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mltt.lagda
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\begin{code}
{-# OPTIONS --rewriting #-}
{-# OPTIONS --guardedness #-}
open import Level using (Level ; 0ℓ ; Lift ; lift ; lower) renaming (suc to lsuc)
open import Agda.Builtin.Bool
open import Data.Bool using (Bool ; _∧_ ; _∨_)
open import Data.Nat using (s≤s) renaming (_<_ to _<ℕ_ ; _≤_ to _≤ℕ_)
open import Data.Nat.Properties
open import Agda.Builtin.Nat
open import Data.Fin using (Fin ; toℕ)
open import Data.Fin.Properties using (toℕ<n)
open import Agda.Builtin.Equality renaming (_≡_ to _≣_)
open import Agda.Builtin.Sigma renaming (fst to π₁ ; snd to π₂)
open import Data.List.Relation.Binary.Subset.Propositional
open import Data.List.Relation.Binary.Subset.Propositional.Properties
using (⊆-refl ; ⊆-trans ; xs⊆x∷xs)
open import Relation.Binary.PropositionalEquality
using (cong ; cong₂ ; subst₂) renaming (trans to ≣trans ; sym to ≣sym ; subst to ≣subst)
open import Data.List using () renaming ([] to nil ; _∷_ to cons)
open import Data.List.Relation.Unary.Any
open import Data.List.Properties
open import Data.Product
open import Data.Empty
open import Data.List.Membership.Propositional
open import Data.List.Membership.Propositional.Properties
open import Data.Sum
open import Relation.Nullary
-- MLTT imports
open import Tools.Nat using (1+)
open import Definition.Untyped hiding (_∷_)
open import Definition.Untyped.Properties using (wk-β ; wk1-sgSubst ; subst-wk)
open import Definition.Typed
--open import Definition.Typed.Properties using (subset*Term ; noNe)
open import Definition.Typed.Weakening renaming (wk to wk⊢)
open import Definition.Typed.Consequences.Substitution using (substType ; substTerm)
--open import Definition.Typed.Consequences.Syntactic using (syntacticEq)
--open import Definition.Typed.Consequences.Canonicity using (sucᵏ)
open import Definition.Typed.EqRelInstance
open import Definition.LogicalRelation --using (Natural-prop)
-- BoxTT imports
open import calculus renaming (Term to BTerm)
open import terms -- renaming (Term to BTerm)
open import util
open import world
open import mod
open import encode
open import choice
open import compatible
open import progress
open import getChoice
open import choiceExt
open import newChoice
module mltt {L : Level}
(W : PossibleWorlds {L})
(M : Mod W)
(C : Choice)
(K : Compatible {L} W C)
(G : GetChoice {L} W C K)
(X : ChoiceExt W C)
(N : NewChoice W C K G)
(EC : Encode)
where
open import worldDef(W)
open import computation(W)(C)(K)(G)(X)(N)(EC)
using (#⇛!sameℕ ; _⇛!_at_ ; _⇓!_at_ ; _#⇛!_at_ ; #⇛!-trans ; ⇛!-trans ; #⇛!-refl)
open import terms2(W)(C)(K)(G)(X)(N)(EC)
using (→∧≡true)
open import terms8(W)(C)(K)(G)(X)(N)(EC)
using (⇓NUM→SUC⇓NUM ; #APPLY2 ; #FST ; #SND ; SUM! ; #SUM! ; #⇛!-FST-PAIR ; #⇛!-SND-PAIR)
open import subst(W)(C)(K)(G)(X)(N)(EC)
open import forcing(W)(M)(C)(K)(G)(X)(N)(EC)
open import props0(W)(M)(C)(K)(G)(X)(N)(EC)
using (eqTypes-mon)
open import props1(W)(M)(C)(K)(G)(X)(N)(EC)
using (TSext-equalTypes-equalInType ; TEQsym-equalTypes ; TEQrefl-equalTypes ; TEQtrans-equalTypes)
open import props2(W)(M)(C)(K)(G)(X)(N)(EC)
using (isTypeNAT! ; eqTypesUniv ; equalTypes→equalInType-UNIV ; equalInType→equalTypes-aux ; eqTypesPI← ; eqTypesSUM← ;
≡CTerm→eqTypes ; ≡CTerm→equalInType ; eqTypesFALSE ; eqTypesTRUE ; ¬equalInType-FALSE ; NUM-equalInType-NAT! ;
equalInType-NAT!→ ; equalInType-local ; equalInType-mon ; equalInType-PI→ ; equalInType-PI ; isFam ;
equalInType-FUN→ ; equalInType-refl ; equalInType-sym ; equalInType-SUM→ ; eqTypesEQ← ; equalInType-EQ)
open import props3(W)(M)(C)(K)(G)(X)(N)(EC)
using (→equalInType-TRUE ; equalInType-EQ→₁)
open import props4(W)(M)(C)(K)(G)(X)(N)(EC)
using (→equalInType-NAT!)
open import props5(W)(M)(C)(K)(G)(X)(N)(EC)
using (≡→equalInType ; eqTypesEQ→ᵣ)
open import props6(W)(M)(C)(K)(G)(X)(N)(EC)
using (_#⇛ₚ_at_ ; equalInType-#⇛ₚ-left-right-rev ; presPure ; →presPure-NATREC₁ ; →presPure-NATREC₂ ; →presPure-NATREC₃ ;
equalTypesPI→ₗ ; equalTypesPI→ᵣ ; eqTypesSUM!← ; SUMeq! ; equalInType-SUM!→ ; equalInType-SUM!)
open import uniMon(W)(M)(C)(K)(G)(X)(N)(EC)
using (equalTypes-uni-mon ; equalInType-uni-mon)
open import sequent(W)(M)(C)(K)(G)(X)(N)(EC)
open import sequent2(W)(M)(C)(K)(G)(X)(N)(EC)
using (valid∈𝕎→valid≡𝕎-UNIV ; valid≡𝕎-sym ; valid≡𝕎-trans ; valid≡𝕎-PI ; valid≡𝕎-SUM! ; valid∈𝕎-mon ; valid≡𝕎-mon ;
valid∈𝕎→valid≡𝕎 ; valid∈-UNIV ; valid≡𝕎→valid∈𝕎ₗ)
open import sequent3(W)(M)(C)(K)(G)(X)(N)(EC)
using (valid∈-PI ; valid∈-SUM! ; valid∈-NAT! ; valid∈-FALSE ; valid∈-UNIT ; valid∈LAMBDA ; valid∈APPLY ; valid∈N0-NAT ;
valid∈SUC-NAT ; valid∈-FALSE→ ; valid∈-AX-UNIT ; valid∈-change-type ; valid≡-change-type ;
valid≡APPLY ; valid≡LAMBDA ; valid≡SUC-NAT ; valid≡-FALSE→ ; valid≡-UNIT)
open import sequent4(W)(M)(C)(K)(G)(X)(N)(EC)
using (valid∈FST ; valid∈SND ; valid∈PAIR ; valid≡FST ; valid≡SND ; valid≡FST-PAIR ; valid≡SND-PAIR ; valid≡PI-ETA)
open import sequent5(W)(M)(C)(K)(G)(X)(N)(EC)
using (valid≡SUM!-ETA ; valid≡NATREC0 ; valid∈NATREC ; valid≡NATREC ; valid≡NATREC-SUC)
∈→ℕ : {n : Nat} {x : Fin n} {A : Term n} {Γ : Con Term n}
→ x ∷ A ∈ Γ
→ Nat
∈→ℕ {.(Nat.suc _)} {.Fin.zero} {.(wk1 _)} {.(_ ∙ _)} here = 0
∈→ℕ {.(Nat.suc _)} {.(Fin.suc _)} {.(wk1 _)} {.(_ ∙ _)} (there i) = Nat.suc (∈→ℕ i)
⟦_⟧T : {n : Nat} {Γ : Con Term n} {σ : Term n}
→ Γ ⊢ σ
→ BTerm
⟦_⟧T {n} {Γ} {.U} (Uⱼ x) = UNIV 1
⟦_⟧T {n} {Γ} {.ℕ} (ℕⱼ x) = NAT!
