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{-# OPTIONS --without-K --safe #-}
module Data.Circuit.Convert where
open import Data.Nat.Base using (ℕ)
open import Data.Circuit.Gate using (Gate; GateLabel; cast-gate; cast-gate-is-id; subst-is-cast-gate)
open import Data.Fin.Base using (Fin)
open import Function.Bundles using (Equivalence)
open import Data.Hypergraph.Edge GateLabel using (Edge)
open import Data.Hypergraph.Base GateLabel using (Hypergraph; sortHypergraph)
open import Data.Hypergraph.Setoid GateLabel using (Hypergraph-Setoid; ≈-Hypergraph)
open import Data.Permutation using (fromList-↭; toList-↭)
open import Data.List using (length)
open import Data.Vec.Functional using (toVec; fromVec; toList; fromList)
open import Function.Base using (_∘_; id; _$_)
import DecorationFunctor.Hypergraph.Labeled as LabeledHypergraph
open LabeledHypergraph using (Hypergraph-same) renaming (Hypergraph to Hypergraph′; Hypergraph-setoid to Hypergraph-Setoid′)
to : {v : ℕ} → Hypergraph v → Hypergraph′ v
to H = record
{ h = length edges
; a = arity ∘ fromList edges
; j = fromVec ∘ ports ∘ fromList edges
; l = label ∘ fromList edges
}
where
open Edge using (arity; ports; label)
open Hypergraph H
from : {v : ℕ} → Hypergraph′ v → Hypergraph v
from {v} H = record
{ edges = toList asEdge
}
where
open Hypergraph′ H
asEdge : Fin h → Edge v
asEdge e = record { label = l e ; ports = toVec (j e) }
open import Data.Product.Base using (proj₁; proj₂)
open import Data.Fin.Permutation using (flip)
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
to-cong : {v : ℕ} {H H′ : Hypergraph v} → ≈-Hypergraph H H′ → Hypergraph-same (to H) (to H′)
to-cong {v} {H} {H′} ≈H = record
{ ↔h = flip ρ
; ≗a = ≗a
; ≗j = ≗j
; ≗l = ≗l
}
where
open Edge using (arity; ports; label)
open import Data.List.Relation.Binary.Permutation.Propositional using (_↭_)
open import Function.Bundles using (_↔_)
open ≈-Hypergraph ≈H
open import Data.Fin.Permutation using (_⟨$⟩ʳ_; _⟨$⟩ˡ_; Permutation′; inverseʳ)
open import Data.Fin.Base using (cast)
open import Data.Fin.Properties using (cast-is-id)
ρ : Fin (length H′.edges) ↔ Fin (length H.edges)
ρ = proj₁ (fromList-↭ ↭edges)
open ≡.≡-Reasoning
edges≗ρ∘edges′ : (i : Fin (length H.edges)) → fromList H.edges i ≡ fromList H′.edges (ρ ⟨$⟩ˡ i)
edges≗ρ∘edges′ i = begin
fromList H.edges i ≡⟨ ≡.cong (fromList H.edges) (inverseʳ ρ) ⟨
fromList H.edges (ρ ⟨$⟩ʳ (ρ ⟨$⟩ˡ i)) ≡⟨ proj₂ (fromList-↭ ↭edges) (ρ ⟨$⟩ˡ i) ⟩
fromList H′.edges (ρ ⟨$⟩ˡ i) ∎
≗a : (e : Fin (Hypergraph′.h (to H)))
→ Hypergraph′.a (to H) e
≡ arity (fromList H′.edges (ρ ⟨$⟩ˡ e))
≗a = ≡.cong arity ∘ edges≗ρ∘edges′
≗j : (e : Fin (Hypergraph′.h (to H)))
(i : Fin (Hypergraph′.a (to H) e))
→ fromVec (ports (fromList H.edges e)) i
≡ fromVec (ports (fromList H′.edges (ρ ⟨$⟩ˡ e))) (cast (≗a e) i)
≗j e i
rewrite edges≗ρ∘edges′ e
rewrite cast-is-id ≡.refl i = ≡.refl
≗l : (e : Fin (Hypergraph′.h (to H)))
→ label (fromList H.edges e)
≡ cast-gate (≡.sym (≗a e)) (label (fromList H′.edges (ρ ⟨$⟩ˡ e)))
≗l e
rewrite edges≗ρ∘edges′ e
rewrite cast-gate-is-id ≡.refl (label (fromList H′.edges (ρ ⟨$⟩ˡ e))) =
≡.refl
module _ {v : ℕ} where
open import Data.Hypergraph.Edge GateLabel using (decTotalOrder; ≈-Edge; ≈-Edge⇒≡)
open import Data.List.Sort (decTotalOrder {v}) using (sort; sort-↭)
open import Data.Permutation.Sort (decTotalOrder {v}) using (sorted-≋)
open import Data.List.Relation.Binary.Permutation.Propositional using (_↭_)
open import Data.Vec.Functional.Relation.Binary.Permutation using () renaming (_↭_ to _↭′_)
