Library ut_sr








The Subject Reduction Property.







Require Import base.



Require Import ut_term.



Require Import ut_red.



Require Import ut_env.



Require Import ut_typ.



Require Import List.






Module Type ut_sr_mod  (X:term_sig) (Y:pts_sig X) (TM:ut_term_mod X) (EM:ut_env_mod X TM) (RM: ut_red_mod X TM).
Interactive Module Type ut_sr_mod started





 Include (ut_typ_mod X Y TM EM RM).



  Import TM EM RM.
Warning: Notation _ →' _ was already used in scope UT_scope





Open Scope UT_scope.













In order to prove SR, we need to be able to talk
about reduction inside a context. 








Reserved Notation " a →e b " (at level 20).



Reserved Notation " a →→e b "(at level 20).



Reserved Notation " a ≡e b " (at level 20).






Inductive Beta_env : Env -> Env -> Prop :=

 | Beta_env_hd   : forall A B Γ , A →  B  -> (A::Γ) →e (B::Γ)

 | Beta_env_cons : forall A Γ Γ', Γ →e Γ' -> (A::Γ) →e (A::Γ')

where " E →e F " := (Beta_env E F) : UT_scope.
Beta_env is defined
Beta_env_ind is defined








Inductive Betas_env : Env -> Env -> Prop :=

 | Betas_env_refl  : forall Γ       , Γ →→e Γ

 | Betas_env_Beta  : forall Γ Γ'    , Γ →e Γ'  -> Γ →→e Γ' 

 | Betas_env_trans : forall Γ Γ' Γ'', Γ →→e Γ' -> Γ' →→e Γ'' -> Γ →→e Γ''

where " E →→e F" := (Betas_env E F) : UT_scope.
Betas_env is defined
Betas_env_ind is defined








Inductive Betac_env : Env -> Env -> Prop :=

 | Betac_env_Betas : forall Γ Γ'    , Γ →→e Γ' -> Γ ≡e Γ'

 | Betac_env_sym   : forall Γ Γ'    , Γ ≡e  Γ' -> Γ' ≡e Γ

 | Betac_env_trans : forall Γ Γ' Γ'', Γ ≡e  Γ' -> Γ' ≡e Γ'' -> Γ ≡e Γ''

where "E ≡e F" := (Betac_env E F) : UT_scope.
Betac_env is defined
Betac_env_ind is defined








Hint Constructors Beta_env Betas_env Betac_env.
Warning: the hint: eapply Betas_env_trans will only be used by eauto
Warning: the hint: eapply Betac_env_trans will only be used by eauto















Some basic properties of Beta inside context.





Lemma Betas_env_nil : forall Γ, nil →→e Γ -> Γ = nil.
1 subgoals, subgoal 1 (ID 28)
  
  ============================
   forall Γ : Env, nil →→e Γ -> Γ = nil

(dependent evars:)





intros. 
1 subgoals, subgoal 1 (ID 30)
  
  Γ : Env
  H : nil →→e Γ
  ============================
   Γ = nil

(dependent evars:)



remember (@nil Term) as N. 
1 subgoals, subgoal 1 (ID 37)
  
  Γ : Env
  N : list Term
  HeqN : N = nil
  H : N →→e Γ
  ============================
   Γ = N

(dependent evars:)



revert HeqN.
1 subgoals, subgoal 1 (ID 39)
  
  Γ : Env
  N : list Term
  H : N →→e Γ
  ============================
   N = nil -> Γ = N

(dependent evars:)





induction H; intros; subst; try discriminate.
3 subgoals, subgoal 1 (ID 69)
  
  ============================
   nil = nil

subgoal 2 (ID 74) is:
 Γ' = nil
subgoal 3 (ID 80) is:
 Γ'' = nil
(dependent evars:)





trivial. 
2 subgoals, subgoal 1 (ID 74)
  
  Γ' : Env
  H : nil →e Γ'
  ============================
   Γ' = nil

subgoal 2 (ID 80) is:
 Γ'' = nil
(dependent evars:)



inversion H. 
1 subgoals, subgoal 1 (ID 80)
  
  Γ' : Env
  Γ'' : Env
  H0 : Γ' →→e Γ''
  IHBetas_env2 : Γ' = nil -> Γ'' = Γ'
  H : nil →→e Γ'
  IHBetas_env1 : nil = nil -> Γ' = nil
  ============================
   Γ'' = nil

