progress
This commit is contained in:
Jeremy Siek
parent
1271286a22
commit
a03eb20083
3 changed files with 196 additions and 40 deletions
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@ -12,6 +12,7 @@ open import plfa.Untyped
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open import plfa.Adequacy
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open import plfa.Denotational
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open import plfa.Soundness
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open import extra.Substitution
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_; _≢_; refl; trans; sym; cong; cong₂; cong-app)
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@ -28,8 +29,6 @@ open import Relation.Nullary using (Dec; yes; no)
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open import Function using (_∘_)
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\end{code}
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## Logical Relation between CBN Closures and Terms
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\begin{code}
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@ -7,10 +7,11 @@ module extra.Confluence where
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\begin{code}
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open import extra.Substitution
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open import plfa.Untyped
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renaming (_—→_ to _——→_; _—↠_ to _——↠_; begin_ to start_; _∎ to _[])
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renaming (_—→_ to _——→_; _—↠_ to _——↠_; begin_ to commence_; _∎ to _fini)
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_; _≢_; refl; trans; sym; cong; cong₂; cong-app)
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{- open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _≡⟨_⟩_; _∎) -}
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open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _≡⟨_⟩_; _∎)
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open import Function using (_∘_)
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\end{code}
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## Reduction without the restrictions
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@ -42,13 +43,13 @@ data _—→_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
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\begin{code}
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infix 2 _—↠_
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infix 1 begin_
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infix 1 start_
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infixr 2 _—→⟨_⟩_
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infix 3 _∎
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infix 3 _[]
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data _—↠_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
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_∎ : ∀ {Γ A} (M : Γ ⊢ A)
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_[] : ∀ {Γ A} (M : Γ ⊢ A)
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--------
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→ M —↠ M
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@ -58,11 +59,11 @@ data _—↠_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
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---------
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→ L —↠ N
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begin_ : ∀ {Γ} {A} {M N : Γ ⊢ A}
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start_ : ∀ {Γ} {A} {M N : Γ ⊢ A}
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→ M —↠ N
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------
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→ M —↠ N
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begin M—↠N = M—↠N
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start M—↠N = M—↠N
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\end{code}
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\begin{code}
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@ -70,7 +71,7 @@ begin M—↠N = M—↠N
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→ L —↠ M
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→ M —↠ N
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→ L —↠ N
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—↠-trans (M ∎) mn = mn
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—↠-trans (M []) mn = mn
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—↠-trans (L —→⟨ r ⟩ lm) mn = L —→⟨ r ⟩ (—↠-trans lm mn)
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\end{code}
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@ -81,21 +82,21 @@ abs-cong : ∀ {Γ} {N N' : Γ , ★ ⊢ ★}
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→ N —↠ N'
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----------
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→ ƛ N —↠ ƛ N'
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abs-cong (M ∎) = ƛ M ∎
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abs-cong (M []) = ƛ M []
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abs-cong (L —→⟨ r ⟩ rs) = ƛ L —→⟨ ζ r ⟩ abs-cong rs