⟦_⟧T {n} {Γ} {.Empty} (Emptyⱼ x) = FALSE
⟦_⟧T {n} {Γ} {.Unit} (Unitⱼ x) = UNIT
⟦_⟧T {n} {Γ} {.(Π _ ▹ _)} ((Πⱼ_▹_) {F} {G} i j) = PI ⟦ i ⟧T ⟦ j ⟧T
⟦_⟧T {n} {Γ} {.(Σ _ ▹ _)} ((Σⱼ_▹_) {F} {G} i j) = SUM! ⟦ i ⟧T ⟦ j ⟧T
⟦_⟧T {n} {Γ} {σ} (univ x) = UNIV 1
∈→⊢ : {n : Nat} {Γ : Con Term n} {x : Fin n} {σ : Term n}
→ ⊢ Γ
→ x ∷ σ ∈ Γ
→ Γ ⊢ σ
∈→⊢ {Nat.suc n} {Γ ∙ A} {.Fin.zero} {.(wk1 _)} (i ∙ x₁) here = wk⊢ (step id) (i ∙ x₁) x₁
∈→⊢ {Nat.suc n} {Γ ∙ B} {Fin.suc k} {.(wk1 _)} (i ∙ x₁) (there {n} {k} {A} j) = wk⊢ (step id) (i ∙ x₁) z
where
z : Γ ⊢ A
z = ∈→⊢ i j
mutual
⊢Π[] : {n : Nat} {Γ : Con Term n} {G : Term (1+ n)} {F a : Term n}
→ Γ ⊢ Π F ▹ G
→ Γ ⊢ a ∷ F
→ Γ ⊢ (G [ a ])
⊢Π[] {n} {Γ} {G} {F} {a} (Πⱼ i ▹ i₁) j = substType i₁ j
⊢Π[] {n} {Γ} {G} {F} {a} (univ x) j = ⊢Π∷[] x j
⊢Π∷[] : {n : Nat} {Γ : Con Term n} {G : Term (1+ n)} {F a A : Term n}
→ Γ ⊢ Π F ▹ G ∷ A
→ Γ ⊢ a ∷ F
→ Γ ⊢ (G [ a ])
⊢Π∷[] {n} {Γ} {G} {F} {a} {.U} (Πⱼ i ▹ i₁) j = univ (substTerm i₁ j)
⊢Π∷[] {n} {Γ} {G} {F} {a} {A} (conv i x) j = ⊢Π∷[] i j
mutual
⊢Σ[] : {n : Nat} {Γ : Con Term n} {G : Term (1+ n)} {F a : Term n}
→ Γ ⊢ Σ F ▹ G
→ Γ ⊢ a ∷ F
→ Γ ⊢ (G [ a ])
⊢Σ[] {n} {Γ} {G} {F} {a} (Σⱼ i ▹ i₁) j = substType i₁ j
⊢Σ[] {n} {Γ} {G} {F} {a} (univ x) j = ⊢Σ∷[] x j
⊢Σ∷[] : {n : Nat} {Γ : Con Term n} {G : Term (1+ n)} {F a A : Term n}
→ Γ ⊢ Σ F ▹ G ∷ A
→ Γ ⊢ a ∷ F
→ Γ ⊢ (G [ a ])
⊢Σ∷[] {n} {Γ} {G} {F} {a} {.U} (Σⱼ i ▹ i₁) j = univ (substTerm i₁ j)
⊢Σ∷[] {n} {Γ} {G} {F} {a} {A} (conv i x) j = ⊢Σ∷[] i j
mutual
→▹▹[]ᵣ : {n : Nat} {Γ : Con Term n} {a F G : Term n}
→ Γ ⊢ a ∷ F
→ Γ ⊢ F ▹▹ G
→ Γ ⊢ G
→▹▹[]ᵣ {n} {Γ} {a} {F} {G} j (Πⱼ i ▹ i₁) = ≣subst (λ x → Γ ⊢ x) (wk1-sgSubst G a) z
where
z : Γ ⊢ (wk1 G [ a ])
z = substType i₁ j
→▹▹[]ᵣ {n} {Γ} {a} {F} {G} j (univ x) = →▹▹∷[]ᵣ j x
→▹▹∷[]ᵣ : {n : Nat} {Γ : Con Term n} {a F G A : Term n}
→ Γ ⊢ a ∷ F
→ Γ ⊢ F ▹▹ G ∷ A
→ Γ ⊢ G
→▹▹∷[]ᵣ {n} {Γ} {a} {F} {G} j (Πⱼ i ▹ i₁) = ≣subst (λ x → Γ ⊢ x) (wk1-sgSubst G a) z
where
z : Γ ⊢ (wk1 G [ a ])
z = univ (substTerm i₁ j)
→▹▹∷[]ᵣ {n} {Γ} {a} {F} {G} j (conv i x) = →▹▹∷[]ᵣ j i
≣liftSubst : {m n : Nat} {σ τ : Subst m n}
→ ((x : Fin n) → σ x ≣ τ x)
→ (x : Fin (1+ n)) → liftSubst σ x ≣ liftSubst τ x
≣liftSubst {m} {n} {σ} {τ} i Fin.zero = refl
≣liftSubst {m} {n} {σ} {τ} i (Fin.suc x) = cong wk1 (i x)
≣liftSubstn : {m n b : Nat} {σ τ : Subst m n}
→ ((x : Fin n) → σ x ≣ τ x)
→ (x : Fin (b + n)) → liftSubstn σ b x ≣ liftSubstn τ b x
≣liftSubstn {m} {n} {Nat.zero} {σ} {τ} i x = i x
≣liftSubstn {m} {n} {1+ b} {σ} {τ} i x = ≣liftSubst (≣liftSubstn i) x
mutual
subst-eta : {m n : Nat} {σ τ : Subst m n} {t : Term n}
→ ((x : Fin n) → σ x ≣ τ x)
→ subst σ t ≣ subst τ t
subst-eta {m} {n} {σ} {τ} {var x} i = i x
subst-eta {m} {n} {σ} {τ} {gen {bs} k c} i = cong (gen k) (subst-eta-gen i)
subst-eta-gen : {m n : Nat} {σ τ : Subst m n} {bs : Data.List.List Nat} {c : GenTs Term n bs}
→ ((x : Fin n) → σ x ≣ τ x)
→ substGen σ c ≣ substGen τ c
subst-eta-gen {m} {n} {σ} {τ} {.nil} {[]} i = refl
subst-eta-gen {m} {n} {σ} {τ} {cons _ _} {GenTs._∷_ {_} {b} t c} i =
cong₂ GenTs._∷_
(subst-eta {b + m} {b + n} {liftSubstn σ b} {liftSubstn τ b} {t} (≣liftSubstn {m} {n} {b} {σ} {τ} i))
(subst-eta-gen i)
▹▹[] : {n : Nat} (F G : Term (1+ n)) (t : Term n)
→ (F ▹▹ G) [ t ] ≣ (F [ t ]) ▹▹ (G [ t ])
▹▹[] {n} F G t = cong₂ Π_▹_ refl (≣trans z (≣sym (wk-β G)))
where
i : (x : Fin (1+ n)) → (liftSubst (sgSubst t) ₛ• step id) x ≣ (sgSubst (wk (step id) t) ₛ• lift (step id)) x
i Fin.zero = refl
i (Fin.suc x) = refl
z : subst (liftSubst (sgSubst t)) (wk (step id) G)
≣ subst (sgSubst (wk (step id) t)) (wk (lift (step id)) G)
z = ≣trans (subst-wk G)
(≣trans (subst-eta
{_} {_}
{liftSubst (sgSubst t) ₛ• step id}
{sgSubst (wk (step id) t) ₛ• lift (step id)} {G} i)
(≣sym (subst-wk G)))
{--
-- a variant of canonicity″
-- not true?