(dependent evars:)



rewrite IHBetas_env1 in *; trivial.
1 subgoals, subgoal 1 (ID 128)
  
  Γ' : Env
  Γ'' : Env
  H0 : nil →→e Γ''
  IHBetas_env2 : nil = nil -> Γ'' = nil
  H : nil →→e nil
  IHBetas_env1 : nil = nil -> Γ' = nil
  ============================
   Γ'' = nil

(dependent evars:)





intuition.
No more subgoals.
(dependent evars:)





Qed.
Betas_env_nil is defined








Lemma Betas_env_nil2 : forall Γ, Γ →→e nil -> Γ = nil.
1 subgoals, subgoal 1 (ID 169)
  
  ============================
   forall Γ : Env, Γ →→e nil -> Γ = nil

(dependent evars:)





intros. 
1 subgoals, subgoal 1 (ID 171)
  
  Γ : Env
  H : Γ →→e nil
  ============================
   Γ = nil

(dependent evars:)



remember (@nil Term) as N. 
1 subgoals, subgoal 1 (ID 178)
  
  Γ : Env
  N : list Term
  HeqN : N = nil
  H : Γ →→e N
  ============================
   Γ = N

(dependent evars:)



revert HeqN.
1 subgoals, subgoal 1 (ID 180)
  
  Γ : Env
  N : list Term
  H : Γ →→e N
  ============================
   N = nil -> Γ = N

(dependent evars:)





induction H; intros; subst; try discriminate.
3 subgoals, subgoal 1 (ID 210)
  
  ============================
   nil = nil

subgoal 2 (ID 215) is:
 Γ = nil
subgoal 3 (ID 221) is:
 Γ = nil
(dependent evars:)





trivial. 
2 subgoals, subgoal 1 (ID 215)
  
  Γ : Env
  H : Γ →e nil
  ============================
   Γ = nil

subgoal 2 (ID 221) is:
 Γ = nil
(dependent evars:)



inversion H. 
1 subgoals, subgoal 1 (ID 221)
  
  Γ : Env
  Γ' : Env
  H : Γ →→e Γ'
  IHBetas_env1 : Γ' = nil -> Γ = Γ'
  H0 : Γ' →→e nil
  IHBetas_env2 : nil = nil -> Γ' = nil
  ============================
   Γ = nil

(dependent evars:)



rewrite IHBetas_env2 in *; trivial.
1 subgoals, subgoal 1 (ID 277)
  
  Γ : Env
  Γ' : Env
  H : Γ →→e nil
  IHBetas_env1 : nil = nil -> Γ = nil
  H0 : nil →→e nil
  IHBetas_env2 : nil = nil -> Γ' = nil
  ============================
   Γ = nil

(dependent evars:)





intuition.
No more subgoals.
(dependent evars:)





Qed.
Betas_env_nil2 is defined








Lemma Betas_env_inv :forall A B Γ Γ', (A::Γ)  →→e ( B::Γ') -> A →→ B /\ Γ →→e Γ'.
1 subgoals, subgoal 1 (ID 322)
  
  ============================
   forall (A B : Term) (Γ Γ' : list Term),
   (A :: Γ) →→e (B :: Γ') -> (A →→ B) /\ Γ →→e Γ'

(dependent evars:)





intros. 
1 subgoals, subgoal 1 (ID 327)
  
  A : Term
  B : Term
  Γ : list Term
  Γ' : list Term
  H : (A :: Γ) →→e (B :: Γ')
  ============================
   (A →→ B) /\ Γ →→e Γ'

(dependent evars:)



remember (A::Γ) as Γ1; remember (B::Γ') as Γ2.
1 subgoals, subgoal 1 (ID 343)
  
  A : Term
  B : Term
  Γ : list Term
  Γ' : list Term
  Γ1 : list Term
  HeqΓ1 : Γ1 = A :: Γ
  Γ2 : list Term
  HeqΓ2 : Γ2 = B :: Γ'
  H : Γ1 →→e Γ2
  ============================
   (A →→ B) /\ Γ →→e Γ'

(dependent evars:)





revert A B Γ Γ' HeqΓ1 HeqΓ2. 
1 subgoals, subgoal 1 (ID 345)
  
  Γ1 : list Term
  Γ2 : list Term
  H : Γ1 →→e Γ2
  ============================
   forall (A B : Term) (Γ Γ' : list Term),
   Γ1 = A :: Γ -> Γ2 = B :: Γ' -> (A →→ B) /\ Γ →→e Γ'