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appL-cong : ∀ {Γ} {L L' M : Γ ⊢ ★}
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→ L —↠ L'
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---------------
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→ L · M —↠ L' · M
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appL-cong {Γ}{L}{L'}{M} (L ∎) = L · M ∎
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appL-cong {Γ}{L}{L'}{M} (L []) = L · M []
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appL-cong {Γ}{L}{L'}{M} (L —→⟨ r ⟩ rs) = L · M —→⟨ ξ₁ r ⟩ appL-cong rs
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appR-cong : ∀ {Γ} {L M M' : Γ ⊢ ★}
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→ M —↠ M'
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---------------
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→ L · M —↠ L · M'
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appR-cong {Γ}{L}{M}{M'} (M ∎) = L · M ∎
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appR-cong {Γ}{L}{M}{M'} (M []) = L · M []
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appR-cong {Γ}{L}{M}{M'} (M —→⟨ r ⟩ rs) = L · M —→⟨ ξ₂ r ⟩ appR-cong rs
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\end{code}
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@ -155,7 +156,7 @@ par-beta : ∀{Γ A}{M N : Γ ⊢ A}
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→ M ⇒ N
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------
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→ M —↠ N
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par-beta {Γ} {A} {.(` _)} (pvar{x = x}) = (` x) ∎
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par-beta {Γ} {A} {.(` _)} (pvar{x = x}) = (` x) []
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par-beta {Γ} {★} {ƛ N} (pabs p) =
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abs-cong (par-beta p)
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par-beta {Γ} {★} {L · M} (papp p₁ p₂) =
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@ -167,7 +168,7 @@ par-beta {Γ} {★} {(ƛ N) · M} (pbeta{N' = N'}{M' = M'} p₁ p₂) =
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a = appL-cong{M = M} (abs-cong ih₁) in
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let b : (ƛ N') · M —↠ (ƛ N') · M'
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b = appR-cong{L = ƛ N'} ih₂ in
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let c = (ƛ N') · M' —→⟨ β ⟩ N' [ M' ] ∎ in
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let c = (ƛ N') · M' —→⟨ β ⟩ N' [ M' ] [] in
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—↠-trans (—↠-trans a b) c
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\end{code}
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@ -190,9 +191,6 @@ par-subst {Γ}{Δ} σ₁ σ₂ = ∀{A}{x : Γ ∋ A} → σ₁ x ⇒ σ₂ x
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rename-subst-commute : ∀{Γ Δ}{N : Γ , ★ ⊢ ★}{M : Γ ⊢ ★}{ρ : Rename Γ Δ }
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→ (rename (ext ρ) N) [ rename ρ M ] ≡ rename ρ (N [ M ])
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rename-subst-commute {Γ}{Δ}{N}{M}{ρ} =
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{-
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let x = commute-subst-rename{σ = subst-zero M}{ρ = ρ} ? in
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-}
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{!!}
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\end{code}
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@ -210,19 +208,178 @@ par-rename {Γ}{Δ}{A}{ρ} (pbeta{Γ}{N}{N'}{M}{M'} p₁ p₂)
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\end{code}
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\begin{code}
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subst-par : ∀{Γ Δ A} {σ₁ σ₂ : Subst Γ Δ} {M M' : Γ ⊢ A}
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→ par-subst σ₁ σ₂ → M ⇒ M'
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--------------------------
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→ subst σ₁ M ⇒ subst σ₂ M'
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subst-par {Γ} {Δ} {A} {σ₁} {σ₂} {` x} s pvar = s
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subst-par {Γ} {Δ} {★} {σ₁} {σ₂} {ƛ N} s (pabs p) =
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let ih = subst-par {σ₁ = exts σ₁}{σ₂ = exts σ₂} {!!} p in
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pabs {!!}
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where
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H : par-subst (exts σ₁ {B = ★}) (exts σ₂)
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H {★} {Z} = pvar
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H {A} {S x} = {!!}
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subst-par {Γ} {Δ} {★} {σ₁} {σ₂} {L · M} s (papp p p₁) = {!!}
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subst-par {Γ} {Δ} {★} {σ₁} {σ₂} {(ƛ N) · M} s (pbeta p p₁) = {!!}
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par-subst-ext : ∀{Γ Δ} {σ τ : Subst Γ Δ}
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→ par-subst σ τ