canonicity2 : {n : Nat} {Γ : Con Term n} {t : Term n}
→ ⊢ Γ
→ Natural-prop Γ t --Natural-prop Γ {!t!} --Γ t
→ ∃ λ k → Γ ⊢ t ≡ sucᵏ k ∷ ℕ
canonicity2 {n} {Γ} {t} g (sucᵣ (ℕₜ n₁ d n≡n prop)) =
let a , b = canonicity2 g prop
in 1+ a , suc-cong (trans (subset*Term (redₜ d)) b)
canonicity2 {n} {Γ} {t} g zeroᵣ = 0 , refl (zeroⱼ g)
canonicity2 {n} {Γ} {t} g (ne (neNfₜ neK ⊢k k≡k)) = {!⊥-elim (noNe ⊢k neK)!}
--}
{--
∷→⊢ : {n : Nat} {Γ : Con Term n} {t : Term n} {σ : Term n}
→ Γ ⊢ t ∷ σ
→ Γ ⊢ σ
∷→⊢ {n} {Γ} {.(Π _ ▹ _)} {.U} (Πⱼ i ▹ i₁) = ∷→⊢ i
∷→⊢ {n} {Γ} {.(Σ _ ▹ _)} {.U} (Σⱼ i ▹ i₁) = ∷→⊢ i
∷→⊢ {n} {Γ} {.ℕ} {.U} (ℕⱼ x) = Uⱼ x
∷→⊢ {n} {Γ} {.Empty} {.U} (Emptyⱼ x) = Uⱼ x
∷→⊢ {n} {Γ} {.Unit} {.U} (Unitⱼ x) = Uⱼ x
∷→⊢ {n} {Γ} {.(var _)} {σ} (var x x₁) = ∈→⊢ x x₁
∷→⊢ {n} {Γ} {.(lam _)} {.(Π _ ▹ _)} (lamⱼ x i) = Πⱼ x ▹ ∷→⊢ i
∷→⊢ {n} {Γ} {.(_ ∘ _)} {.(G [ a ])} ((_∘ⱼ_) {g} {a} {F} {G} i i₁) =
⊢Π[] x i₁
where
x : Γ ⊢ Π F ▹ G
x = ∷→⊢ i
∷→⊢ {n} {Γ} {.(prod _ _)} {.(Σ _ ▹ _)} (prodⱼ x x₁ i i₁) = Σⱼ x ▹ x₁
∷→⊢ {n} {Γ} {.(fst _)} {σ} (fstⱼ x x₁ i) = x
∷→⊢ {n} {Γ} {.(snd _)} {.(G [ fst t ])} (sndⱼ{F} {G} {t} x x₁ i) =
⊢Σ[] z (fstⱼ x x₁ i)
where
z : Γ ⊢ Σ F ▹ G
z = ∷→⊢ i
∷→⊢ {n} {Γ} {.Definition.Untyped.zero} {.ℕ} (zeroⱼ x) = ℕⱼ x
∷→⊢ {n} {Γ} {.(Definition.Untyped.suc _)} {.ℕ} (sucⱼ i) = ∷→⊢ i
∷→⊢ {n} {Γ} {.(natrec _ _ _ _)} {.(G [ k ])} (natrecⱼ {G} {s} {z} {k} x i i₁ i₂) = {!|!}
-- canonicity could be useful, but it's only for empty contexts
{-- where
-- not the way to go
y1 : Γ ⊢ Π ℕ ▹ (G ▹▹ G [ Definition.Untyped.suc (var Fin.zero) ]↑)
y1 = ∷→⊢ i₁
y2 : Γ ⊢ ((G ▹▹ G [ Definition.Untyped.suc (var Fin.zero) ]↑) [ k ])
y2 = ⊢Π[] y1 i₂
y3 : Γ ⊢ G [ k ] ▹▹ G [ Definition.Untyped.suc (var Fin.zero) ]↑ [ k ]
y3 = ≣subst (λ z → Γ ⊢ z) (▹▹[] G (G [ Definition.Untyped.suc (var Fin.zero) ]↑) k) y2
y4 : Γ ⊢ (G [ Definition.Untyped.suc (var Fin.zero) ]↑) [ k ]
y4 = →▹▹[]ᵣ {!!} y3--}
∷→⊢ {n} {Γ} {.(Emptyrec σ _)} {σ} (Emptyrecⱼ x i) = x
∷→⊢ {n} {Γ} {.star} {.Unit} (starⱼ x) = Unitⱼ x
∷→⊢ {n} {Γ} {t} {σ} (conv {t} {A} {B} i x) =
π₂ (syntacticEq x)
where
y : Γ ⊢ A
y = ∷→⊢ i
--}
-- Conversion of an untyped term
-- TODO: replace the recursive functions below by a call to this function
⟦_⟧ᵤ : {n : Nat} (t : Term n)
→ BTerm
⟦_⟧ᵤ {n} (var x) = VAR (toℕ x)
⟦_⟧ᵤ {n} (gen {.nil} Ukind c) = UNIV 1
⟦_⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) = PI ⟦ t ⟧ᵤ ⟦ t₁ ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) = LAMBDA ⟦ t ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) = APPLY ⟦ t ⟧ᵤ ⟦ t₁ ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) = SUM! ⟦ t ⟧ᵤ ⟦ t₁ ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) = PAIR ⟦ t ⟧ᵤ ⟦ t₁ ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) = FST ⟦ t ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) = SND ⟦ t ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.nil} Natkind []) = NAT!
⟦_⟧ᵤ {n} (gen {.nil} Zerokind []) = N0
⟦_⟧ᵤ {n} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) = SUC ⟦ t ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) = NATREC ⟦ t₃ ⟧ᵤ ⟦ t₁ ⟧ᵤ ⟦ t₂ ⟧ᵤ
⟦_⟧ᵤ {n} (gen {.nil} Unitkind []) = UNIT
⟦_⟧ᵤ {n} (gen {.nil} Starkind []) = AX
⟦_⟧ᵤ {n} (gen {.nil} Emptykind []) = FALSE
⟦_⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) = ⟦ t₁ ⟧ᵤ
¬names-FST : (t : BTerm) → ¬names (FST t) ≣ ¬names t
¬names-FST t with ¬names t
... | true = refl
... | false = refl
¬names-SND : (t : BTerm) → ¬names (SND t) ≣ ¬names t
¬names-SND t with ¬names t
... | true = refl
... | false = refl
noseq-FST : (t : BTerm) → noseq (FST t) ≣ noseq t
noseq-FST t with noseq t
... | true = refl
... | false = refl
noseq-SND : (t : BTerm) → noseq (SND t) ≣ noseq t
noseq-SND t with noseq t
... | true = refl
... | false = refl
¬enc-FST : (t : BTerm) → ¬enc (FST t) ≣ ¬enc t
¬enc-FST t with ¬enc t
... | true = refl
... | false = refl
¬enc-SND : (t : BTerm) → ¬enc (SND t) ≣ ¬enc t
¬enc-SND t with ¬enc t
... | true = refl
... | false = refl
→¬Names-SUM! : {a b : BTerm}
→ ¬Names a
→ ¬Names b
→ ¬Names (SUM! a b)
→¬Names-SUM! {a} {b} na nb
rewrite na | nb = refl
→¬Seq-SUM! : {a b : BTerm}
→ ¬Seq a
→ ¬Seq b
→ ¬Seq (SUM! a b)
→¬Seq-SUM! {a} {b} na nb
rewrite na | nb = refl
→¬Enc-SUM! : {a b : BTerm}
→ ¬Enc a
→ ¬Enc b
→ ¬Enc (SUM! a b)
→¬Enc-SUM! {a} {b} na nb
rewrite na | nb = refl
¬Names⟦⟧ᵤ : {n : Nat} (t : Term n)
→ ¬Names ⟦ t ⟧ᵤ
¬Names⟦⟧ᵤ {n} (var x) = refl
¬Names⟦⟧ᵤ {n} (gen {.nil} Ukind c) = refl
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Names⟦⟧ᵤ t) (¬Names⟦⟧ᵤ t₁)
¬Names⟦⟧ᵤ {n} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) = ¬Names⟦⟧ᵤ t
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Names⟦⟧ᵤ t) (¬Names⟦⟧ᵤ t₁)