(dependent evars:)



induction H; intros; subst; try discriminate.
3 subgoals, subgoal 1 (ID 392)
  
  A : Term
  B : Term
  Γ0 : list Term
  Γ' : list Term
  HeqΓ2 : A :: Γ0 = B :: Γ'
  ============================
   (A →→ B) /\ Γ0 →→e Γ'

subgoal 2 (ID 402) is:
 (A →→ B) /\ Γ0 →→e Γ'0
subgoal 3 (ID 414) is:
 (A →→ B) /\ Γ0 →→e Γ'0
(dependent evars:)





injection HeqΓ2; intros; subst; clear HeqΓ2. 
3 subgoals, subgoal 1 (ID 435)
  
  B : Term
  Γ' : list Term
  ============================
   (B →→ B) /\ Γ' →→e Γ'

subgoal 2 (ID 402) is:
 (A →→ B) /\ Γ0 →→e Γ'0
subgoal 3 (ID 414) is:
 (A →→ B) /\ Γ0 →→e Γ'0
(dependent evars:)



intuition.
2 subgoals, subgoal 1 (ID 402)
  
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H : (A :: Γ0) →e (B :: Γ'0)
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

subgoal 2 (ID 414) is:
 (A →→ B) /\ Γ0 →→e Γ'0
(dependent evars:)





inversion H; subst; clear H. 
3 subgoals, subgoal 1 (ID 583)
  
  A : Term
  B : Term
  Γ'0 : list Term
  H1 : A → B
  ============================
   (A →→ B) /\ Γ'0 →→e Γ'0

subgoal 2 (ID 584) is:
 (B →→ B) /\ Γ0 →→e Γ'0
subgoal 3 (ID 414) is:
 (A →→ B) /\ Γ0 →→e Γ'0
(dependent evars:)



intuition. 
2 subgoals, subgoal 1 (ID 584)
  
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H1 : Γ0 →e Γ'0
  ============================
   (B →→ B) /\ Γ0 →→e Γ'0

subgoal 2 (ID 414) is:
 (A →→ B) /\ Γ0 →→e Γ'0
(dependent evars:)



intuition.
1 subgoals, subgoal 1 (ID 414)
  
  Γ' : Env
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : Γ' →→e (B :: Γ'0)
  IHBetas_env2 : forall (A B0 : Term) (Γ Γ'1 : list Term),
                 Γ' = A :: Γ ->
                 B :: Γ'0 = B0 :: Γ'1 -> (A →→ B0) /\ Γ →→e Γ'1
  H : (A :: Γ0) →→e Γ'
  IHBetas_env1 : forall (A0 B : Term) (Γ Γ'0 : list Term),
                 A :: Γ0 = A0 :: Γ -> Γ' = B :: Γ'0 -> (A0 →→ B) /\ Γ →→e Γ'0
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

(dependent evars:)





destruct Γ'. 
2 subgoals, subgoal 1 (ID 622)
  
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : nil →→e (B :: Γ'0)
  IHBetas_env2 : forall (A B0 : Term) (Γ Γ' : list Term),
                 nil = A :: Γ -> B :: Γ'0 = B0 :: Γ' -> (A →→ B0) /\ Γ →→e Γ'
  H : (A :: Γ0) →→e nil
  IHBetas_env1 : forall (A0 B : Term) (Γ Γ' : list Term),
                 A :: Γ0 = A0 :: Γ -> nil = B :: Γ' -> (A0 →→ B) /\ Γ →→e Γ'
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

subgoal 2 (ID 630) is:
 (A →→ B) /\ Γ0 →→e Γ'0
(dependent evars:)



apply Betas_env_nil in H0. 
2 subgoals, subgoal 1 (ID 632)
  
