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binders-disjointness.agda
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binders-disjointness.agda
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open import List
open import Nat
open import Prelude
open import contexts
open import core
open import freshness
open import patterns-core
-- this module contains various judgements which
-- state that two term do not share any binders, or
-- do not share any hole binders. to avoid duplication,
-- each judgement is generic in its second argument.
-- we insert these assumptions as needed to follow
-- Barendregt's convention
module binders-disjointness where
mutual
data binders-disjoint-p {T : Set} {{_ : UnboundIn T}} :
pattrn → T → Set where
BDPUnit : ∀{t} →
binders-disjoint-p unit t
BDPNum : ∀{n t} →
binders-disjoint-p (N n) t
BDPVar : ∀{x t} →
unbound-in x t →
binders-disjoint-p (X x) t
BDPInl : ∀{p t} →
binders-disjoint-p p t →
binders-disjoint-p (inl p) t
BDPInr : ∀{p t} →
binders-disjoint-p p t →
binders-disjoint-p (inr p) t
BDPPair : ∀{p1 p2 t} →
binders-disjoint-p p1 t →
binders-disjoint-p p2 t →
binders-disjoint-p ⟨ p1 , p2 ⟩ t
BDPWild : ∀{t} →
binders-disjoint-p wild t
BDPEHole : ∀{w t} →
binders-disjoint-p ⦇-⦈[ w ] t
BDPHole : ∀{p w τ t} →
binders-disjoint-p p t →
binders-disjoint-p ⦇⌜ p ⌟⦈[ w , τ ] t
data binders-disjoint-r {T : Set} {{_ : UnboundIn T}} :
rule → T → Set where
BDRule : ∀{p e t} →
binders-disjoint-p p t →
binders-disjoint e t →
binders-disjoint-r (p => e) t
data binders-disjoint-rs {T : Set} {{_ : UnboundIn T}} :
rules → T → Set where
BDNoRules : ∀{t} →
binders-disjoint-rs nil t
BDRules : ∀{r rs t} →
binders-disjoint-r r t →
binders-disjoint-rs rs t →
binders-disjoint-rs (r / rs) t
data binders-disjoint-zrs {T : Set} {{_ : UnboundIn T}} :
zrules → T → Set where
BDZRules : ∀{rs-pre r rs-post t} →
binders-disjoint-rs rs-pre t →
binders-disjoint-rs (r / rs-post) t →
binders-disjoint-zrs (rs-pre / r / rs-post) t
data binders-disjoint-σ {T : Set} {{_ : UnboundIn T}} :
subst-env → T → Set where
BDσId : ∀{Γ t} →
binders-disjoint-σ (Id Γ) t
BDσSubst : ∀{d y σ t} →
binders-disjoint d t →
unbound-in y t →
binders-disjoint-σ σ t →
binders-disjoint-σ (Subst d y σ) t
data binders-disjoint {T : Set} {{_ : UnboundIn T}} :
ihexp → T → Set where
BDUnit : ∀{t} →
binders-disjoint unit t
BDNum : ∀{n t} →
binders-disjoint (N n) t
BDVar : ∀{x t} →
binders-disjoint (X x) t
BDLam : ∀{x τ e t} →
unbound-in x t →
binders-disjoint e t →
binders-disjoint (·λ x ·[ τ ] e) t
BDAp : ∀{e1 e2 t} →
binders-disjoint e1 t →
binders-disjoint e2 t →
binders-disjoint (e1 ∘ e2) t
BDInl : ∀{e τ t} →
binders-disjoint e t →
binders-disjoint (inl τ e) t
BDInr : ∀{e τ t} →
binders-disjoint e t →
binders-disjoint (inr τ e) t
BDMatch : ∀{e τ rs t} →
binders-disjoint e t →
binders-disjoint-zrs rs t →
binders-disjoint (match e ·: τ of rs) t
BDPair : ∀{e1 e2 t} →
binders-disjoint e1 t →
binders-disjoint e2 t →
binders-disjoint ⟨ e1 , e2 ⟩ t
BDFst : ∀{e t} →
binders-disjoint e t →
binders-disjoint (fst e) t
BDSnd : ∀{e t} →
binders-disjoint e t →
binders-disjoint (snd e) t
BDEHole : ∀{u σ t} →
