higher-alg/UniverseTrans.agda at master · ericfinster/higher-alg · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
{-# OPTIONS --without-K --rewriting --type-in-type #-}

open import Base

module UniverseTrans where

  --
  --  Basic Signature
  --

  Frm : ℕ → Set
  Cell : {n : ℕ} → Frm n → Set

  data Tree : {n : ℕ} → Frm n → Set
  Pos : {n : ℕ} {f : Frm n} (σ : Tree f) → Set

  Typ : {n : ℕ} {f : Frm n} (σ : Tree f) (p : Pos σ) → Frm n
  Inh : {n : ℕ} {f : Frm n} (σ : Tree f) (p : Pos σ) → Cell (Typ σ p)

  Frm↓ : {n : ℕ} → Frm n → Set
  Cell↓ : {n : ℕ} {f : Frm n} → Frm↓ f → Cell f → Set

  data Tree↓ : {n : ℕ} {f : Frm n} → Frm↓ f → Tree f → Set
  -- Pos↓ : {n : ℕ} {f : Frm n} {f↓ : Frm↓ f} {σ : Tree f}
  --   → (σ↓ : Tree↓ f↓ σ) → Set

  --
  --  Equivalences
  --

  record Eqv (A B : Set) : Set where
    field
      Rel : A → B → Set
      To : A → B
      ToRel : (a : A) → Rel a (To a)

  open Eqv

  --
  --  Base definitions
  --

  Frm O = ⊤
  Frm (S n) = Σ (Frm n) (λ f →
    Tree {n} f × Cell {n} f)

  Cell {O} unit = Set
  Cell {S n} (f , σ , τ) =
    (f↓ : Frm↓ f) → Eqv (Tree↓ f↓ σ) (Cell↓ f↓ τ)

  η : {n : ℕ} (f : Frm n)
    → Cell {n} f
    → Tree {n} f

  μ-frm : {n : ℕ} (f : Frm n) (σ : Tree {n} f)
    → (δ : (p : Pos σ) → Tree {n} (Typ σ p))
    → (ε : (p : Pos σ) → Tree {S n} (Typ σ p , δ p , Inh σ p))
    → Frm n

  μ : {n : ℕ} (f : Frm n) (σ : Tree {n} f)
    → (δ : (p : Pos σ) → Tree {n} (Typ σ p))
    → (ε : (p : Pos σ) → Tree {S n} (Typ σ p , δ p , Inh σ p))
    → Tree (μ-frm f σ δ ε)

  μ-trans : {n : ℕ} (f : Frm n) (σ : Tree {n} f)
    → (δ : (p : Pos σ) → Tree {n} (Typ σ p))
    → (ε : (p : Pos σ) → Tree {S n} (Typ σ p , δ p , Inh σ p))
    → Cell {n} f
    → Cell {n} (μ-frm f σ δ ε)

  γ : {n : ℕ} (f : Frm n) (σ : Tree {n} f) (τ : Cell {n} f)
    → (θ : Tree {S n} (f , σ , τ))
    → (δ : (p : Pos σ) → Tree {n} (Typ σ p))
    → (ε : (p : Pos σ) → Tree {S n} (Typ σ p , δ p , Inh σ p))
    → Tree {S n} (μ-frm f σ δ ε , μ f σ δ ε , μ-trans f σ δ ε τ)

  data Tree where

    nil : Tree {O} unit
    cns : (A : Set) (B : A → Tree {O} unit)
      → Tree {O} unit

    lf : {n : ℕ} (f : Frm n) (τ : Cell {n} f)
      → Tree {S n} (f , η f τ , τ)
    nd : {n : ℕ} (f : Frm n)
      → (σ : Tree {n} f) (τ : Cell {n} f)
      → (θ : Cell {S n} (f , σ , τ))
      → (δ : (p : Pos σ) → Tree {n} (Typ σ p))
      → (ε : (p : Pos σ) → Tree {S n} (Typ σ p , δ p , Inh σ p))
      → Tree {S n} (μ-frm f σ δ ε , μ f σ δ ε , μ-trans f σ δ ε τ)

  Pos nil = ⊥
  Pos (cns A B) = ⊤ ⊔ Σ A (λ a → Pos (B a))
  Pos (lf f τ) = ⊥
  Pos (nd f σ τ θ δ ε) = ⊤ ⊔ Σ (Pos σ) (λ p → Pos (ε p))