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : B :: Γ'0 = nil
  IHBetas_env2 : forall (A B0 : Term) (Γ Γ' : list Term),
                 nil = A :: Γ -> B :: Γ'0 = B0 :: Γ' -> (A →→ B0) /\ Γ →→e Γ'
  H : (A :: Γ0) →→e nil
  IHBetas_env1 : forall (A0 B : Term) (Γ Γ' : list Term),
                 A :: Γ0 = A0 :: Γ -> nil = B :: Γ' -> (A0 →→ B) /\ Γ →→e Γ'
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

subgoal 2 (ID 630) is:
 (A →→ B) /\ Γ0 →→e Γ'0
(dependent evars:)



discriminate.
1 subgoals, subgoal 1 (ID 630)
  
  t : Term
  Γ' : list Term
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : (t :: Γ') →→e (B :: Γ'0)
  IHBetas_env2 : forall (A B0 : Term) (Γ Γ'1 : list Term),
                 t :: Γ' = A :: Γ ->
                 B :: Γ'0 = B0 :: Γ'1 -> (A →→ B0) /\ Γ →→e Γ'1
  H : (A :: Γ0) →→e (t :: Γ')
  IHBetas_env1 : forall (A0 B : Term) (Γ Γ'0 : list Term),
                 A :: Γ0 = A0 :: Γ ->
                 t :: Γ' = B :: Γ'0 -> (A0 →→ B) /\ Γ →→e Γ'0
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

(dependent evars:)





destruct (IHBetas_env1 A t Γ0 Γ') as ( ? & ?); trivial. 
1 subgoals, subgoal 1 (ID 650)
  
  t : Term
  Γ' : list Term
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : (t :: Γ') →→e (B :: Γ'0)
  IHBetas_env2 : forall (A B0 : Term) (Γ Γ'1 : list Term),
                 t :: Γ' = A :: Γ ->
                 B :: Γ'0 = B0 :: Γ'1 -> (A →→ B0) /\ Γ →→e Γ'1
  H : (A :: Γ0) →→e (t :: Γ')
  IHBetas_env1 : forall (A0 B : Term) (Γ Γ'0 : list Term),
                 A :: Γ0 = A0 :: Γ ->
                 t :: Γ' = B :: Γ'0 -> (A0 →→ B) /\ Γ →→e Γ'0
  H1 : A →→ t
  H2 : Γ0 →→e Γ'
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

(dependent evars:)



clear IHBetas_env1.
1 subgoals, subgoal 1 (ID 651)
  
  t : Term
  Γ' : list Term
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : (t :: Γ') →→e (B :: Γ'0)
  IHBetas_env2 : forall (A B0 : Term) (Γ Γ'1 : list Term),
                 t :: Γ' = A :: Γ ->
                 B :: Γ'0 = B0 :: Γ'1 -> (A →→ B0) /\ Γ →→e Γ'1
  H : (A :: Γ0) →→e (t :: Γ')
  H1 : A →→ t
  H2 : Γ0 →→e Γ'
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

(dependent evars:)





destruct (IHBetas_env2 t B Γ' Γ'0) as ( ? & ?); trivial. 
1 subgoals, subgoal 1 (ID 664)
  
  t : Term
  Γ' : list Term
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : (t :: Γ') →→e (B :: Γ'0)
  IHBetas_env2 : forall (A B0 : Term) (Γ Γ'1 : list Term),
                 t :: Γ' = A :: Γ ->
                 B :: Γ'0 = B0 :: Γ'1 -> (A →→ B0) /\ Γ →→e Γ'1
  H : (A :: Γ0) →→e (t :: Γ')
  H1 : A →→ t
  H2 : Γ0 →→e Γ'
  H3 : t →→ B
  H4 : Γ' →→e Γ'0
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

(dependent evars:)



clear IHBetas_env2.
1 subgoals, subgoal 1 (ID 665)
  
  t : Term
  Γ' : list Term
  A : Term
  B : Term
  Γ0 : list Term
  Γ'0 : list Term
  H0 : (t :: Γ') →→e (B :: Γ'0)
  H : (A :: Γ0) →→e (t :: Γ')
  H1 : A →→ t
  H2 : Γ0 →→e Γ'
  H3 : t →→ B
  H4 : Γ' →→e Γ'0
  ============================
   (A →→ B) /\ Γ0 →→e Γ'0

(dependent evars:)





eauto.
No more subgoals.
(dependent evars: ?669 using , ?681 using ,)





Qed.
Betas_env_inv is defined








Lemma Betas_env_hd : forall A B Γ , A→→B -> (A::Γ) →→e (B::Γ).
1 subgoals, subgoal 1 (ID 697)
  
  ============================
   forall (A B : Term) (Γ : list Term), A →→ B -> (A :: Γ) →→e (B :: Γ)

(dependent evars:)





intros.
1 subgoals, subgoal 1 (ID 701)
  
