Definition shiftn k f :=
  fun n ⇒ if Nat.ltb n k then n else k + (f (n - k)).

Fixpoint rename f t : term :=
  match t with
  | tRel i ⇒ tRel (f i)
  | tEvar ev args ⇒ tEvar ev (List.map (rename f) args)
  | tLambda na T M ⇒ tLambda na (rename f T) (rename (shiftn 1 f) M)
  | tApp u v ⇒ tApp (rename f u) (rename f v)
  | tProd na A B ⇒ tProd na (rename f A) (rename (shiftn 1 f) B)
  | tLetIn na b t b' ⇒ tLetIn na (rename f b) (rename f t) (rename (shiftn 1 f) b')
  | tCase ind p c brs ⇒
    let brs' := List.map (on_snd (rename f)) brs in
    tCase ind (rename f p) (rename f c) brs'
  | tProj p c ⇒ tProj p (rename f c)
  | tFix mfix idx ⇒
    let mfix' := List.map (map_def (rename f) (rename (shiftn (List.length mfix) f))) mfix in
    tFix mfix' idx
  | tCoFix mfix idx ⇒
    let mfix' := List.map (map_def (rename f) (rename (shiftn (List.length mfix) f))) mfix in
    tCoFix mfix' idx
  | x ⇒ x
  end.

Fixpoint lift n k t : term :=
  match t with
  | tRel i ⇒ if Nat.leb k i then tRel (n + i) else tRel i
  | tEvar ev args ⇒ tEvar ev (List.map (lift n k) args)
  | tLambda na T M ⇒ tLambda na (lift n k T) (lift n (S k) M)
  | tApp u v ⇒ tApp (lift n k u) (lift n k v)
  | tProd na A B ⇒ tProd na (lift n k A) (lift n (S k) B)
  | tLetIn na b t b' ⇒ tLetIn na (lift n k b) (lift n k t) (lift n (S k) b')
  | tCase ind p c brs ⇒
    let brs' := List.map (on_snd (lift n k)) brs in
    tCase ind (lift n k p) (lift n k c) brs'
  | tProj p c ⇒ tProj p (lift n k c)
  | tFix mfix idx ⇒
    let k' := List.length mfix + k in
    let mfix' := List.map (map_def (lift n k) (lift n k')) mfix in
    tFix mfix' idx
  | tCoFix mfix idx ⇒
    let k' := List.length mfix + k in
    let mfix' := List.map (map_def (lift n k) (lift n k')) mfix in
    tCoFix mfix' idx
  | x ⇒ x
  end.

Notation lift0 n := (lift n 0).

Fixpoint subst s k u :=
  match u with
  | tRel n ⇒
    if Nat.leb k n then
      match nth_error s (n - k) with
      | Some b ⇒ lift0 k b
      | None ⇒ tRel (n - List.length s)
      end
    else tRel n
  | tEvar ev args ⇒ tEvar ev (List.map (subst s k) args)
  | tLambda na T M ⇒ tLambda na (subst s k T) (subst s (S k) M)
  | tApp u v ⇒ tApp (subst s k u) (subst s k v)
  | tProd na A B ⇒ tProd na (subst s k A) (subst s (S k) B)
  | tLetIn na b ty b' ⇒ tLetIn na (subst s k b) (subst s k ty) (subst s (S k) b')
  | tCase ind p c brs ⇒
    let brs' := List.map (on_snd (subst s k)) brs in
    tCase ind (subst s k p) (subst s k c) brs'
  | tProj p c ⇒ tProj p (subst s k c)
  | tFix mfix idx ⇒
    let k' := List.length mfix + k in
    let mfix' := List.map (map_def (subst s k) (subst s k')) mfix in
    tFix mfix' idx
  | tCoFix mfix idx ⇒
    let k' := List.length mfix + k in
    let mfix' := List.map (map_def (subst s k) (subst s k')) mfix in
    tCoFix mfix' idx
  | x ⇒ x
  end.

Notation subst0 t := (subst t 0).
Definition subst1 t k u := subst [t] k u.
Notation subst10 t := (subst1 t 0).
Notation "M { j := N }" := (subst1 N j M) (at level 10, right associativity).

