2024-09-21 00:36:50 +02:00
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From Coq Require Import Lists.List.
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Import ListNotations.
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Require Import Atom.
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2024-09-21 13:00:57 +02:00
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Require Import debruijn.
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Require Import subtype.
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Require Import morph.
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Require Import typing.
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2024-09-21 00:36:50 +02:00
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Open Scope ladder_expr_scope.
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Inductive is_value : expr_DeBruijn -> Prop :=
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| V_TAbs : forall e,
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is_value [{ Λ e }]
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| V_Abs : forall σ e,
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is_value [{ λ σ ↦ e }]
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| V_Morph : forall σ e,
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is_value [{ λ σ ↦morph e }]
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| V_Ascend : forall τ e,
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is_value e ->
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is_value [{ e as τ }]
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| V_Descend : forall τ e,
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is_value e ->
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is_value [{ e des τ }]
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.
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Reserved Notation " s '-->eval' t " (at level 40).
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Inductive eval : expr_DeBruijn -> expr_DeBruijn -> Prop :=
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| E_App1 : forall e1 e1' e2,
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e1 -->eval e1' ->
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[{ e1 e2 }] -->eval [{ e1' e2 }]
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| E_App2 : forall v1 e2 e2',
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(is_value v1) ->
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e2 -->eval e2' ->
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[{ v1 e2 }] -->eval [{ v1 e2' }]
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| E_TypApp : forall e e',
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e -->eval e' ->
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[{ Λ e }] -->eval [{ Λ e' }]
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| E_TypAppLam : forall e τ,
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[{ (Λ e) # τ }] -->eval (expr_open_type τ e)
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| E_AppLam : forall τ e a,
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[{ (λ τ ↦ e) a }] -->eval (expr_open a e)
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| E_AppMorph : forall τ e a,
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[{ (λ τ ↦morph e) a }] -->eval (expr_open a e)
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| E_Let : forall e a,
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[{ let a in e }] -->eval (expr_open a e)
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| E_StripAscend : forall τ e,
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[{ e as τ }] -->eval e
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| E_StripDescend : forall τ e,
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[{ e des τ }] -->eval e
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| E_Ascend : forall τ e e',
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(e -->eval e') ->
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[{ e as τ }] -->eval [{ e' as τ }]
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| E_AscendCollapse : forall τ' τ e,
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[{ (e as τ) as τ' }] -->eval [{ e as (τ'~τ) }]
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| E_DescendCollapse : forall τ' τ e,
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(τ':<=τ) ->
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[{ (e des τ') des τ }] -->eval [{ e des τ }]
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where "s '-->eval' t" := (eval s t).
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