  A : Term
  B : Term
  Γ : list Term
  H : A →→ B
  ============================
   (A :: Γ) →→e (B :: Γ)

(dependent evars:)





induction H.
3 subgoals, subgoal 1 (ID 714)
  
  Γ : list Term
  M : Term
  ============================
   (M :: Γ) →→e (M :: Γ)

subgoal 2 (ID 717) is:
 (M :: Γ) →→e (N :: Γ)
subgoal 3 (ID 725) is:
 (M :: Γ) →→e (P :: Γ)
(dependent evars:)





constructor.
2 subgoals, subgoal 1 (ID 717)
  
  Γ : list Term
  M : Term
  N : Term
  H : M → N
  ============================
   (M :: Γ) →→e (N :: Γ)

subgoal 2 (ID 725) is:
 (M :: Γ) →→e (P :: Γ)
(dependent evars:)





constructor; apply Beta_env_hd; trivial.
1 subgoals, subgoal 1 (ID 725)
  
  Γ : list Term
  M : Term
  N : Term
  P : Term
  H : M →→ N
  H0 : N →→ P
  IHBetas1 : (M :: Γ) →→e (N :: Γ)
  IHBetas2 : (N :: Γ) →→e (P :: Γ)
  ============================
   (M :: Γ) →→e (P :: Γ)

(dependent evars:)





apply Betas_env_trans with (N::Γ); intuition.
No more subgoals.
(dependent evars:)





Qed.
Betas_env_hd is defined








Lemma Betas_env_cons : forall A Γ Γ', Γ →→e Γ' -> (A::Γ) →→e (A::Γ').
1 subgoals, subgoal 1 (ID 739)
  
  ============================
   forall (A : Term) (Γ Γ' : Env), Γ →→e Γ' -> (A :: Γ) →→e (A :: Γ')

(dependent evars:)





intros.
1 subgoals, subgoal 1 (ID 743)
  
  A : Term
  Γ : Env
  Γ' : Env
  H : Γ →→e Γ'
  ============================
   (A :: Γ) →→e (A :: Γ')

(dependent evars:)





induction H.
3 subgoals, subgoal 1 (ID 756)
  
  A : Term
  Γ : Env
  ============================
   (A :: Γ) →→e (A :: Γ)

subgoal 2 (ID 759) is:
 (A :: Γ) →→e (A :: Γ')
subgoal 3 (ID 767) is:
 (A :: Γ) →→e (A :: Γ'')
(dependent evars:)





constructor.
2 subgoals, subgoal 1 (ID 759)
  
  A : Term
  Γ : Env
  Γ' : Env
  H : Γ →e Γ'
  ============================
   (A :: Γ) →→e (A :: Γ')

subgoal 2 (ID 767) is:
 (A :: Γ) →→e (A :: Γ'')
(dependent evars:)





constructor; apply Beta_env_cons; trivial.
1 subgoals, subgoal 1 (ID 767)
  
  A : Term
  Γ : Env
  Γ' : Env
  Γ'' : Env
  H : Γ →→e Γ'
  H0 : Γ' →→e Γ''
  IHBetas_env1 : (A :: Γ) →→e (A :: Γ')
  IHBetas_env2 : (A :: Γ') →→e (A :: Γ'')
  ============================
   (A :: Γ) →→e (A :: Γ'')

(dependent evars:)





apply Betas_env_trans with (A::Γ'); trivial.
No more subgoals.
(dependent evars:)





Qed.
Betas_env_cons is defined








Lemma Betas_env_comp : forall A B Γ Γ', A→→B -> Γ →→e Γ' -> (A::Γ) →→e (B::Γ').
1 subgoals, subgoal 1 (ID 782)
  
  ============================
   forall (A B : Term) (Γ Γ' : Env),
   A →→ B -> Γ →→e Γ' -> (A :: Γ) →→e (B :: Γ')

(dependent evars:)





intros.
1 subgoals, subgoal 1 (ID 788)
  
  A : Term
  B : Term
  Γ : Env
  Γ' : Env
  H : A →→ B
  H0 : Γ →→e Γ'
  ============================
   (A :: Γ) →→e (B :: Γ')

(dependent evars:)





apply Betas_env_trans with (A::Γ').
2 subgoals, subgoal 1 (ID 790)
  