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→¬Names-SUM! {⟦ t ⟧ᵤ} {⟦ t₁ ⟧ᵤ} (¬Names⟦⟧ᵤ t) (¬Names⟦⟧ᵤ t₁)
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Names⟦⟧ᵤ t) (¬Names⟦⟧ᵤ t₁)
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) =
≣trans (¬names-FST ⟦ t ⟧ᵤ) (¬Names⟦⟧ᵤ t)
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) =
≣trans (¬names-SND ⟦ t ⟧ᵤ) (¬Names⟦⟧ᵤ t)
¬Names⟦⟧ᵤ {n} (gen {.nil} Natkind []) = refl
¬Names⟦⟧ᵤ {n} (gen {.nil} Zerokind []) = refl
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) = ¬Names⟦⟧ᵤ t
¬Names⟦⟧ᵤ {n} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) =
→∧≡true (¬Names⟦⟧ᵤ t₃) (→∧≡true (¬Names⟦⟧ᵤ t₁) (¬Names⟦⟧ᵤ t₂))
¬Names⟦⟧ᵤ {n} (gen {.nil} Unitkind []) = refl
¬Names⟦⟧ᵤ {n} (gen {.nil} Starkind []) = refl
¬Names⟦⟧ᵤ {n} (gen {.nil} Emptykind []) = refl
¬Names⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) = ¬Names⟦⟧ᵤ t₁
¬Seq⟦⟧ᵤ : {n : Nat} (t : Term n)
→ ¬Seq ⟦ t ⟧ᵤ
¬Seq⟦⟧ᵤ {n} (var x) = refl
¬Seq⟦⟧ᵤ {n} (gen {.nil} Ukind c) = refl
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Seq⟦⟧ᵤ t) (¬Seq⟦⟧ᵤ t₁)
¬Seq⟦⟧ᵤ {n} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) = ¬Seq⟦⟧ᵤ t
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Seq⟦⟧ᵤ t) (¬Seq⟦⟧ᵤ t₁)
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→¬Seq-SUM! {⟦ t ⟧ᵤ} {⟦ t₁ ⟧ᵤ} (¬Seq⟦⟧ᵤ t) (¬Seq⟦⟧ᵤ t₁)
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Seq⟦⟧ᵤ t) (¬Seq⟦⟧ᵤ t₁)
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) =
≣trans (noseq-FST ⟦ t ⟧ᵤ) (¬Seq⟦⟧ᵤ t)
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) =
≣trans (noseq-SND ⟦ t ⟧ᵤ) (¬Seq⟦⟧ᵤ t)
¬Seq⟦⟧ᵤ {n} (gen {.nil} Natkind []) = refl
¬Seq⟦⟧ᵤ {n} (gen {.nil} Zerokind []) = refl
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) = ¬Seq⟦⟧ᵤ t
¬Seq⟦⟧ᵤ {n} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) =
→∧≡true (¬Seq⟦⟧ᵤ t₃) (→∧≡true (¬Seq⟦⟧ᵤ t₁) (¬Seq⟦⟧ᵤ t₂))
¬Seq⟦⟧ᵤ {n} (gen {.nil} Unitkind []) = refl
¬Seq⟦⟧ᵤ {n} (gen {.nil} Starkind []) = refl
¬Seq⟦⟧ᵤ {n} (gen {.nil} Emptykind []) = refl
¬Seq⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) = ¬Seq⟦⟧ᵤ t₁
¬Enc⟦⟧ᵤ : {n : Nat} (t : Term n)
→ ¬Enc ⟦ t ⟧ᵤ
¬Enc⟦⟧ᵤ {n} (var x) = refl
¬Enc⟦⟧ᵤ {n} (gen {.nil} Ukind c) = refl
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Enc⟦⟧ᵤ t) (¬Enc⟦⟧ᵤ t₁)
¬Enc⟦⟧ᵤ {n} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) = ¬Enc⟦⟧ᵤ t
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Enc⟦⟧ᵤ t) (¬Enc⟦⟧ᵤ t₁)
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→¬Enc-SUM! {⟦ t ⟧ᵤ} {⟦ t₁ ⟧ᵤ} (¬Enc⟦⟧ᵤ t) (¬Enc⟦⟧ᵤ t₁)
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
→∧≡true (¬Enc⟦⟧ᵤ t) (¬Enc⟦⟧ᵤ t₁)
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) =
≣trans (¬enc-FST ⟦ t ⟧ᵤ) (¬Enc⟦⟧ᵤ t)
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) =
≣trans (¬enc-SND ⟦ t ⟧ᵤ) (¬Enc⟦⟧ᵤ t)
¬Enc⟦⟧ᵤ {n} (gen {.nil} Natkind []) = refl
¬Enc⟦⟧ᵤ {n} (gen {.nil} Zerokind []) = refl
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) = ¬Enc⟦⟧ᵤ t
¬Enc⟦⟧ᵤ {n} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) =
→∧≡true (¬Enc⟦⟧ᵤ t₃) (→∧≡true (¬Enc⟦⟧ᵤ t₁) (¬Enc⟦⟧ᵤ t₂))
¬Enc⟦⟧ᵤ {n} (gen {.nil} Unitkind []) = refl
¬Enc⟦⟧ᵤ {n} (gen {.nil} Starkind []) = refl
¬Enc⟦⟧ᵤ {n} (gen {.nil} Emptykind []) = refl
¬Enc⟦⟧ᵤ {n} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) = ¬Enc⟦⟧ᵤ t₁
⟦_⟧Γ : {n : Nat} (Γ : Con Term n) → hypotheses
⟦_⟧Γ {.0} ε = Data.List.[]
⟦_⟧Γ {.(1+ _)} (Γ ∙ x) = ⟦ Γ ⟧Γ Data.List.∷ʳ mkHyp ⟦ x ⟧ᵤ
{--
-- intreptation of σ as a BoxTT type
⟦_⟧∈ₜ : {n : Nat} {Γ : Con Term n} {j : Fin n} {σ : Term n}
→ ⊢ Γ
→ j ∷ σ ∈ Γ
→ BTerm
⟦_⟧∈ₜ {n} {Γ} {j} {σ} i k = {!!}
--}
-- Converts an MLTT type (σ here) to its BoxTT type
⟦_⟧ₜ : {n : Nat} {Γ : Con Term n} {t : Term n} {σ : Term n}
→ Γ ⊢ t ∷ σ
→ BTerm
⟦_⟧ₜ {n} {Γ} {t} {σ} i = ⟦ σ ⟧ᵤ
{--
⟦_⟧ₜ {n} {Γ} {.(Π _ ▹ _)} {.U} ((Πⱼ_▹_) {F} {G} A B) = UNIV 1
⟦_⟧ₜ {n} {Γ} {.(Σ _ ▹ _)} {.U} ((Σⱼ_▹_) {F} {G} A B) = UNIV 1
⟦_⟧ₜ {n} {Γ} {.ℕ} {.U} (ℕⱼ x) = UNIV 1
⟦_⟧ₜ {n} {Γ} {.Empty} {.U} (Emptyⱼ x) = UNIV 1
⟦_⟧ₜ {n} {Γ} {.Unit} {.U} (Unitⱼ x) = UNIV 1
⟦_⟧ₜ {n} {Γ} {var j} {σ} (var x x₁) = {!!} --VAR (toℕ j)
⟦_⟧ₜ {n} {Γ} {.(lam _)} {.(Π _ ▹ _)} (lamⱼ {F} {G} {u} x i) = PI ⟦ i ⟧ₜ ⟦ i ⟧ₜ
⟦_⟧ₜ {n} {Γ} {.(_ ∘ _)} {.(G [ a ])} ((_∘ⱼ_) {g} {a} {F} {G} i i₁) = ⟦ i₁ ⟧ₜ
⟦_⟧ₜ {n} {Γ} {.(prod _ _)} {.(Σ _ ▹ _)} (prodⱼ x x₁ i i₁) = SUM ⟦ i ⟧ₜ ⟦ i₁ ⟧ₜ
⟦_⟧ₜ {n} {Γ} {.(fst _)} {σ} (fstⱼ x x₁ i) = ⟦ i ⟧ₜ
⟦_⟧ₜ {n} {Γ} {.(snd _)} {.(G [ fst t ])} (sndⱼ {F} {G} {t} x x₁ i) = ⟦ i ⟧ₜ
⟦_⟧ₜ {n} {Γ} {.Definition.Untyped.zero} {.ℕ} (zeroⱼ x) = NAT!
⟦_⟧ₜ {n} {Γ} {.(Definition.Untyped.suc _)} {.ℕ} (sucⱼ i) = NAT!