binders-disjoint-σ σ t →
binders-disjoint ⦇-⦈⟨ u , σ ⟩ t
BDHole : ∀{e u σ t} →
binders-disjoint-σ σ t →
binders-disjoint e t →
binders-disjoint ⦇⌜ e ⌟⦈⟨ u , σ ⟩ t
mutual
data hole-binders-disjoint-p {T : Set} {{_ : HoleUnboundIn T}} :
pattrn → T → Set where
HBDPUnit : ∀{t} →
hole-binders-disjoint-p unit t
HBDPNum : ∀{n t} →
hole-binders-disjoint-p (N n) t
HBDPVar : ∀{x t} →
hole-binders-disjoint-p (X x) t
HBDPInl : ∀{p t} →
hole-binders-disjoint-p p t →
hole-binders-disjoint-p (inl p) t
HBDPInr : ∀{p t} →
hole-binders-disjoint-p p t →
hole-binders-disjoint-p (inr p) t
HBDPPair : ∀{p1 p2 t} →
hole-binders-disjoint-p p1 t →
hole-binders-disjoint-p p2 t →
hole-binders-disjoint-p ⟨ p1 , p2 ⟩ t
HBDPWild : ∀{t} →
hole-binders-disjoint-p wild t
HBDPEHole : ∀{w t} →
hole-unbound-in w t →
hole-binders-disjoint-p ⦇-⦈[ w ] t
HBDPHole : ∀{p w τ t} →
hole-unbound-in w t →
hole-binders-disjoint-p p t →
hole-binders-disjoint-p ⦇⌜ p ⌟⦈[ w , τ ] t
data hole-binders-disjoint-r {T : Set} {{_ : HoleUnboundIn T}} :
rule → T → Set where
HBDRule : ∀{p e t} →
hole-binders-disjoint-p p t →
hole-binders-disjoint e t →
hole-binders-disjoint-r (p => e) t
data hole-binders-disjoint-rs {T : Set} {{_ : HoleUnboundIn T}} :
rules → T → Set where
HBDNoRules : ∀{t} →
hole-binders-disjoint-rs nil t
HBDRules : ∀{r rs t} →
hole-binders-disjoint-r r t →
hole-binders-disjoint-rs rs t →
hole-binders-disjoint-rs (r / rs) t
data hole-binders-disjoint-zrs {T : Set} {{_ : HoleUnboundIn T}} :
zrules → T → Set where
HBDZRules : ∀{rs-pre r rs-post t} →
hole-binders-disjoint-rs rs-pre t →
hole-binders-disjoint-rs (r / rs-post) t →
hole-binders-disjoint-zrs (rs-pre / r / rs-post) t
data hole-binders-disjoint-σ {T : Set} {{_ : HoleUnboundIn T}} :
subst-env → T → Set where
HBDσId : ∀{Γ t} →
hole-binders-disjoint-σ (Id Γ) t
HBDσSubst : ∀{d y σ t} →
hole-binders-disjoint d t →
hole-binders-disjoint-σ σ t →
hole-binders-disjoint-σ (Subst d y σ) t
data hole-binders-disjoint {T : Set} {{_ : HoleUnboundIn T}} :
ihexp → T → Set where
HBDUnit : ∀{t} →
hole-binders-disjoint unit t
HBDNum : ∀{n t} →
hole-binders-disjoint (N n) t
HBDVar : ∀{x t} →
hole-binders-disjoint (X x) t
HBDLam : ∀{x τ e t} →
hole-binders-disjoint e t →
hole-binders-disjoint (·λ x ·[ τ ] e) t
HBDAp : ∀{e1 e2 t} →
hole-binders-disjoint e1 t →
hole-binders-disjoint e2 t →
hole-binders-disjoint (e1 ∘ e2) t
HBDInl : ∀{e τ t} →
hole-binders-disjoint e t →
hole-binders-disjoint (inl τ e) t
HBDInr : ∀{e τ t} →
hole-binders-disjoint e t →
hole-binders-disjoint (inr τ e) t
HBDMatch : ∀{e τ rs t} →
hole-binders-disjoint e t →
hole-binders-disjoint-zrs rs t →
hole-binders-disjoint (match e ·: τ of rs) t
HBDPair : ∀{e1 e2 t} →
hole-binders-disjoint e1 t →
hole-binders-disjoint e2 t →
hole-binders-disjoint ⟨ e1 , e2 ⟩ t
HBDFst : ∀{e t} →
hole-binders-disjoint e t →
hole-binders-disjoint (fst e) t
HBDSnd : ∀{e t} →
hole-binders-disjoint e t →
hole-binders-disjoint (snd e) t
HBDEHole : ∀{u σ t} →
hole-binders-disjoint-σ σ t →
hole-binders-disjoint ⦇-⦈⟨ u , σ ⟩ t
HBDHole : ∀{e u σ t} →
hole-binders-disjoint-σ σ t →
hole-binders-disjoint e t →
hole-binders-disjoint ⦇⌜ e ⌟⦈⟨ u , σ ⟩ t