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→ par-subst (exts σ {B = ★}) (exts τ)
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par-subst-ext s {x = Z} = pvar
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par-subst-ext s {x = S x} = par-rename s
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\end{code}
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\begin{code}
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ids : ∀{Γ} → Subst Γ Γ
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ids {A} x = ` x
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cons : ∀{Γ Δ A} → (Δ ⊢ A) → Subst Γ Δ → Subst (Γ , A) Δ
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cons {Γ} {Δ} {A} M σ {B} Z = M
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cons {Γ} {Δ} {A} M σ {B} (S x) = σ x
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seq : ∀{Γ Δ Σ} → Subst Γ Δ → Subst Δ Σ → Subst Γ Σ
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seq σ τ = (subst τ) ∘ σ
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ren : ∀{Γ Δ} → Rename Γ Δ → Subst Γ Δ
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ren ρ = ids ∘ ρ
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ren-ext : ∀ {Γ Δ}{B C : Type} {ρ : Rename Γ Δ} {x : Γ , B ∋ C}
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→ (ren (ext ρ)) x ≡ exts (ren ρ) x
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ren-ext {x = Z} = refl
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ren-ext {x = S x} = refl
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rename-seq-ren : ∀ {Γ Δ}{A} {ρ : Rename Γ Δ}{M : Γ ⊢ A}
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→ rename ρ M ≡ subst (ren ρ) M
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rename-seq-ren {M = ` x} = refl
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rename-seq-ren {ρ = ρ}{M = ƛ N} =
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cong ƛ_ G
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where IH : rename (ext ρ) N ≡ subst (ren (ext ρ)) N
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IH = rename-seq-ren {ρ = ext ρ}{M = N}
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G : rename (ext ρ) N ≡ subst (exts (ren ρ)) N
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G =
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begin
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rename (ext ρ) N
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≡⟨ IH ⟩
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subst (ren (ext ρ)) N
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≡⟨ subst-equal {M = N} ren-ext ⟩
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subst (exts (ren ρ)) N
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∎
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rename-seq-ren {M = L · M} = cong₂ _·_ rename-seq-ren rename-seq-ren
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exts-cons-shift : ∀{Γ Δ : Context} {A : Type}{B : Type} {σ : Subst Γ Δ}{x}
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→ exts σ {A}{B} x ≡ cons (` Z) (seq σ (ren S_)) x
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exts-cons-shift {x = Z} = refl
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exts-cons-shift {x = S x} = rename-seq-ren
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subst-cons-Z : ∀{Γ Δ : Context}{A : Type}{M : Δ ⊢ A}{σ : Subst Γ Δ}
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→ subst (cons M σ) (` Z) ≡ M
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subst-cons-Z = refl
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seq-inc-cons : ∀{Γ Δ : Context} {A B : Type} {M : Δ ⊢ A} {σ : Subst Γ Δ}
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{x : Γ ∋ B}
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→ seq (ren S_) (cons M σ) x ≡ σ x
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seq-inc-cons = refl
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ids-id : ∀{Γ : Context}{A : Type} {M : Γ ⊢ A}
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→ subst ids M ≡ M
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ids-id = subst-id λ {A} {x} → refl
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cons-ext : ∀{Γ Δ : Context} {A B : Type} {σ : Subst (Γ , A) Δ} {x : Γ , A ∋ B}
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→ cons (subst σ (` Z)) (seq (ren S_) σ) x ≡ σ x
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cons-ext {x = Z} = refl
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cons-ext {x = S x} = refl
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id-seq : ∀{Γ Δ : Context} {B : Type} {σ : Subst Γ Δ} {x : Γ ∋ B}
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→ (seq ids σ) x ≡ σ x
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id-seq = refl
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seq-id : ∀{Γ Δ : Context} {B : Type} {σ : Subst Γ Δ} {x : Γ ∋ B}
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→ (seq σ ids) x ≡ σ x
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seq-id {Γ}{σ = σ}{x = x} =
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begin
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(seq σ ids) x
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≡⟨⟩
