j k z = hcomp (λ l → λ { (i = i0) → hcomp (λ m → λ { (j = i0) → push base base (~ m ∧ ~ l ∨ z)
; (j = i1) → push base base (~ m ∧ ~ l ∨ z)
; (l = i0) → push (loop j) base (~ m)
; (l = i1) → push base base (~ m ∧ z)
})
(push base base (~ l ∨ z))
; (i = i1) → push base (loop k) (~ l ∨ z)
; (j = i0) → push base (loop k) (i ∧ ~ l ∨ z)
; (j = i1) → push base (loop k) (i ∧ ~ l ∨ z)
; (k = i0) → hcomp (λ m → λ { (i = i1) → push base base (~ l ∨ z)
; (j = i0) → push base base ((i ∨ ~ m) ∧ (~ l ∨ z))
; (j = i1) → push base base ((i ∨ ~ m) ∧ (~ l ∨ z))
; (l = i0) → push (loop j) base (i ∨ ~ m)
; (l = i1) → push base base ((i ∨ ~ m) ∧ z)})
(push base base (~ l ∨ z))
; (k = i1) → hcomp (λ m → λ { (i = i1) → push base base (~ l ∨ z)
; (j = i0) → push base base ((i ∨ ~ m) ∧ (~ l ∨ z))
; (j = i1) → push base base ((i ∨ ~ m) ∧ (~ l ∨ z))
; (l = i0) → push (loop j) base (i ∨ ~ m)
; (l = i1) → push base base ((i ∨ ~ m) ∧ z)})
(push base base (~ l ∨ z))
}) (push (loop j) (loop k) i)The idea is, when z = i0, it's the same as p0 above. And when z = i1, we don't know what it will be, but as it's mostly of the form - ∨ z, it will be simpler when z = i1 since i1 dominates when we compute ∨.
What is it when z = i1? Well it's
p1 : (i j k : I) → join S¹ S¹
p1 i j k = hcomp (λ l → λ { (i = i0) → hcomp (λ m → λ { (j = i0) → push base base (~ m)
; (j = i1) → push base base (~ m)
; (l = i0) → push (loop j) base (~ m)
; (l = i1) → push base base (~ m)
})
(inr base)
; (i = i1) → inr (loop k)
; (j = i0) → push base (loop k) i
; (j = i1) → push base (loop k) i
; (k = i0) → hcomp (λ m → λ { (i = i1) → inr base
; (j = i0) → push base base (i ∨ ~ m)
; (j = i1) → push base base (i ∨ ~ m)
; (l = i0) → push (loop j) base (i ∨ ~ m)
; (l = i1) → push base base (i ∨ ~ m)})
(inr base)
; (k = i1) → hcomp (λ m → λ { (i = i1) → inr base
; (j = i0) → push base base (i ∨ ~ m)
; (j = i1) → push base base (i ∨ ~ m)
; (l = i0) → push (loop j) base (i ∨ ~ m)
; (l = i1) → push base base (i ∨ ~ m)})
(inr base)
}) (push (loop j) (loop k) i)I hope you agree this is actually (slightly) simpler! If we replace some hcomp by hfill (with the variable) maybe we can even remove some hcomp's. However in the example above end points no longer match so it's not a term in Ω³ (it's because I allowed those related to (l = i1) to flow). But we can write a (Agda or Haskell) program to keep trying various possibilities and measure the complexity of the results, so in the end we let the term "flow" to a simpler one.
I think this should be doable for test0To3 or test0To4, they are kinda small but still too complicated for human.
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