⟦_⟧ₜ {n} {Γ} {.(natrec _ _ _ _)} {.(G [ k ])} (natrecⱼ {G} {s} {z} {k} x i i₁ i₂) = ⟦ i₂ ⟧ₜ
⟦_⟧ₜ {n} {Γ} {.(Emptyrec σ _)} {σ} (Emptyrecⱼ x i) = ⟦ i ⟧ₜ
⟦_⟧ₜ {n} {Γ} {.star} {.Unit} (starⱼ x) = VAR n
⟦_⟧ₜ {n} {Γ} {t} {σ} (conv i x) = ⟦ i ⟧ₜ
--}
∈fvars-SUM!→ : {v : Var} {a b : BTerm}
→ v ∈ fvars (SUM! a b)
→ v ∈ fvars a ⊎ v ∈ lowerVars (fvars b)
∈fvars-SUM!→ {v} {a} {b} i
with ∈-++⁻ ((fvars a Data.List.++ lowerVars (fvars b)) Data.List.++ nil) i
∈fvars-SUM!→ {v} {a} {b} i | inj₁ p
with ∈-++⁻ (fvars a Data.List.++ lowerVars (fvars b)) p
∈fvars-SUM!→ {v} {a} {b} i | inj₁ p | inj₁ q
with ∈-++⁻ (fvars a) q
... | inj₁ r = inj₁ r
... | inj₂ r = inj₂ r
∈fvars-SUM!→ {v} {a} {b} i | inj₁ p | inj₂ ()
∈fvars-SUM!→ {v} {a} {b} i | inj₂ ()
fvarsᵤ : {n : Nat} (t : Term n)
→ (v : Var) → v ∈ fvars (⟦ t ⟧ᵤ) → v <ℕ n
fvarsᵤ {n} (var x) v (here px) rewrite px = toℕ<n x
fvarsᵤ {n} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) v i
with ∈-++⁻ (fvars ⟦ t ⟧ᵤ) i
... | inj₁ k = fvarsᵤ t _ k
... | inj₂ k = s≤s-inj (fvarsᵤ t₁ _ (∈lowerVars→ v (fvars ⟦ t₁ ⟧ᵤ) k))
fvarsᵤ {n} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) v i =
s≤s-inj (fvarsᵤ t _ (∈lowerVars→ v (fvars ⟦ t ⟧ᵤ) i))
fvarsᵤ {n} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) v i
with ∈-++⁻ (fvars ⟦ t ⟧ᵤ) i
... | inj₁ k = fvarsᵤ t _ k
... | inj₂ k = fvarsᵤ t₁ _ k
fvarsᵤ {n} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) v i
with ∈fvars-SUM!→ {v} {⟦ t ⟧ᵤ} {⟦ t₁ ⟧ᵤ} i
... | inj₁ k = fvarsᵤ t _ k
... | inj₂ k = s≤s-inj (fvarsᵤ t₁ _ (∈lowerVars→ v (fvars ⟦ t₁ ⟧ᵤ) k))
fvarsᵤ {n} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) v i
with ∈-++⁻ (fvars ⟦ t ⟧ᵤ) i
... | inj₁ k = fvarsᵤ t _ k
... | inj₂ k = fvarsᵤ t₁ _ k
fvarsᵤ {n} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) v i
with ∈-++⁻ (fvars ⟦ t ⟧ᵤ) i
... | inj₁ k = fvarsᵤ t _ k
... | inj₂ ()
fvarsᵤ {n} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) v i
with ∈-++⁻ (fvars ⟦ t ⟧ᵤ) i
... | inj₁ k = fvarsᵤ t _ k
... | inj₂ ()
fvarsᵤ {n} (gen {.nil} Natkind []) v ()
fvarsᵤ {n} (gen {.nil} Zerokind []) v ()
fvarsᵤ {n} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) v i = fvarsᵤ t _ i
fvarsᵤ {n} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) v i
with ∈-++⁻ (fvars ⟦ t₃ ⟧ᵤ) i
... | inj₁ k = fvarsᵤ t₃ _ k
... | inj₂ k with ∈-++⁻ (fvars ⟦ t₁ ⟧ᵤ) k
... | inj₁ k₁ = fvarsᵤ t₁ _ k₁
... | inj₂ k₁ = fvarsᵤ t₂ _ k₁
fvarsᵤ {n} (gen {.nil} Unitkind []) v ()
fvarsᵤ {n} (gen {.nil} Starkind []) v ()
fvarsᵤ {n} (gen {.nil} Emptykind []) v ()
fvarsᵤ {n} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) v i = fvarsᵤ t₁ _ i
{--
fvarsᵤ {n} {Γ} {.(Π _ ▹ _)} {.U} (Πⱼ i ▹ i₁) v ()
fvarsᵤ {n} {Γ} {.(Σ _ ▹ _)} {.U} (Σⱼ i ▹ i₁) v ()
fvarsᵤ {n} {Γ} {.ℕ} {.U} (ℕⱼ x) v ()
fvarsᵤ {n} {Γ} {.Empty} {.U} (Emptyⱼ x) v ()
fvarsᵤ {n} {Γ} {.Unit} {.U} (Unitⱼ x) v ()
fvarsᵤ {n} {Γ} {.(var _)} {σ} (var x x₁) v (here px) rewrite px = {!!}
fvarsᵤ {n} {Γ} {.(lam _)} {.(Π _ ▹ _)} (lamⱼ x i) = {!!}
fvarsᵤ {n} {Γ} {.(_ ∘ _)} {.(_ [ _ ])} (i ∘ⱼ i₁) = {!!}
fvarsᵤ {n} {Γ} {.(prod _ _)} {.(Σ _ ▹ _)} (prodⱼ x x₁ i i₁) = {!!}
fvarsᵤ {n} {Γ} {.(fst _)} {σ} (fstⱼ x x₁ i) = {!!}
fvarsᵤ {n} {Γ} {.(snd _)} {.(_ [ fst _ ])} (sndⱼ x x₁ i) = {!!}
fvarsᵤ {n} {Γ} {.Definition.Untyped.zero} {.ℕ} (zeroⱼ x) = {!!}
fvarsᵤ {n} {Γ} {.(Definition.Untyped.suc _)} {.ℕ} (sucⱼ i) = {!!}
fvarsᵤ {n} {Γ} {.(natrec _ _ _ _)} {.(_ [ _ ])} (natrecⱼ x i i₁ i₂) = {!!}
fvarsᵤ {n} {Γ} {.(Emptyrec σ _)} {σ} (Emptyrecⱼ x i) = {!!}
fvarsᵤ {n} {Γ} {.star} {.Unit} (starⱼ x) = {!!}
fvarsᵤ {n} {Γ} {t} {σ} (conv i x) = {!!}
--}
¬∈[]→ : {A : Set} (l : Data.List.List A) → ((v : A) → ¬ (v ∈ l)) → l ≣ Data.List.[]
¬∈[]→ {A} Data.List.[] i = refl
¬∈[]→ {A} (x Data.List.∷ l) i = ⊥-elim (i x (here refl))
#⟦⟧ᵤ0 : (t : Term 0) → # ⟦ t ⟧ᵤ
#⟦⟧ᵤ0 t = ¬∈[]→ (fvars ⟦ t ⟧ᵤ) j
where
j : (v : Var) → ¬ v ∈ fvars ⟦ t ⟧ᵤ
j v k = m<n⇒n≢0 z refl
where
z : v <ℕ 0
z = fvarsᵤ t v k
⟦_⟧ᵤ₀ : (t : Term 0) → CTerm
⟦_⟧ᵤ₀ t = ct ⟦ t ⟧ᵤ (#⟦⟧ᵤ0 t)
⟦_⟧ₜ₀ : {t : Term 0} {σ : Term 0}
→ ε ⊢ t ∷ σ
→ CTerm
⟦_⟧ₜ₀ {t} {σ} i =
ct ⟦ σ ⟧ᵤ (¬∈[]→ (fvars ⟦ σ ⟧ᵤ) j)
where
j : (v : Var) → ¬ v ∈ fvars ⟦ σ ⟧ᵤ
j v k = m<n⇒n≢0 z refl
where
z : v <ℕ 0
z = fvarsᵤ σ v k
⟦_⟧≡ₜ₀ : {t u : Term 0} {σ : Term 0}
→ ε ⊢ t ≡ u ∷ σ
→ CTerm
⟦_⟧≡ₜ₀ {t} {u} {σ} i =
ct ⟦ σ ⟧ᵤ (¬∈[]→ (fvars ⟦ σ ⟧ᵤ) j)
where
j : (v : Var) → ¬ v ∈ fvars ⟦ σ ⟧ᵤ
j v k = m<n⇒n≢0 z refl
where
z : v <ℕ 0
z = fvarsᵤ σ v k
-- Converts an MLTT term (t here) into a BoxTT term
⟦_⟧ : {n : Nat} {Γ : Con Term n} {t : Term n} {σ : Term n}
→ Γ ⊢ t ∷ σ
→ BTerm
⟦_⟧ {n} {Γ} {t} {σ} i = ⟦ t ⟧ᵤ
{--
⟦_⟧ {n} {Γ} {.(Π _ ▹ _)} {U} ((Πⱼ_▹_) {F} {G} A B) = PI ⟦ A ⟧ ⟦ B ⟧
⟦_⟧ {n} {Γ} {.(Σ _ ▹ _)} {U} ((Σⱼ_▹_) {F} {G} A B) = SUM ⟦ A ⟧ ⟦ B ⟧
⟦_⟧ {n} {Γ} {ℕ} {U} (ℕⱼ x) = NAT!