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subst ids (σ x)
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≡⟨ ids-id ⟩
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σ x
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∎
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seq-assoc : ∀{Γ Δ Σ Ψ : Context}{B} {σ : Subst Γ Δ} {τ : Subst Δ Σ}
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{θ : Subst Σ Ψ} {x : Γ ∋ B}
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→ seq (seq σ τ) θ x ≡ seq σ (seq τ θ) x
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seq-assoc{Γ}{Δ}{Σ}{Ψ}{B}{σ}{τ}{θ}{x} =
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begin
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seq (seq σ τ) θ x
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≡⟨⟩
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subst θ (subst τ (σ x))
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≡⟨ subst-subst{M = σ x} ⟩
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(subst ((subst θ) ∘ τ)) (σ x)
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≡⟨⟩
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seq σ (seq τ θ) x
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∎
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seq-cons : ∀{Γ Δ Σ : Context} {A B} {σ : Subst Γ Δ} {τ : Subst Δ Σ}
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{M : Δ ⊢ A} {x : Γ , A ∋ B}
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→ seq (cons M σ) τ x ≡ cons (subst τ M) (seq σ τ) x
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seq-cons {x = Z} = refl
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seq-cons {x = S x} = refl
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cons-zero-S : ∀{Γ}{A B}{x : Γ , A ∋ B} → cons (` Z) (ren S_) x ≡ ids x
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cons-zero-S {x = Z} = refl
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cons-zero-S {x = S x} = refl
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\end{code}
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\begin{code}
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subst-zero-cons-ids : ∀{Γ}{A B : Type}{M : Γ ⊢ B}{x : Γ , B ∋ A}
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→ subst-zero M x ≡ cons M ids x
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subst-zero-cons-ids {x = Z} = refl
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subst-zero-cons-ids {x = S x} = refl
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\end{code}
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\begin{code}
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subst-commute : ∀{Γ Δ}{N : Γ , ★ ⊢ ★}{M : Γ ⊢ ★}{σ : Subst Γ Δ }
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→ (subst (exts σ) N) [ subst σ M ] ≡ subst σ (N [ M ])
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subst-commute {Γ}{Δ}{N}{M}{σ} =
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begin
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(subst (exts σ) N) [ subst σ M ]
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≡⟨⟩
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subst (subst-zero (subst σ M)) (subst (exts σ) N)
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≡⟨ subst-equal{M = subst (exts σ) N}
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(λ {A}{x} → subst-zero-cons-ids{A = A}{x = x}) ⟩
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subst (cons (subst σ M) ids) (subst (exts σ) N)
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≡⟨ subst-subst{M = N} ⟩
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subst (seq (exts σ) (cons (subst σ M) ids)) N
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≡⟨ {!!} ⟩
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subst (seq (cons (` Z) (seq σ (ren S_))) (cons (subst σ M) ids)) N
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≡⟨ {!!} ⟩
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subst (cons (subst (cons (subst σ M) ids) (` Z))
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(seq (seq σ (ren S_)) (cons (subst σ M) ids))) N
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≡⟨ {!!} ⟩
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subst (cons (subst σ M)
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(seq (seq σ (ren S_)) (cons (subst σ M) ids))) N
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≡⟨ {!!} ⟩
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subst (cons (subst σ M)
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(seq σ (seq (ren S_) (cons (subst σ M) ids)))) N
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≡⟨ {!!} ⟩
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subst (cons (subst σ M) (seq σ ids)) N
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≡⟨ {!!} ⟩
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subst (cons (subst σ M) σ) N
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≡⟨ {!!} ⟩
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subst (cons (subst σ M) (seq ids σ)) N
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≡⟨ {!!} ⟩
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subst (seq (cons M ids) σ) N
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≡⟨ sym (subst-subst{M = N}) ⟩