  Typ nil ()
  Typ (cns A B) (inl unit) = unit
  Typ (cns A B) (inr (p , q)) = Typ (B p) q
  Typ (lf f τ) ()
  Typ (nd f σ τ θ δ ε) (inl unit) = f , σ , τ
  Typ (nd f σ τ θ δ ε) (inr (p , q)) = Typ (ε p) q

  Inh nil ()
  Inh (cns A B) (inl unit) = A
  Inh (cns A B) (inr (p , q)) = Inh (B p) q
  Inh (lf f τ) ()
  Inh (nd f σ τ θ δ ε) (inl unit) = θ
  Inh (nd f σ τ θ δ ε) (inr (p , q)) = Inh (ε p) q

  postulate

    -- μ laws
    μ-frm-unit-r : {n : ℕ} (f : Frm n) (σ : Tree {n} f)
      → μ-frm f σ (λ p → η (Typ σ p) (Inh σ p)) (λ p → lf (Typ σ p) (Inh σ p)) ↦ f
    {-# REWRITE μ-frm-unit-r #-}

    μ-unit-r : {n : ℕ} (f : Frm n) (σ : Tree {n} f)
      → μ f σ (λ p → η (Typ σ p) (Inh σ p)) (λ p → lf (Typ σ p) (Inh σ p)) ↦ σ
    {-# REWRITE μ-unit-r #-}

    μ-trans-unit-r : {n : ℕ} (f : Frm n) (σ : Tree {n} f) (τ : Cell {n} f)
      → μ-trans f σ (λ p → η (Typ σ p) (Inh σ p)) (λ p → lf (Typ σ p) (Inh σ p)) τ ↦ τ
    {-# REWRITE μ-trans-unit-r #-}

  η {O} unit A = cns A (λ _ → nil)
  η {S n} (f , σ , τ) θ =
    let η-dec p = η (Typ σ p) (Inh σ p)
        lf-dec p = lf (Typ σ p) (Inh σ p)
    in nd f σ τ θ η-dec lf-dec

  μ-frm {O} f σ δ ε = unit
  μ-frm {S n} (f , .(η f τ) , τ) (lf .f .τ) δ₁ ε₁ = f , η f τ , τ
  μ-frm {S n} (._ , ._ , ._) (nd f σ τ θ δ₀ ε₀) δ₁ ε₁ =
    let δ₀' p = {!!}
    in μ-frm f σ δ₀ {!!} , μ f σ δ₀ {!!} , μ-trans f σ δ₀ {!!} τ

  -- Okay, I think I understand the idea: it's that in the recursive
  -- call, you should modify δ₀ and ε₀ by a multiplicaiton ...

  μ {O} unit nil δ ε = nil
  μ {O} unit (cns A B) δ ε = {!!}
  μ {S n} (f , .(η f τ) , τ) (lf .f .τ) δ₁ ε₁ = lf f τ
  μ {S n} (._ , ._ , ._) (nd f σ τ θ δ₀ ε₀) δ₁ ε₁ =
    let w = δ₁ (inl unit)
        δ₁' p q = δ₁ (inr (p , q))
        ε₁' p q = ε₁ (inr (p , q))
        -- Shouldn't we have to modify δ₀ as well then?
        ε₀' p = μ (Typ σ p , δ₀ p , Inh σ p) (ε₀ p) (δ₁' p) (ε₁' p)
    in {! γ f σ τ w δ₀ ε₀'!}

  -- Here's another possibility: because we now use the extra data
  -- when computing μ, it's possible that γ should no longer just
  -- be multiplication in the free monad.  Rather, it should carry
  -- along the higher information as well, calling back to μ when
  -- it arrives at a leaf.  Could this allow for its type to be
  -- more flexible????

  μ-trans = {!!}

  γ = {!!}

  --
  --  Total definitions
  --

  Frm↓ {O} unit = ⊤
  Frm↓ {S n} (f , σ , τ) = Σ (Frm↓ f) (λ f↓ →
    Tree↓ f↓ σ × Cell↓ f↓ τ)

  Cell↓ {O} unit A = A
  Cell↓ {S n} {f , σ , τ} (f↓ , σ↓ , τ↓) E =
    Rel (E f↓) σ↓ τ↓

  data Tree↓ where