⟦_⟧ {n} {Γ} {Empty} {U} (Emptyⱼ x) = FALSE
⟦_⟧ {n} {Γ} {Unit} {U} (Unitⱼ x) = UNIT
⟦_⟧ {n} {Γ} {var _} {σ} (var x i) = VAR (∈→ℕ i)
⟦_⟧ {n} {Γ} {.(lam _)} {.(Π _ ▹ _)} (lamⱼ {F} {G} {u} x i) = LAMBDA ⟦ i ⟧
⟦_⟧ {n} {Γ} {.(_ ∘ _)} {.(G [ a ])} ((_∘ⱼ_) {g} {a} {F} {G} x x₁) = APPLY ⟦ x ⟧ ⟦ x₁ ⟧
⟦_⟧ {n} {Γ} {.(prod _ _)} {.(Σ _ ▹ _)} (prodⱼ x x₁ x₂ x₃) = PAIR ⟦ x₂ ⟧ ⟦ x₃ ⟧
⟦_⟧ {n} {Γ} {.(fst _)} {σ} (fstⱼ x x₁ x₂) = FST ⟦ x₂ ⟧
⟦_⟧ {n} {Γ} {.(snd _)} {.(G [ fst u ])} (sndⱼ {F} {G} {u} x x₁ x₂) = SND ⟦ x₂ ⟧
⟦_⟧ {n} {Γ} {.Definition.Untyped.zero} {.ℕ} (zeroⱼ x) = NUM 0
⟦_⟧ {n} {Γ} {.(Definition.Untyped.suc _)} {.ℕ} (sucⱼ x) = SUC ⟦ x ⟧
⟦_⟧ {n} {Γ} {.(natrec _ _ _ _)} {.(G [ k ])} (natrecⱼ {G} {s} {z} {k} x x₁ x₂ x₃) = NATREC ⟦ x₃ ⟧ ⟦ x₁ ⟧ ⟦ x₂ ⟧
⟦_⟧ {n} {Γ} {.(Emptyrec σ _)} {σ} (Emptyrecⱼ x x₁) = BOT
⟦_⟧ {n} {Γ} {.star} {.Unit} (starⱼ x) = AX
⟦_⟧ {n} {Γ} {t} {σ} (conv x x₁) = ⟦ x ⟧
--}
⟦_⟧₀ : {t : Term 0} {σ : Term 0}
→ ε ⊢ t ∷ σ
→ CTerm
⟦_⟧₀ {t} {σ} i =
ct ⟦ t ⟧ᵤ (¬∈[]→ (fvars ⟦ t ⟧ᵤ) j)
where
j : (v : Var) → ¬ v ∈ fvars ⟦ t ⟧ᵤ
j v k = m<n⇒n≢0 z refl
where
z : v <ℕ 0
z = fvarsᵤ t v k
⟦_⟧≡ₗ₀ : {t u : Term 0} {σ : Term 0}
→ ε ⊢ t ≡ u ∷ σ
→ CTerm
⟦_⟧≡ₗ₀ {t} {u} {σ} i =
ct ⟦ t ⟧ᵤ (¬∈[]→ (fvars ⟦ t ⟧ᵤ) j)
where
j : (v : Var) → ¬ v ∈ fvars ⟦ t ⟧ᵤ
j v k = m<n⇒n≢0 z refl
where
z : v <ℕ 0
z = fvarsᵤ t v k
⟦_⟧≡ᵣ₀ : {t u : Term 0} {σ : Term 0}
→ ε ⊢ t ≡ u ∷ σ
→ CTerm
⟦_⟧≡ᵣ₀ {t} {u} {σ} i =
ct ⟦ u ⟧ᵤ (¬∈[]→ (fvars ⟦ u ⟧ᵤ) j)
where
j : (v : Var) → ¬ v ∈ fvars ⟦ u ⟧ᵤ
j v k = m<n⇒n≢0 z refl
where
z : v <ℕ 0
z = fvarsᵤ u v k
{--
NAT!∈UNIV : (i : Nat) (w : 𝕎·) (j : Nat)
→ equalInType i w (#UNIV j) #NAT! #NAT!
NAT!∈UNIV i w j = {!!}
--}
length⟦⟧Γ : {n : Nat} {Γ : Con Term n}
→ Data.List.length ⟦ Γ ⟧Γ ≣ n
length⟦⟧Γ {0} {ε} = refl
length⟦⟧Γ {1+ n} {Γ ∙ x} =
≣trans (length-++ ⟦ Γ ⟧Γ)
(≣trans (+-comm (Data.List.length ⟦ Γ ⟧Γ) 1)
(cong Nat.suc (length⟦⟧Γ {n} {Γ})))
coveredΓ : {n : Nat} (Γ : Con Term n) (σ : Term n)
→ coveredH ⟦ Γ ⟧Γ ⟦ σ ⟧ᵤ
coveredΓ {n} Γ σ {x} i = →∈hdom q
where
h : x <ℕ n
h = fvarsᵤ {n} σ x i
q : x <ℕ Data.List.length ⟦ Γ ⟧Γ
q rewrite length⟦⟧Γ {n} {Γ} = h
sub-VAR0 : (t : BTerm) → sub t (VAR 0) ≣ t
sub-VAR0 t = shiftDownUp t 0
sub-VAR+ : (t : BTerm) (n : Nat) → sub t (VAR (1+ n)) ≣ VAR n
sub-VAR+ t n = refl
shiftUpN : (m n : Nat) (t : BTerm) → BTerm
shiftUpN m 0 t = t
shiftUpN m (Nat.suc n) t = shiftUp m (shiftUpN m n t)
shiftUpN-UNIV : (k m : Nat) (i : Nat) → shiftUpN k m (UNIV i) ≣ UNIV i
shiftUpN-UNIV k 0 i = refl
shiftUpN-UNIV k (Nat.suc m) i rewrite shiftUpN-UNIV k m i = refl
shiftUpN-PI : (k m : Nat) (a b : BTerm) → shiftUpN k m (PI a b) ≣ PI (shiftUpN k m a) (shiftUpN (Nat.suc k) m b)
shiftUpN-PI k 0 a b = refl
shiftUpN-PI k (Nat.suc m) a b rewrite shiftUpN-PI k m a b = refl
⟦wk⟧ᵤ-var1 : (m n : Nat) (x : Fin (m + n))
→ 1+ (toℕ x) ≤ℕ m
→ toℕ (wkVar (liftn (step id) m) x) ≣ toℕ x
⟦wk⟧ᵤ-var1 (1+ m) n Fin.zero p = refl
⟦wk⟧ᵤ-var1 (1+ m) n (Fin.suc x) p = cong 1+ (⟦wk⟧ᵤ-var1 m n x (s≤s-inj p))
⟦wk⟧ᵤ-var2 : (m n : Nat) (x : Fin (m + n))
→ m <ℕ 1+ (toℕ x)
→ toℕ (wkVar (liftn (step id) m) x) ≣ 1+ (toℕ x)
⟦wk⟧ᵤ-var2 Nat.zero n x p = refl
⟦wk⟧ᵤ-var2 (1+ m) n Fin.zero p = ⊥-elim (m+n≮m 1 m p)
⟦wk⟧ᵤ-var2 (1+ m) n (Fin.suc x) p = cong 1+ (⟦wk⟧ᵤ-var2 m n x (s≤s-inj p))
⟦wk⟧ᵤ : {n m : Nat} (t : Term (m + n)) → ⟦ wk (liftn (step id) m) t ⟧ᵤ ≣ shiftUp m ⟦ t ⟧ᵤ
⟦wk⟧ᵤ {n} {m} (var x) with toℕ x <? m
... | yes p = cong VAR (⟦wk⟧ᵤ-var1 m n x p)
... | no p = cong VAR (⟦wk⟧ᵤ-var2 m n x (≰⇒> p))
⟦wk⟧ᵤ {n} {m} (gen {.nil} Ukind []) = refl
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) =
cong₂ PI (⟦wk⟧ᵤ {n} {m} t) (⟦wk⟧ᵤ {n} {1+ m} t₁)
⟦wk⟧ᵤ {n} {m} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) =
cong LAMBDA (⟦wk⟧ᵤ t)
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
cong₂ APPLY (⟦wk⟧ᵤ {n} {m} t) (⟦wk⟧ᵤ {n} {m} t₁)
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) =
cong₂ SUM! (⟦wk⟧ᵤ {n} {m} t) (⟦wk⟧ᵤ {n} {1+ m} t₁)
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) =
cong₂ PAIR (⟦wk⟧ᵤ {n} {m} t) (⟦wk⟧ᵤ {n} {m} t₁)
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) =
cong FST (⟦wk⟧ᵤ t)
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) =
cong SND (⟦wk⟧ᵤ t)
⟦wk⟧ᵤ {n} {m} (gen {.nil} Natkind []) = refl
⟦wk⟧ᵤ {n} {m} (gen {.nil} Zerokind []) = refl
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) =