  A : Term
  B : Term
  Γ : Env
  Γ' : Env
  H : A →→ B
  H0 : Γ →→e Γ'
  ============================
   (A :: Γ) →→e (A :: Γ')

subgoal 2 (ID 791) is:
 (A :: Γ') →→e (B :: Γ')
(dependent evars:)





apply Betas_env_cons; trivial.
1 subgoals, subgoal 1 (ID 791)
  
  A : Term
  B : Term
  Γ : Env
  Γ' : Env
  H : A →→ B
  H0 : Γ →→e Γ'
  ============================
   (A :: Γ') →→e (B :: Γ')

(dependent evars:)





apply Betas_env_hd; trivial.
No more subgoals.
(dependent evars:)





Qed.
Betas_env_comp is defined















There is two ways to prove SR: either we prove that reduction inside the
context and inside the term "at the same time" is valid, or we first do this
for the context only at first, but we need to weaken a little the hypothesis.
  Here we choose the second solution. First step toward full SR: if we do reductions inside the context
  of a valid judgment, and if the resulting context is well-formed, 
  then the judgement is still valid with the resulting context. 





Lemma Beta_env_sound :   forall Γ M T, Γ ⊢ M : T -> forall Γ', Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : T .
1 subgoals, subgoal 1 (ID 798)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T -> forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : T

(dependent evars:)





induction 1; intros.
6 subgoals, subgoal 1 (ID 894)
  
  Γ : Env
  s : X.Sorts
  t : X.Sorts
  H : Y.Ax s t
  H0 : Γ ⊣
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ →e Γ'
  ============================
   Γ' ⊢ !s : !t

subgoal 2 (ID 897) is:
 Γ' ⊢ #v : A
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)





constructor; trivial.
5 subgoals, subgoal 1 (ID 897)
  
  Γ : Env
  A : Term
  v : nat
  H : Γ ⊣
  H0 : A ↓ v ⊂ Γ
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ →e Γ'
  ============================
   Γ' ⊢ #v : A

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)





revert A v H0 H H1.
5 subgoals, subgoal 1 (ID 914)
  
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  ============================
   forall (A : Term) (v : nat), A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)





induction H2; intros. 
6 subgoals, subgoal 1 (ID 936)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  A0 : Term
  v : nat
  H0 : A0 ↓ v ⊂ A :: Γ
  H1 : A :: Γ ⊣
  H2 : B :: Γ ⊣
  ============================
   B :: Γ ⊢ #v : A0

subgoal 2 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)



destruct H0 as (AA&  ? & ?). 
6 subgoals, subgoal 1 (ID 952)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  A0 : Term
  v : nat
  AA : Term
  H0 : A0 = AA ↑ (S v)
  H3 : AA ↓ v ∈ A :: Γ
  H1 : A :: Γ ⊣
  H2 : B :: Γ ⊣
  ============================
   B :: Γ ⊢ #v : A0

subgoal 2 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)



inversion H3; subst; clear H3.
7 subgoals, subgoal 1 (ID 1041)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  H1 : A :: Γ ⊣
  H2 : B :: Γ ⊣
  ============================
   B :: Γ ⊢ #0 : A ↑ 1

subgoal 2 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)





inversion H1; subst; clear H1. 
7 subgoals, subgoal 1 (ID 1091)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  H2 : B :: Γ ⊣
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  ============================
   B :: Γ ⊢ #0 : A ↑ 1

subgoal 2 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)



inversion H2; subst; clear H2.
7 subgoals, subgoal 1 (ID 1140)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ B : !s0
  ============================
   B :: Γ ⊢ #0 : A ↑ 1

subgoal 2 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)





apply Cnv with ( B  ↑ 1 ) s. 
9 subgoals, subgoal 1 (ID 1141)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ B : !s0
  ============================
   B ↑ 1 ≡ A ↑ 1

subgoal 2 (ID 1142) is:
 B :: Γ ⊢ #0 : B ↑ 1
subgoal 3 (ID 1143) is:
 B :: Γ ⊢ A ↑ 1 : !s
subgoal 4 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 5 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 6 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 7 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 8 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 9 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)



intuition.
8 subgoals, subgoal 1 (ID 1142)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ B : !s0
  ============================
   B :: Γ ⊢ #0 : B ↑ 1

subgoal 2 (ID 1143) is:
 B :: Γ ⊢ A ↑ 1 : !s
subgoal 3 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 4 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 5 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 6 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 7 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 8 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)





constructor. 
9 subgoals, subgoal 1 (ID 1165)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ B : !s0
  ============================
   B :: Γ ⊣