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subst σ (subst (cons M ids) N)
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≡⟨ {!!} ⟩
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subst σ (N [ M ])
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∎
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\end{code}
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\begin{code}
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subst-par : ∀{Γ Δ A} {σ τ : Subst Γ Δ} {M M' : Γ ⊢ A}
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→ par-subst σ τ → M ⇒ M'
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--------------------------
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→ subst σ M ⇒ subst τ M'
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subst-par {Γ} {Δ} {A} {σ} {τ} {` x} s pvar = s
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subst-par {Γ} {Δ} {★} {σ} {τ} {ƛ N} s (pabs p) =
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pabs (subst-par {σ = exts σ} {τ = exts τ}
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(λ {A}{x} → par-subst-ext s {A}{x}) p)
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subst-par {Γ} {Δ} {★} {σ} {τ} {L · M} s (papp p₁ p₂) =
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papp (subst-par s p₁) (subst-par s p₂)
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subst-par {Γ} {Δ} {★} {σ} {τ} {(ƛ N) · M} s (pbeta{N' = N'}{M' = M'} p₁ p₂)
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with pbeta (subst-par{σ = exts σ}{τ = exts τ}{M = N}
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(λ {A}{x} → par-subst-ext s {A}{x}) p₁)
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(subst-par (λ {A}{x} → s{A}{x}) p₂)
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... | G rewrite subst-commute{N = N'}{M = M'}{σ = τ} =
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G
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\end{code}
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@ -6,7 +6,7 @@ module extra.Substitution where
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\begin{code}
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open import plfa.Untyped
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using (Context; _⊢_; ★; _∋_; ∅; _,_; Z; S_; `_; ƛ_; _·_; rename; subst;
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using (Type; Context; _⊢_; ★; _∋_; ∅; _,_; Z; S_; `_; ƛ_; _·_; rename; subst;
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ext; exts; _[_]; subst-zero)
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renaming (_∎ to _[])
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open import plfa.Denotational using (Rename)
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@ -152,10 +152,10 @@ subst-exts {A = ★}{x = S x}{σ₁}{σ₂} = G
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\begin{code}
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subst-subst : ∀{Γ Δ Σ}{M : Γ ⊢ ★} {σ₁ : Subst Γ Δ}{σ₂ : Subst Δ Σ}
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subst-subst : ∀{Γ Δ Σ}{A}{M : Γ ⊢ A} {σ₁ : Subst Γ Δ}{σ₂ : Subst Δ Σ}
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→ ((subst σ₂) ∘ (subst σ₁)) M ≡ subst (subst σ₂ ∘ σ₁) M
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subst-subst {M = ` x} = refl
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subst-subst {Γ}{Δ}{Σ}{ƛ N}{σ₁}{σ₂} = G
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subst-subst {Γ}{Δ}{Σ}{A}{ƛ N}{σ₁}{σ₂} = G
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where
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G : ((subst σ₂) ∘ subst σ₁) (ƛ N) ≡ (ƛ subst (exts ((subst σ₂) ∘ σ₁)) N)
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G =
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@ -205,15 +205,15 @@ rename-subst {M = L · M} =
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\begin{code}
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is-id-subst : ∀{Γ} → Subst Γ Γ → Set
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is-id-subst {Γ} σ = ∀{x : Γ ∋ ★} → σ x ≡ ` x
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is-id-subst {Γ} σ = ∀{A}{x : Γ ∋ A} → σ x ≡ ` x
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is-id-exts : ∀{Γ} {σ : Subst Γ Γ}
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→ is-id-subst σ
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→ is-id-subst (exts σ {B = ★})
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is-id-exts id {Z} = refl
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is-id-exts{Γ}{σ} id {S x} rewrite id {x} = refl
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||||
is-id-exts id {x = Z} = refl
|
||||
is-id-exts{Γ}{σ} id {x = S x} rewrite id {x = x} = refl
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subst-id : ∀{Γ} {M : Γ ⊢ ★} {σ : Subst Γ Γ}
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subst-id : ∀{Γ : Context}{A : Type} {M : Γ ⊢ A} {σ : Subst Γ Γ}
|
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→ is-id-subst σ
|
||||
→ subst σ M ≡ M
|
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subst-id {M = ` x} {σ} id = id
|
||||
|
|
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Reference in a new issue