Fixpoint closedn k (t : term) : bool :=
  match t with
  | tRel i ⇒ Nat.ltb i k
  | tEvar ev args ⇒ List.forallb (closedn k) args
  | tLambda _ T M | tProd _ T M ⇒ closedn k T && closedn (S k) M
  | tApp u v ⇒ closedn k u && closedn k v
  | tLetIn na b t b' ⇒ closedn k b && closedn k t && closedn (S k) b'
  | tCase ind p c brs ⇒
    let brs' := List.forallb (test_snd (closedn k)) brs in
    closedn k p && closedn k c && brs'
  | tProj p c ⇒ closedn k c
  | tFix mfix idx ⇒
    let k' := List.length mfix + k in
    List.forallb (test_def (closedn k) (closedn k')) mfix
  | tCoFix mfix idx ⇒
    let k' := List.length mfix + k in
    List.forallb (test_def (closedn k) (closedn k')) mfix
  | x ⇒ true
  end.

Notation closed t := (closedn 0 t).






Notation subst_rec N M k := (subst N k M) (only parsing).

Lemma lift_rel_ge :
  ∀ k n p, p ≤ n → lift k p (tRel n) = tRel (k + n).

Lemma lift_rel_lt : ∀ k n p, p > n → lift k p (tRel n) = tRel n.

Lemma lift_rel_alt : ∀ n k i, lift n k (tRel i) = tRel (if Nat.leb k i then n + i else i).

Lemma subst_rel_lt : ∀ u n k, k > n → subst u k (tRel n) = tRel n.

Lemma subst_rel_gt :
  ∀ u n k, n ≥ k + length u → subst u k (tRel n) = tRel (n - length u).

Lemma subst_rel_eq :
  ∀ (u : list term) n i t p,
    List.nth_error u i = Some t → p = n + i →
    subst u n (tRel p) = lift0 n t.


Lemma on_snd_eq_id_spec {A B} (f : B → B) (x : A × B) :
  f (snd x) = snd x ↔
  on_snd f x = x.

Lemma map_def_eq_spec {A B : Set} (f f' g g' : A → B) (x : def A) :
  f (dtype x) = g (dtype x) →
  f' (dbody x) = g' (dbody x) →
  map_def f f' x = map_def g g' x.

Lemma map_def_id_spec {A : Set} (f f' : A → A) (x : def A) :
  f (dtype x) = (dtype x) →
  f' (dbody x) = (dbody x) →
  map_def f f' x = x.







Lemma lift0_id : ∀ M k, lift 0 k M = M.

Lemma lift0_p : ∀ M, lift0 0 M = M.

Lemma simpl_lift :
  ∀ M n k p i,
    i ≤ k + n →
    k ≤ i → lift p i (lift n k M) = lift (p + n) k M.

Lemma simpl_lift0 : ∀ M n, lift0 (S n) M = lift0 1 (lift0 n M).

Lemma permute_lift :
  ∀ M n k p i,
    i ≤ k →
    lift p i (lift n k M) = lift n (k + p) (lift p i M).

Lemma permute_lift0 :
  ∀ M k, lift0 1 (lift 1 k M) = lift 1 (S k) (lift0 1 M).

Lemma lift_isApp n k t : ¬ isApp t = true → ¬ isApp (lift n k t) = true.

Lemma map_non_nil {A B} (f : A → B) l : l ≠ nil → map f l ≠ nil.

Lemma isLambda_lift n k (bod : term) :
  isLambda bod = true → isLambda (lift n k bod) = true.



Lemma simpl_subst_rec :
  ∀ M N n p k,
    p ≤ n + k →
    k ≤ p → subst N p (lift (List.length N + n) k M) = lift n k M.

Lemma simpl_subst :
  ∀ N M n p, p ≤ n → subst N p (lift0 (length N + n) M) = lift0 n M.

Lemma lift_mkApps n k t l : lift n k (mkApps t l) = mkApps (lift n k t) (map (lift n k) l).

Lemma commut_lift_subst_rec :
  ∀ M N n p k,
    k ≤ p →
    lift n k (subst N p M) = subst N (p + n) (lift n k M).