subgoal 2 (ID 1166) is:
 B ↑ 1 ↓ 0 ⊂ B :: Γ
subgoal 3 (ID 1143) is:
 B :: Γ ⊢ A ↑ 1 : !s
subgoal 4 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 5 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 6 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 7 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 8 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 9 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars:)



econstructor;apply H1. 
8 subgoals, subgoal 1 (ID 1166)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ B : !s0
  ============================
   B ↑ 1 ↓ 0 ⊂ B :: Γ

subgoal 2 (ID 1143) is:
 B :: Γ ⊢ A ↑ 1 : !s
subgoal 3 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 4 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 5 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 6 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 7 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 8 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



exists B; intuition.
7 subgoals, subgoal 1 (ID 1143)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ B : !s0
  ============================
   B :: Γ ⊢ A ↑ 1 : !s

subgoal 2 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





change !s with !s ↑ 1. 
7 subgoals, subgoal 1 (ID 1187)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ B : !s0
  ============================
   B :: Γ ⊢ A ↑ 1 : !s ↑ 1

subgoal 2 (ID 1042) is:
 B :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



apply thinning with s0 ;trivial.
6 subgoals, subgoal 1 (ID 1042)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  AA : Term
  H1 : A :: Γ ⊣
  H2 : B :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   B :: Γ ⊢ #(S n) : AA ↑ (S (S n))

subgoal 2 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





constructor. 
7 subgoals, subgoal 1 (ID 1192)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  AA : Term
  H1 : A :: Γ ⊣
  H2 : B :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   B :: Γ ⊣

subgoal 2 (ID 1193) is:
 AA ↑ (S (S n)) ↓ S n ⊂ B :: Γ
subgoal 3 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



trivial. 
6 subgoals, subgoal 1 (ID 1193)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  AA : Term
  H1 : A :: Γ ⊣
  H2 : B :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   AA ↑ (S (S n)) ↓ S n ⊂ B :: Γ

subgoal 2 (ID 941) is:
 A :: Γ' ⊢ #v : A0
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



exists AA; intuition.
5 subgoals, subgoal 1 (ID 941)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  A0 : Term
  v : nat
  H0 : A0 ↓ v ⊂ A :: Γ
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  ============================
   A :: Γ' ⊢ #v : A0

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





destruct H0 as (AA&  ? & ?). 
5 subgoals, subgoal 1 (ID 1221)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  A0 : Term
  v : nat
  AA : Term
  H0 : A0 = AA ↑ (S v)
  H3 : AA ↓ v ∈ A :: Γ
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  ============================
   A :: Γ' ⊢ #v : A0

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



inversion H3; subst; clear H3.
6 subgoals, subgoal 1 (ID 1310)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  ============================
   A :: Γ' ⊢ #0 : A ↑ 1

subgoal 2 (ID 1311) is:
 A :: Γ' ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





constructor. 
7 subgoals, subgoal 1 (ID 1314)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  ============================
   A :: Γ' ⊣

subgoal 2 (ID 1315) is:
 A ↑ 1 ↓ 0 ⊂ A :: Γ'
subgoal 3 (ID 1311) is:
 A :: Γ' ⊢ #(S n) : AA ↑ (S (S n))
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



trivial. 
6 subgoals, subgoal 1 (ID 1315)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  ============================
   A ↑ 1 ↓ 0 ⊂ A :: Γ'

subgoal 2 (ID 1311) is:
 A :: Γ' ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



exists A; intuition.
5 subgoals, subgoal 1 (ID 1311)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   A :: Γ' ⊢ #(S n) : AA ↑ (S (S n))

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





change (S(S n)) with (1+S n). 
5 subgoals, subgoal 1 (ID 1331)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   A :: Γ' ⊢ #(S n) : AA ↑ (1 + S n)

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



change #(S n) with (#n  ↑ 1).
5 subgoals, subgoal 1 (ID 1333)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   A :: Γ' ⊢ #n ↑ 1 : AA ↑ (1 + S n)

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





rewrite <- lift_lift.
5 subgoals, subgoal 1 (ID 1334)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  H1 : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   A :: Γ' ⊢ #n ↑ 1 : AA ↑ (S n) ↑ 1

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





inversion H1; subst; clear H1.
5 subgoals, subgoal 1 (ID 1383)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  s : X.Sorts
  H3 : Γ' ⊢ A : !s
  ============================
   A :: Γ' ⊢ #n ↑ 1 : AA ↑ (S n) ↑ 1