cong SUC (⟦wk⟧ᵤ {n} {m} t)
⟦wk⟧ᵤ {n} {m} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) =
cong₃ NATREC (⟦wk⟧ᵤ {n} {m} t₃) (⟦wk⟧ᵤ {n} {m} t₁) (⟦wk⟧ᵤ {n} {m} t₂)
⟦wk⟧ᵤ {n} {m} (gen {.nil} Unitkind []) = refl
⟦wk⟧ᵤ {n} {m} (gen {.nil} Starkind []) = refl
⟦wk⟧ᵤ {n} {m} (gen {.nil} Emptykind []) = refl
⟦wk⟧ᵤ {n} {m} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) = ⟦wk⟧ᵤ t₁
⟦[]⟧ᵤ'-var1 : {n m : Nat} (x : Fin (m + 1+ n)) (u : Term n)
→ toℕ x ≣ m
→ ⟦ liftSubstn (consSubst var u) m x ⟧ᵤ ≣ shiftUpN 0 m ⟦ u ⟧ᵤ
⟦[]⟧ᵤ'-var1 {n} {0} Fin.zero u e = refl
⟦[]⟧ᵤ'-var1 {n} {1+ m} (Fin.suc x) u e
rewrite ≣sym (⟦[]⟧ᵤ'-var1 x u (suc-injective e))
= ⟦wk⟧ᵤ (liftSubstn (consSubst var u) m x)
sucIf≤-predIf≤-prop1 : (x m : Nat)
→ ¬ x ≣ m
→ x ≤ℕ m
→ sucIf≤ 0 (predIf≤ m x) ≣ 1+ x
sucIf≤-predIf≤-prop1 0 m p q with 0 <? 0
... | yes a = refl
... | no a = refl
sucIf≤-predIf≤-prop1 (1+ x) m p q with 1+ x ≤? m
... | yes a = refl
... | no a = ⊥-elim (a q)
sucIf≤-predIf≤-prop2 : (x m : Nat)
→ ¬ x ≣ m
→ m <ℕ x
→ sucIf≤ 0 (predIf≤ m x) ≣ x
sucIf≤-predIf≤-prop2 0 m p q with 0 <? 0
... | yes a = ⊥-elim (n≮n 0 a)
... | no a = ⊥-elim (m+n≮m 0 m q)
sucIf≤-predIf≤-prop2 (1+ x) m p q with 1+ x ≤? m
... | yes a = ⊥-elim (n≮n m (≤-trans q a))
... | no a = refl
⟦[]⟧ᵤ'-var2 : {n m : Nat} (x : Fin (m + 1+ n)) (u : Term n)
→ ¬ toℕ x ≣ m
→ ⟦ liftSubstn (consSubst var u) m x ⟧ᵤ ≣ VAR (predIf≤ m (toℕ x))
⟦[]⟧ᵤ'-var2 {n} {0} Fin.zero u p = ⊥-elim (p refl)
⟦[]⟧ᵤ'-var2 {n} {0} (Fin.suc x) u p = refl
⟦[]⟧ᵤ'-var2 {n} {1+ m} Fin.zero u p = refl
⟦[]⟧ᵤ'-var2 {n} {1+ m} (Fin.suc x) u p with 1+ (toℕ x) ≤? 1+ m
... | yes q =
≣trans (⟦wk⟧ᵤ {_} {0} (liftSubstn (consSubst var u) m x))
(≣trans (cong (shiftUp 0) (⟦[]⟧ᵤ'-var2 x u λ z → p (cong 1+ z)))
(cong VAR (sucIf≤-predIf≤-prop1 (toℕ x) m (λ z → p (cong 1+ z)) (s≤s-inj q))))
... | no q =
≣trans (⟦wk⟧ᵤ {_} {0} (liftSubstn (consSubst var u) m x))
(≣trans (cong (shiftUp 0) (⟦[]⟧ᵤ'-var2 x u λ z → p (cong 1+ z)))
(cong VAR (sucIf≤-predIf≤-prop2 (toℕ x) m (λ z → p (cong 1+ z)) (≰⇒> (λ z → q (s≤s z))))))
⟦[]⟧ᵤ' : {n m : Nat} (G : Term (m + 1+ n)) (u : Term n)
→ ⟦ subst (liftSubstn (sgSubst u) m) G ⟧ᵤ ≣ subn m (shiftUpN 0 m ⟦ u ⟧ᵤ) ⟦ G ⟧ᵤ
⟦[]⟧ᵤ' {n} {m} (var x) u with toℕ x ≟ m
... | yes p = ⟦[]⟧ᵤ'-var1 x u p
... | no p = ⟦[]⟧ᵤ'-var2 x u p
⟦[]⟧ᵤ' {n} {m} (gen {.nil} Ukind []) u = refl
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ PI (⟦[]⟧ᵤ' t u) (⟦[]⟧ᵤ' {n} {1+ m} t₁ u)
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) u =
cong LAMBDA (⟦[]⟧ᵤ' {n} {1+ m} t u)
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ APPLY (⟦[]⟧ᵤ' t u) (⟦[]⟧ᵤ' t₁ u)
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ SUM! (⟦[]⟧ᵤ' t u) (⟦[]⟧ᵤ' {n} {1+ m} t₁ u)
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ PAIR (⟦[]⟧ᵤ' t u) (⟦[]⟧ᵤ' t₁ u)
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) u =
cong FST (⟦[]⟧ᵤ' t u)
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) u =
cong SND (⟦[]⟧ᵤ' t u)
⟦[]⟧ᵤ' {n} {m} (gen {.nil} Natkind []) u = refl
⟦[]⟧ᵤ' {n} {m} (gen {.nil} Zerokind []) u = refl
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) u =
cong SUC (⟦[]⟧ᵤ' t u)
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) u =
cong₃ NATREC (⟦[]⟧ᵤ' t₃ u) (⟦[]⟧ᵤ' t₁ u) (⟦[]⟧ᵤ' t₂ u)
⟦[]⟧ᵤ' {n} {m} (gen {.nil} Unitkind []) u = refl
⟦[]⟧ᵤ' {n} {m} (gen {.nil} Starkind []) u = refl
⟦[]⟧ᵤ' {n} {m} (gen {.nil} Emptykind []) u = refl
⟦[]⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) u = ⟦[]⟧ᵤ' t₁ u
⟦[]⟧ᵤ-as-subn : {n : Nat} (G : Term (1+ n)) (u : Term n)
→ ⟦ G [ u ] ⟧ᵤ ≣ subn 0 ⟦ u ⟧ᵤ ⟦ G ⟧ᵤ
⟦[]⟧ᵤ-as-subn {n} G u = ⟦[]⟧ᵤ' {n} {0} G u
⟦[]⟧ᵤ-as-sub : {n : Nat} (G : Term (1+ n)) (u : Term n)
→ ⟦ G [ u ] ⟧ᵤ ≣ sub ⟦ u ⟧ᵤ ⟦ G ⟧ᵤ
⟦[]⟧ᵤ-as-sub {n} G u = ≣trans (⟦[]⟧ᵤ-as-subn G u) (≣sym (sub≡subn ⟦ u ⟧ᵤ ⟦ G ⟧ᵤ))
⟦[]↑⟧ᵤ'-var1 : {n m : Nat} (x : Fin (m + 1+ n)) (u : Term (1+ n))
→ toℕ x ≣ m
→ ⟦ liftSubstn (consSubst (λ z → var (Fin.suc z)) u) m x ⟧ᵤ ≣ shiftUpN 0 m ⟦ u ⟧ᵤ
⟦[]↑⟧ᵤ'-var1 {n} {0} Fin.zero u e = refl
⟦[]↑⟧ᵤ'-var1 {n} {1+ m} (Fin.suc x) u e
rewrite ≣sym (⟦[]↑⟧ᵤ'-var1 x u (suc-injective e))
= ⟦wk⟧ᵤ {m + 1+ n} {0} (liftSubstn (consSubst (λ z → var (Fin.suc z)) u) m x)
sucIf≤0 : (n : Nat) → sucIf≤ 0 n ≣ 1+ n
sucIf≤0 n with n <? 0
... | no p = refl