Lemma commut_lift_subst :
  ∀ M N k, subst N (S k) (lift0 1 M) = lift0 1 (subst N k M).

Lemma distr_lift_subst_rec :
  ∀ M N n p k,
    lift n (p + k) (subst N p M) =
    subst (List.map (lift n k) N) p (lift n (p + length N + k) M).

Lemma distr_lift_subst :
  ∀ M N n k,
    lift n k (subst0 N M) = subst0 (map (lift n k) N) (lift n (length N + k) M).

Lemma distr_lift_subst10 :
  ∀ M N n k,
    lift n k (subst10 N M) = subst10 (lift n k N) (lift n (S k) M).

Lemma subst_mkApps u k t l :
  subst u k (mkApps t l) = mkApps (subst u k t) (map (subst u k) l).

Lemma subst1_mkApps u k t l : subst1 u k (mkApps t l) = mkApps (subst1 u k t) (map (subst1 u k) l).

Lemma distr_subst_rec :
  ∀ M N (P : list term) n p,
    subst P (p + n) (subst N p M) =
    subst (map (subst P n) N) p (subst P (p + length N + n) M).

Lemma distr_subst :
  ∀ P N M k,
    subst P k (subst0 N M) = subst0 (map (subst P k) N) (subst P (length N + k) M).

Lemma lift_closed n k t : closedn k t → lift n k t = t.

Lemma closed_upwards {k t} k' : closedn k t → k' ≥ k → closedn k' t.

Lemma subst_empty k a : subst [] k a = a.

Lemma lift_to_extended_list_k Γ k : ∀ k',
    to_extended_list_k Γ (k' + k) = map (lift0 k') (to_extended_list_k Γ k).

Lemma simpl_subst_k (N : list term) (M : term) :
  ∀ k p, p = #|N| → subst N k (lift p k M) = M.

Lemma subst_app_decomp l l' k t :
  subst (l ++ l') k t = subst l' k (subst (List.map (lift0 (length l')) l) k t).

Lemma subst_app_simpl l l' k t :
  subst (l ++ l') k t = subst l k (subst l' (k + length l) t).

Lemma isLambda_subst (s : list term) k (bod : term) :
  isLambda bod = true → isLambda (subst s k bod) = true.

Lemma map_vass_map_def g l n k :
  (mapi (fun i (d : def term) ⇒ vass (dname d) (lift0 i (dtype d)))
        (map (map_def (lift n k) g) l)) =
  (mapi (fun i d ⇒ map_decl (lift n (i + k)) d) (mapi (fun i (d : def term) ⇒ vass (dname d) (lift0 i (dtype d))) l)).

Definition fix_context (m : mfixpoint term) : context :=
  List.rev (mapi (fun i d ⇒ vass d.(dname) (lift0 i d.(dtype))) m).

Lemma shiftn_ext n f f' : (∀ i, f i = f' i) → ∀ t, shiftn n f t = shiftn n f' t.

Lemma rename_ext f f' : (∀ i, f i = f' i) → (∀ t, rename f t = rename f' t).

Definition lift_renaming n k :=
  fun i ⇒
    if Nat.leb k i then n + i
    else i.

Lemma shiftn_lift_renaming n m k : ∀ i, shiftn m (lift_renaming n k) i = lift_renaming n (m + k) i.

Lemma lift_rename n k t : lift n k t = rename (lift_renaming n k) t.

Definition up k (s : nat → term) :=
  fun i ⇒
    if k <=? i then rename (add k) (s (i - k))
    else tRel i.


Notation "`=1`" := (pointwise_relation _ Logic.eq) (at level 80).
Infix "=1" := (pointwise_relation _ Logic.eq) (at level 90).

Lemma shiftn_compose n f f' : shiftn n f ∘ shiftn n f' =1 shiftn n (f ∘ f').

Lemma rename_compose f f' : rename f ∘ rename f' =1 rename (f ∘ f').

Lemma up_up k k' s : up k (up k' s) =1 up (k + k') s.

