93 lines
2.4 KiB
Coq
93 lines
2.4 KiB
Coq
Require Import Autosubst2.core Autosubst2.unscoped compile Autosubst2.syntax ssreflect.
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From Hammer Require Import Tactics.
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Reserved Notation "Γ ⊢ a ∈ A" (at level 70).
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Reserved Notation "⊢ Γ" (at level 70).
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Inductive lookup : nat -> list Tm -> Tm -> Prop :=
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| here A Γ : lookup 0 (cons A Γ) (ren_Tm shift A)
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| there i Γ A B :
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lookup i Γ A ->
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lookup (S i) (cons B Γ) (ren_Tm shift A).
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Lemma lookup_deter i Γ A B :
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lookup i Γ A ->
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lookup i Γ B ->
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A = B.
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Proof. move => h. move : B. induction h; hauto lq:on inv:lookup. Qed.
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Lemma here' A Γ U : U = ren_Tm shift A -> lookup 0 (A :: Γ) U.
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Proof. move => ->. apply here. Qed.
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Lemma there' i Γ A B U : U = ren_Tm shift A -> lookup i Γ A ->
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lookup (S i) (cons B Γ) U.
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Proof. move => ->. apply there. Qed.
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Derive Inversion lookup_inv with (forall i Γ A, lookup i Γ A).
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Inductive Wt : list Tm -> Tm -> Tm -> Prop :=
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| T_Var i Γ A :
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⊢ Γ ->
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lookup i Γ A ->
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Γ ⊢ VarTm i ∈ A
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| T_Bind Γ i p A B :
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Γ ⊢ A ∈ Univ i ->
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cons A Γ ⊢ B ∈ Univ i ->
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Γ ⊢ TBind p A B ∈ Univ i
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| T_Abs Γ a A B i :
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Γ ⊢ TBind TPi A B ∈ (Univ i) ->
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(cons A Γ) ⊢ a ∈ B ->
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Γ ⊢ Abs a ∈ TBind TPi A B
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| T_App Γ b a A B :
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Γ ⊢ b ∈ TBind TPi A B ->
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Γ ⊢ a ∈ A ->
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Γ ⊢ App b a ∈ subst_Tm (scons a VarTm) B
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| T_Pair Γ (a b : Tm) A B i :
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Γ ⊢ TBind TSig A B ∈ (Univ i) ->
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Γ ⊢ a ∈ A ->
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Γ ⊢ b ∈ subst_Tm (scons a VarTm) B ->
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Γ ⊢ Pair a b ∈ TBind TSig A B
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| T_Proj1 Γ (a : Tm) A B :
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Γ ⊢ a ∈ TBind TSig A B ->
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Γ ⊢ Proj PL a ∈ A
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| T_Proj2 Γ (a : Tm) A B :
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Γ ⊢ a ∈ TBind TSig A B ->
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Γ ⊢ Proj PR a ∈ subst_Tm (scons (Proj PL a) VarTm) B
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| T_Univ Γ i :
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⊢ Γ ->
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Γ ⊢ Univ i ∈ Univ (S i)
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| T_Conv Γ (a : Tm) A B i :
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Γ ⊢ a ∈ A ->
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Γ ⊢ B ∈ Univ i ->
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Join.R A B ->
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Γ ⊢ a ∈ B
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with Wff : list Tm -> Prop :=
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| Wff_Nil :
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⊢ nil
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| Wff_Cons Γ (A : Tm) i :
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⊢ Γ ->
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Γ ⊢ A ∈ Univ i ->
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(* -------------------------------- *)
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⊢ (cons A Γ)
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where
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"Γ ⊢ a ∈ A" := (Wt Γ a A) and "⊢ Γ" := (Wff Γ).
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Scheme wf_ind := Induction for Wff Sort Prop
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with wt_ind := Induction for Wt Sort Prop.
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Combined Scheme wt_mutual from wf_ind, wt_ind.
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(* Lemma lem : *)
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(* (forall n (Γ : fin n -> Tm n), ⊢ Γ -> ...) /\ *)
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(* (forall n Γ (a A : Tm n), Γ ⊢ a ∈ A -> ...) /\ *)
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(* Proof. apply wt_mutual. ... *)
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