⟦[]↑⟧ᵤ'-var2 : {n m : Nat} (x : Fin (m + 1+ n)) (u : Term (1+ n))
→ ¬ toℕ x ≣ m
→ ⟦ liftSubstn (consSubst (λ z → var (Fin.suc z)) u) m x ⟧ᵤ ≣ VAR (toℕ x)
⟦[]↑⟧ᵤ'-var2 {n} {0} Fin.zero u p = ⊥-elim (p refl)
⟦[]↑⟧ᵤ'-var2 {n} {0} (Fin.suc x) u p = refl
⟦[]↑⟧ᵤ'-var2 {n} {1+ m} Fin.zero u p = refl
⟦[]↑⟧ᵤ'-var2 {n} {1+ m} (Fin.suc x) u p =
≣trans (⟦wk⟧ᵤ {_} {0} (liftSubstn (consSubst (λ z → var (Fin.suc z)) u) m x))
(≣trans (cong (shiftUp 0) (⟦[]↑⟧ᵤ'-var2 x u (λ z → p (cong 1+ z))))
(cong VAR (sucIf≤0 (toℕ x))))
⟦[]↑⟧ᵤ' : {n m : Nat} (G : Term (m + 1+ n)) (u : Term (1+ n))
→ ⟦ subst (liftSubstn (consSubst (wk1Subst idSubst) u) m) G ⟧ᵤ ≣ subi m (shiftUpN 0 m ⟦ u ⟧ᵤ) ⟦ G ⟧ᵤ
⟦[]↑⟧ᵤ' {n} {m} (var x) u with toℕ x ≟ m
... | yes p = ⟦[]↑⟧ᵤ'-var1 x u p
... | no p = ⟦[]↑⟧ᵤ'-var2 x u p
⟦[]↑⟧ᵤ' {n} {m} (gen {.nil} Ukind []) u = refl
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 1 nil))} Pikind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ PI (⟦[]↑⟧ᵤ' t u) (⟦[]↑⟧ᵤ' {n} {1+ m} t₁ u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 1 nil)} Lamkind (t GenTs.∷ [])) u =
cong LAMBDA (⟦[]↑⟧ᵤ' {n} {1+ m} t u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 0 nil))} Appkind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ APPLY (⟦[]↑⟧ᵤ' t u) (⟦[]↑⟧ᵤ' t₁ u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 1 nil))} Sigmakind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ SUM! (⟦[]↑⟧ᵤ' t u) (⟦[]↑⟧ᵤ' {n} {1+ m} t₁ u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 0 nil))} Prodkind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
cong₂ PAIR (⟦[]↑⟧ᵤ' t u) (⟦[]↑⟧ᵤ' t₁ u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 nil)} Fstkind (t GenTs.∷ [])) u =
cong FST (⟦[]↑⟧ᵤ' t u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 nil)} Sndkind (t GenTs.∷ [])) u =
cong SND (⟦[]↑⟧ᵤ' t u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.nil} Natkind []) u = refl
⟦[]↑⟧ᵤ' {n} {m} (gen {.nil} Zerokind []) u = refl
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 nil)} Suckind (t GenTs.∷ [])) u =
cong SUC (⟦[]↑⟧ᵤ' t u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 1 (cons 0 (cons 0 (cons 0 nil))))} Natreckind (t GenTs.∷ (t₁ GenTs.∷ (t₂ GenTs.∷ (t₃ GenTs.∷ []))))) u =
cong₃ NATREC (⟦[]↑⟧ᵤ' t₃ u) (⟦[]↑⟧ᵤ' t₁ u) (⟦[]↑⟧ᵤ' t₂ u)
⟦[]↑⟧ᵤ' {n} {m} (gen {.nil} Unitkind []) u = refl
⟦[]↑⟧ᵤ' {n} {m} (gen {.nil} Starkind []) u = refl
⟦[]↑⟧ᵤ' {n} {m} (gen {.nil} Emptykind []) u = refl
⟦[]↑⟧ᵤ' {n} {m} (gen {.(cons 0 (cons 0 nil))} Emptyreckind (t GenTs.∷ (t₁ GenTs.∷ []))) u =
⟦[]↑⟧ᵤ' t₁ u
⟦[]↑⟧ᵤ : {n m : Nat} (G : Term (1+ n)) (u : Term (1+ n))
→ ⟦ G [ u ]↑ ⟧ᵤ ≣ subi 0 ⟦ u ⟧ᵤ ⟦ G ⟧ᵤ
⟦[]↑⟧ᵤ {n} {m} G u = ⟦[]↑⟧ᵤ' {n} {0} G u
⟦▹▹⟧ᵤ : {n : Nat} (A B : Term n)
→ ⟦ A ▹▹ B ⟧ᵤ ≣ FUN ⟦ A ⟧ᵤ ⟦ B ⟧ᵤ
⟦▹▹⟧ᵤ {n} A B = cong₂ PI refl (⟦wk⟧ᵤ {n} {0} B)
shiftDown-subv-subsN1# : (s : Sub) (u t : BTerm) (#u : # u)
→ shiftDown 0 (subv 0 u (subsN 1 s t))
≣ subs (s Data.List.∷ʳ ct u #u) t
shiftDown-subv-subsN1# s u t #u =
≣trans c (sub-subsN1 (ct u #u) s t)
where
c : shiftDown 0 (subv 0 u (subsN 1 s t)) ≣ shiftDown 0 (subv 0 (shiftUp 0 u) (subsN 1 s t))
c rewrite #shiftUp 0 (ct u #u) = refl
⟦wk1⟧ᵤ : {n : Nat} (t : Term n) → ⟦ wk1 t ⟧ᵤ ≣ shiftUp 0 ⟦ t ⟧ᵤ
⟦wk1⟧ᵤ {n} t = ⟦wk⟧ᵤ {n} {0} t
valid∈VAR : {n : Nat} {Γ : Con Term n} {σ : Term n} {x : Fin n}
→ x ∷ σ ∈ Γ
→ (i : Nat) (w : 𝕎·) → valid∈ i w ⟦ Γ ⟧Γ (VAR (toℕ x)) ⟦ σ ⟧ᵤ
valid∈VAR {1+ n} {Γ ∙ A} {.(wk1 A)} {.Fin.zero} here i w s1 s2 cc1 cc2 ce1 ce2 es eh
rewrite ⟦wk1⟧ᵤ {n} A =
c1 , c2
where
c1 : equalTypes i w (#subs s1 (shiftUp 0 ⟦ A ⟧ᵤ) cc1) (#subs s2 (shiftUp 0 ⟦ A ⟧ᵤ) cc2)
c1 with ≡hyps∷ʳ→ i w s1 s2 ⟦ Γ ⟧Γ ⟦ Γ ⟧Γ ⟦ A ⟧ᵤ ⟦ A ⟧ᵤ eh
... | t1 , t2 , ss1 , ss2 , cA , cB , e1 , e2 , eH , eT
rewrite e1 | e2
= ≡CTerm→eqTypes (CTerm≡ (≣sym (subs∷ʳ-shiftUp ss1 t1 ⟦ A ⟧ᵤ))) (CTerm≡ (≣sym (subs∷ʳ-shiftUp ss2 t2 ⟦ A ⟧ᵤ))) eT
c2 : equalInType i w (#subs s1 (shiftUp 0 ⟦ A ⟧ᵤ) cc1) (#subs s1 (VAR 0) ce1) (#subs s2 (VAR 0) ce2)
c2 with ≡subs∷ʳ→ i w s1 s2 ⟦ Γ ⟧Γ ⟦ A ⟧ᵤ es
... | t1 , t2 , ss1 , ss2 , cA , e1 , e2 , eS , eT
rewrite e1 | e2
= ≡→equalInType (CTerm≡ (≣sym (subs∷ʳ-shiftUp ss1 t1 ⟦ A ⟧ᵤ)))
(≣sym (CTerm≡ (subs∷ʳ-VAR0 ss1 t1)))
(≣sym (CTerm≡ (subs∷ʳ-VAR0 ss2 t2)))
eT
valid∈VAR {1+ n} {Γ ∙ B} {.(wk1 _)} {Fin.suc x} (there {_} {_} {A} j) i w s1 s2 cc1 cc2 ce1 ce2 es eh
rewrite ⟦wk1⟧ᵤ {n} A