Fixpoint inst s u :=
  match u with
  | tRel n ⇒ s n
  | tEvar ev args ⇒ tEvar ev (List.map (inst s) args)
  | tLambda na T M ⇒ tLambda na (inst s T) (inst (up 1 s) M)
  | tApp u v ⇒ tApp (inst s u) (inst s v)
  | tProd na A B ⇒ tProd na (inst s A) (inst (up 1 s) B)
  | tLetIn na b ty b' ⇒ tLetIn na (inst s b) (inst s ty) (inst (up 1 s) b')
  | tCase ind p c brs ⇒
    let brs' := List.map (on_snd (inst s)) brs in
    tCase ind (inst s p) (inst s c) brs'
  | tProj p c ⇒ tProj p (inst s c)
  | tFix mfix idx ⇒
    let mfix' := List.map (map_def (inst s) (inst (up (List.length mfix) s))) mfix in
    tFix mfix' idx
  | tCoFix mfix idx ⇒
    let mfix' := List.map (map_def (inst s) (inst (up (List.length mfix) s))) mfix in
    tCoFix mfix' idx
  | x ⇒ x
  end.

Definition subst_fn (l : list term) :=
  fun i ⇒
    match List.nth_error l i with
    | None ⇒ tRel (i - List.length l)
    | Some t ⇒ t
    end.

Lemma up_ext k s s' : s =1 s' → up k s =1 up k s'.

Lemma inst_ext s s' : s =1 s' → inst s =1 inst s'.

Definition ren (f : nat → nat) : nat → term :=
  fun i ⇒ tRel (f i).

Lemma ren_shiftn n f : up n (ren f) =1 ren (shiftn n f).

Instance proper_inst : Proper (`=1` ==> Logic.eq ==> Logic.eq) inst.

Instance proper_ext_eq {A B} : Proper (`=1` ==> `=1` ==> iff) (@pointwise_relation A _ (@Logic.eq B)).

Lemma rename_inst f : rename f =1 inst (ren f).


Instance rename_proper : Proper (`=1` ==> Logic.eq ==> Logic.eq) rename.

Definition substitution := nat → term.

Notation "t '.[' σ ]" := (inst σ t) (at level 6, format "t .[ σ ]") : sigma_scope.

Definition subst_cons (t : term) (f : nat → term) :=
  fun i ⇒
    match i with
    | 0 ⇒ t
    | S n ⇒ f n
    end.

Notation " t ⋅ s " := (subst_cons t s) (at level 90) : sigma_scope.

Instance subst_cons_proper : Proper (Logic.eq ==> `=1` ==> `=1`) subst_cons.

Definition shift : nat → term := tRel ∘ S.
Notation "↑" := shift : sigma_scope.

Definition subst_compose (σ τ : nat → term) :=
  fun i ⇒ (σ i).[τ].

Infix "∘" := subst_compose : sigma_scope.

Instance subst_compose_proper : Proper (`=1` ==> `=1` ==> `=1`) subst_compose.

Definition Up σ : substitution := tRel 0 ⋅ (σ ∘ ↑).
Notation "⇑ s" := (Up s) (at level 20).

Lemma up_Up σ : up 1 σ =1 ⇑ σ.

Definition ids (x : nat) := tRel x.

Definition ren_id (x : nat) := x.

Lemma ren_id_ids : ren ren_id =1 ids.

Lemma shiftn_id n : shiftn n ren_id =1 ren_id.

Lemma rename_ren_id : rename ren_id =1 id.

Lemma subst_ids t : t.[ids] = t.


Lemma compose_ids_r σ : σ ∘ ids =1 σ.

Lemma compose_ids_l σ : ids ∘ σ =1 σ.


Definition shiftk (k : nat) (x : nat) := tRel (k + x).
Notation "↑^ k" := (shiftk k) (at level 30, k at level 2, format "↑^ k") : sigma_scope.

Lemma shiftk_0 : shiftk 0 =1 ids.

Definition subst_consn {A} (l : list A) (σ : nat → A) :=
  fun i ⇒
    match List.nth_error l i with
    | None ⇒ σ (i - List.length l)
    | Some t ⇒ t
    end.