open import Data.List.Relation.Binary.Pointwise using (Pointwise; Pointwise-≡⇒≡; map)
open import Data.Product.Base using (_,_)
open import Data.Hypergraph.Label using (HypergraphLabel)
open HypergraphLabel GateLabel using (isCastable)
open import Data.Castable using (IsCastable)
open IsCastable isCastable using (≈-reflexive; ≈-sym; ≈-trans)
from-cong
: {H H′ : Hypergraph′ v}
→ Hypergraph-same H H′
→ ≈-Hypergraph (from H) (from H′)
from-cong {H} {H′} ≈H = record
{ ≡sorted = ≡sorted
}
where
module H = Hypergraph′ H
module H′ = Hypergraph′ H′
open Hypergraph′
open Hypergraph-same ≈H using (↔h; ≗a; ≗l; ≗j; inverseˡ) renaming (from to f; to to t)
asEdge : (H : Hypergraph′ v) → Fin (h H) → Edge v
asEdge H e = record { label = l H e ; ports = toVec (j H e) }
to-from : (e : Fin H′.h) → t (f e) ≡ e
to-from e = inverseˡ ≡.refl
a∘to-from : (e : Fin H′.h) → H′.a (t (f e)) ≡ H′.a e
a∘to-from = ≡.cong H′.a ∘ to-from
≗a′ : (e : Fin H′.h) → H.a (f e) ≡ H′.a e
≗a′ e = ≡.trans (≗a (f e)) (a∘to-from e)
l≗ : (e : Fin H.h) → cast-gate (≗a e) (H.l e) ≡ H′.l (t e)
l≗ e = ≈-sym (≡.sym (≗l e))
l∘to-from : (e : Fin H′.h) → cast-gate (a∘to-from e) (H′.l (t (f e))) ≡ H′.l e
l∘to-from e rewrite to-from e = ≈-reflexive ≡.refl
≗l′ : (e : Fin H′.h) → cast-gate (≗a′ e) (H.l (f e)) ≡ H′.l e
≗l′ e = ≈-trans {H.a _} (l≗ (f e)) (l∘to-from e)
import Data.Vec.Relation.Binary.Equality.Cast as VecCast
open import Data.Vec using (cast) renaming (map to vecmap)
open import Data.Vec.Properties using (tabulate-cong; tabulate-∘; map-cast)
open import Data.Fin.Base using () renaming (cast to fincast)
open import Data.Fin.Properties using () renaming (cast-trans to fincast-trans; cast-is-id to fincast-is-id)
j∘to-from
: (e : Fin H′.h) (i : Fin (H′.a (t (f e))))
→ H′.j (t (f e)) i
≡ H′.j e (fincast (a∘to-from e) i)
j∘to-from e i rewrite to-from e = ≡.cong (H′.j e) (≡.sym (fincast-is-id ≡.refl i))
open ≡.≡-Reasoning
≗j′ : (e : Fin H′.h) (i : Fin (H.a (f e))) → H.j (f e) i ≡ H′.j e (fincast (≗a′ e) i)
≗j′ e i = begin
H.j (f e) i ≡⟨ ≗j (f e) i ⟩
H′.j (t (f e)) (fincast _ i) ≡⟨ j∘to-from e (fincast _ i) ⟩
H′.j e (fincast (a∘to-from e) (fincast _ i)) ≡⟨ ≡.cong (H′.j e) (fincast-trans (≗a (f e)) _ i) ⟩
H′.j e (fincast (≗a′ e) i) ∎
cast-toVec
: {n m : ℕ}
{A : Set}
(m≡n : m ≡ n)
(f : Fin n → A)
→ cast m≡n (toVec (f ∘ fincast m≡n)) ≡ toVec f
cast-toVec m≡n f rewrite m≡n = begin
cast _ (toVec (f ∘ (fincast _))) ≡⟨ VecCast.cast-is-id ≡.refl (toVec (f ∘ fincast ≡.refl)) ⟩
toVec (f ∘ fincast _) ≡⟨ tabulate-∘ f (fincast ≡.refl) ⟩
vecmap f (toVec (fincast _)) ≡⟨ ≡.cong (vecmap f) (tabulate-cong (fincast-is-id ≡.refl)) ⟩
vecmap f (toVec id) ≡⟨ tabulate-∘ f id ⟨
toVec f ∎
≗p′ : (e : Fin H′.h) → cast (≗a′ e) (toVec (H.j (f e))) ≡ toVec (H′.j e)
≗p′ e = begin
cast (≗a′ e) (toVec (H.j (f e))) ≡⟨ ≡.cong (cast (≗a′ e)) (tabulate-cong (≗j′ e)) ⟩
cast _ (toVec (H′.j e ∘ fincast _)) ≡⟨ cast-toVec (≗a′ e) (H′.j e) ⟩
toVec (H′.j e) ∎
H∘ρ≗H′ : (e : Fin H′.h) → asEdge H (flip ↔h Data.Fin.Permutation.⟨$⟩ʳ e) ≡ asEdge H′ e
H∘ρ≗H′ e = ≈-Edge⇒≡ record
{ ≡arity = ≗a′ e
; ≡label = ≗l′ e
; ≡ports = ≗p′ e
}
≡sorted : sortHypergraph (from H) ≡ sortHypergraph (from H′)
≡sorted =
≡.cong (λ x → record { edges = x } ) $
Pointwise-≡⇒≡ $
map ≈-Edge⇒≡ $
sorted-≋ $
toList-↭ $
flip ↔h , H∘ρ≗H′
equiv : (v : ℕ) → Equivalence (Hypergraph-Setoid v) (Hypergraph-Setoid′ v)
equiv v = record
{ to = to
; from = from
; to-cong = to-cong
; from-cong = from-cong
}
|