Notation " t ⋅n s " := (subst_consn t s) (at level 40) : sigma_scope.

Lemma subst_consn_nil {A} (σ : nat → A) : nil ⋅n σ =1 σ.

Lemma subst_consn_subst_cons t l σ : (t :: l) ⋅n σ =1 (t ⋅ subst_consn l σ).

Lemma subst_consn_tip t σ : [t] ⋅n σ =1 (t ⋅ σ).

Instance subst_consn_proper {A} : Proper (Logic.eq ==> `=1` ==> `=1`) (@subst_consn A).

Instance subst_consn_proper_ext {A} : Proper (Logic.eq ==> `=1` ==> Logic.eq ==> Logic.eq) (@subst_consn A).

Fixpoint idsn n : list term :=
  match n with
  | 0 ⇒ []
  | S n ⇒ idsn n ++ [tRel n]
  end.

Definition Upn n σ := idsn n ⋅n (σ ∘ ↑^n).
Notation "⇑^ n σ" := (Upn n σ) (at level 30, n at level 2, format "⇑^ n σ") : sigma_scope.

Lemma Upn_eq n σ : Upn n σ = idsn n ⋅n (σ ∘ ↑^n).

Lemma Upn_proper : Proper (Logic.eq ==> `=1` ==> `=1`) Upn.

Definition subst_cons_gen {A} (t : A) (f : nat → A) :=
  fun i ⇒
    match i with
    | 0 ⇒ t
    | S n ⇒ f n
    end.

Instance subst_cons_gen_proper {A} : Proper (Logic.eq ==> `=1` ==> `=1`) (@subst_cons_gen A).

Lemma subst_consn_subst_cons_gen {A} (t : A) l σ : subst_consn (t :: l) σ =1 (subst_cons_gen t (l ⋅n σ)).

Lemma subst_consn_app {A} {l l' : list A} {σ} : (l ++ l') ⋅n σ =1 l ⋅n (l' ⋅n σ).

Lemma subst_consn_ge {A} {l : list A} {i σ} : #|l| ≤ i → (l ⋅n σ) i = σ (i - #|l|).

Lemma subst_consn_lt {A} {l : list A} {i} :
  i < #|l| →
  { x : _ & (List.nth_error l i = Some x) ∧ (∀ σ, (l ⋅n σ) i = x) }%type.

Lemma idsn_length n : #|idsn n| = n.

Lemma idsn_lt {n i} : i < n → nth_error (idsn n) i = Some (tRel i).

Lemma nth_error_idsn_Some :
  ∀ n k,
    k < n →
    nth_error (idsn n) k = Some (tRel k).

Lemma nth_error_idsn_None :
  ∀ n k,
    k ≥ n →
    nth_error (idsn n) k = None.

Lemma up_Upn {n σ} : up n σ =1 ⇑^n σ.

Lemma inst_app {s t σ} : (tApp s t).[σ] = tApp s.[σ] t.[σ].

Lemma inst_lam {na t b σ} : (tLambda na t b).[σ] = tLambda na t.[σ] b.[⇑ σ].

Lemma inst_prod {na t b σ} : (tProd na t b).[σ] = tProd na t.[σ] b.[⇑ σ].

Lemma inst_letin {na t b b' σ} : (tLetIn na t b b').[σ] = tLetIn na t.[σ] b.[σ] b'.[⇑ σ].

Lemma inst_fix {mfix idx σ} : (tFix mfix idx).[σ] =
                              tFix (map (map_def (inst σ) (inst (⇑^#|mfix| σ))) mfix) idx.

Lemma inst_cofix {mfix idx σ} : (tCoFix mfix idx).[σ] =
                                tCoFix (map (map_def (inst σ) (inst (⇑^#|mfix| σ))) mfix) idx.

Lemma inst_mkApps :
  ∀ t l σ,
    (mkApps t l).[σ] = mkApps t.[σ] (map (inst σ) l).


Lemma subst_cons_0 t σ : (tRel 0).[t ⋅ σ] = t.
Lemma subst_cons_shift t σ : ↑ ∘ (t ⋅ σ) = σ.

Lemma shiftk_shift n : ↑^(S n) =1 ↑^n ∘ ↑.

Lemma shiftk_shift_l n : ↑^(S n) =1 ↑ ∘ ↑^n.

Lemma Upn_1_Up σ : ⇑^1 σ =1 ⇑ σ.

Lemma subst_subst_consn s σ τ : (s ⋅ σ) ∘ τ =1 (s.[τ] ⋅ σ ∘ τ).


Lemma ren_shift : ↑ =1 ren S.

Lemma compose_ren f g : ren f ∘ ren g =1 ren (Basics.compose g f).

Lemma subst_cons_ren i f : (tRel i ⋅ ren f) =1 ren (subst_cons_gen i f).

Fixpoint ren_ids (n : nat) :=
  match n with
  | 0 ⇒ []
  | S n ⇒ ren_ids n ++ [n]
  end.

Lemma ren_ids_length n : #|ren_ids n| = n.

Lemma ren_ids_lt {n i} : i < n → nth_error (ren_ids n) i = Some i.

Infix "=2" := (Logic.eq ==> (pointwise_relation _ Logic.eq))%signature (at level 80).

Definition compose2 {A B C} (g : B → C) (f : A → B) : A → C :=
  fun x ⇒ g (f x).
Infix "∘'" := compose2 (at level 90).


Lemma subst_consn_subst_cons' {A} (t : A) l : subst_consn (t :: l) =2 (subst_cons_gen t ∘ subst_consn l)%prog.

Lemma subst_consn_ids_ren n f : (idsn n ⋅n ren f) =1 ren (ren_ids n ⋅n f).

Lemma ren_shiftk n : ↑^n =1 ren (add n).

Lemma ren_subst_consn_comm:
  ∀ (f : nat → nat) (σ : nat → term) (n : nat),
    ren (subst_consn (ren_ids n) (Init.Nat.add n ∘ f)%prog) ∘ subst_consn (idsn n) (σ ∘ ↑^n) =1
    subst_consn (idsn n) (ren f ∘ σ ∘ ↑^n).

Lemma rename_inst_assoc t f σ : t.[ren f].[σ] = t.[ren f ∘ σ].

Lemma inst_rename_assoc_n:
  ∀ (f : nat → nat) (σ : nat → term) (n : nat),
    subst_consn (idsn n) (σ ∘ ↑^n) ∘ ren (subst_consn (ren_ids n) (Init.Nat.add n ∘ f)%prog) =1
    subst_consn (idsn n) (σ ∘ ren f ∘ ↑^n).

Lemma inst_rename_assoc t f σ : t.[σ].[ren f] = t.[σ ∘ ren f].

Lemma rename_subst_compose1 r s s' : ren r ∘ (s ∘ s') =1 ren r ∘ s ∘ s'.

Lemma rename_subst_compose2 r s s' : s ∘ (ren r ∘ s') =1 s ∘ ren r ∘ s'.

Lemma rename_subst_compose3 r s s' : s ∘ (s' ∘ ren r) =1 s ∘ s' ∘ ren r.

Lemma Up_Up_assoc:
  ∀ s s' : nat → term, (⇑ s) ∘ (⇑ s') =1 ⇑ (s ∘ s').


Lemma up_up_assoc:
  ∀ (s s' : nat → term) (n : nat), up n s ∘ up n s' =1 up n (s ∘ s').

Lemma inst_assoc t s s' : t.[s].[s'] = t.[s ∘ s'].


Lemma subst_compose_assoc s s' s'' : (s ∘ s') ∘ s'' =1 s ∘ (s' ∘ s'').


Lemma Upn_0 σ : ⇑^0 σ =1 σ.

Lemma Upn_Up σ n : ⇑^(S n) σ =1 ⇑^n ⇑ σ.

Lemma Upn_1 σ : ⇑^1 σ =1 ⇑ σ.

Lemma subst_cons_0_shift : tRel 0 ⋅ ↑ =1 ids.


Lemma subst_cons_0s_shifts σ : (σ 0) ⋅ (↑ ∘ σ) =1 σ.



Lemma subst_inst_aux s k t : subst s k t = inst (up k (subst_fn s)) t.

Lemma subst_fn_subst_consn s : subst_fn s =1 subst_consn s ids.

Theorem subst_inst s k t : subst s k t = inst (⇑^k (subst_consn s ids)) t.

Lemma term_forall_ctx_list_ind :
  ∀ P : context → term → Type,
    (∀ Γ (n : nat), P Γ (tRel n)) →
    (∀ Γ (i : ident), P Γ (tVar i)) →
    (∀ Γ (n : nat) (l : list term), All (P Γ) l → P Γ (tEvar n l)) →
    (∀ Γ s, P Γ (tSort s)) →
    (∀ Γ (n : name) (t : term), P Γ t → ∀ t0 : term, P (vass n t :: Γ) t0 → P Γ (tProd n t t0)) →
    (∀ Γ (n : name) (t : term), P Γ t → ∀ t0 : term, P (vass n t :: Γ) t0 → P Γ (tLambda n t t0)) →
    (∀ Γ (n : name) (t : term),
        P Γ t → ∀ t0 : term, P Γ t0 → ∀ t1 : term, P (vdef n t t0 :: Γ) t1 → P Γ (tLetIn n t t0 t1)) →
    (∀ Γ (t u : term), P Γ t → P Γ u → P Γ (tApp t u)) →
    (∀ Γ (s : String.string) (u : list Level.t), P Γ (tConst s u)) →
    (∀ Γ (i : inductive) (u : list Level.t), P Γ (tInd i u)) →
    (∀ Γ (i : inductive) (n : nat) (u : list Level.t), P Γ (tConstruct i n u)) →
    (∀ Γ (p : inductive × nat) (t : term),
        P Γ t → ∀ t0 : term, P Γ t0 → ∀ l : list (nat × term),
            tCaseBrsProp (P Γ) l → P Γ (tCase p t t0 l)) →
    (∀ Γ (s : projection) (t : term), P Γ t → P Γ (tProj s t)) →
    (∀ Γ (m : mfixpoint term) (n : nat), tFixProp (P Γ) (P (Γ ,,, fix_context m)) m → P Γ (tFix m n)) →
    (∀ Γ (m : mfixpoint term) (n : nat), tFixProp (P Γ) (P (Γ ,,, fix_context m)) m → P Γ (tCoFix m n)) →
    ∀ Γ (t : term), P Γ t.

Definition lift_decl n k d := (map_decl (lift n k) d).

Definition lift_context n k (Γ : context) : context :=
  fold_context (fun k' ⇒ lift n (k' + k)) Γ.

Lemma lift_decl0 k d : map_decl (lift 0 k) d = d.

Lemma lift0_context k Γ : lift_context 0 k Γ = Γ.

Lemma lift_context_length n k Γ : #|lift_context n k Γ| = #|Γ|.

Definition lift_context_snoc0 n k Γ d : lift_context n k (d :: Γ) = lift_context n k Γ ,, lift_decl n (#|Γ| + k) d.

Lemma lift_context_snoc n k Γ d : lift_context n k (Γ ,, d) = lift_context n k Γ ,, lift_decl n (#|Γ| + k) d.

Lemma lift_context_alt n k Γ :
  lift_context n k Γ =
  mapi (fun k' d ⇒ lift_decl n (Nat.pred #|Γ| - k' + k) d) Γ.

Lemma lift_context_app n k Γ Δ :
  lift_context n k (Γ ,,, Δ) = lift_context n k Γ ,,, lift_context n (#|Γ| + k) Δ.

Lemma nth_error_lift_context:
  ∀ (Γ' Γ'' : context) (v : nat),
    v < #|Γ'| → ∀ nth k,
    nth_error Γ' v = Some nth →
    nth_error (lift_context #|Γ''| k Γ') v = Some (lift_decl #|Γ''| (#|Γ'| - S v + k) nth).

Lemma nth_error_lift_context_eq:
  ∀ (Γ' Γ'' : context) (v : nat) k,
    nth_error (lift_context #|Γ''| k Γ') v =
    option_map (lift_decl #|Γ''| (#|Γ'| - S v + k)) (nth_error Γ' v).