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)





apply thinning with s; trivial. 
5 subgoals, subgoal 1 (ID 1384)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  s : X.Sorts
  H3 : Γ' ⊢ A : !s
  ============================
   Γ' ⊢ #n : AA ↑ (S n)

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



apply IHBeta_env. 
7 subgoals, subgoal 1 (ID 1386)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  s : X.Sorts
  H3 : Γ' ⊢ A : !s
  ============================
   AA ↑ (S n) ↓ n ⊂ Γ

subgoal 2 (ID 1387) is:
 Γ ⊣
subgoal 3 (ID 1388) is:
 Γ' ⊣
subgoal 4 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using ,)



eauto. 
6 subgoals, subgoal 1 (ID 1387)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  s : X.Sorts
  H3 : Γ' ⊢ A : !s
  ============================
   Γ ⊣

subgoal 2 (ID 1388) is:
 Γ' ⊣
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using ,)



inversion H; subst; clear H. 
6 subgoals, subgoal 1 (ID 1443)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  n : nat
  H7 : AA ↓ n ∈ Γ
  s : X.Sorts
  H3 : Γ' ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   Γ ⊣

subgoal 2 (ID 1388) is:
 Γ' ⊣
subgoal 3 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using ,)



eauto.
5 subgoals, subgoal 1 (ID 1388)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A
  AA : Term
  H : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  s : X.Sorts
  H3 : Γ' ⊢ A : !s
  ============================
   Γ' ⊣

subgoal 2 (ID 900) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using ,)





eauto.
4 subgoals, subgoal 1 (ID 900)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   Γ' ⊢ Π (A), B : !u

subgoal 2 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 3 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 4 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using ,)





econstructor. 
6 subgoals, subgoal 1 (ID 1561)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   Y.Rel ?1559 ?1560 u

subgoal 2 (ID 1562) is:
 Γ' ⊢ A : !?1559
subgoal 3 (ID 1563) is:
 A :: Γ' ⊢ B : !?1560
subgoal 4 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 open, ?1560 open,)



apply H. 
5 subgoals, subgoal 1 (ID 1562)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   Γ' ⊢ A : !s

subgoal 2 (ID 1563) is:
 A :: Γ' ⊢ B : !t
subgoal 3 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using ,)



apply IHtyp1; eauto. 
4 subgoals, subgoal 1 (ID 1563)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   A :: Γ' ⊢ B : !t

subgoal 2 (ID 903) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 3 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 4 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using ,)



apply IHtyp2; eauto.
3 subgoals, subgoal 1 (ID 903)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ →e Γ'
  ============================
   Γ' ⊢ λ [A], M : Π (A), B

subgoal 2 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 3 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using ,)





econstructor. 
6 subgoals, subgoal 1 (ID 1594)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ →e Γ'
  ============================
   Y.Rel ?1591 ?1592 ?1593

subgoal 2 (ID 1595) is:
 Γ' ⊢ A : !?1591
subgoal 3 (ID 1596) is:
 A :: Γ' ⊢ B : !?1592
subgoal 4 (ID 1597) is:
 A :: Γ' ⊢ M : B
subgoal 5 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 open, ?1592 open, ?1593 open,)



apply H. 
5 subgoals, subgoal 1 (ID 1595)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ →e Γ'
  ============================
   Γ' ⊢ A : !s1

subgoal 2 (ID 1596) is:
 A :: Γ' ⊢ B : !s2
subgoal 3 (ID 1597) is:
 A :: Γ' ⊢ M : B
subgoal 4 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using ,)



apply IHtyp1; eauto. 
4 subgoals, subgoal 1 (ID 1596)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ →e Γ'
  ============================
   A :: Γ' ⊢ B : !s2

subgoal 2 (ID 1597) is:
 A :: Γ' ⊢ M : B
subgoal 3 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 4 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using ,)



apply IHtyp2; eauto. 
3 subgoals, subgoal 1 (ID 1597)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> (A :: Γ) →e Γ' -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ →e Γ'
  ============================
   A :: Γ' ⊢ M : B

subgoal 2 (ID 906) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 3 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using ,)



apply IHtyp3; eauto.
2 subgoals, subgoal 1 (ID 906)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : Π (A), B
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ N : A
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ →e Γ'
  ============================
   Γ' ⊢ M · N : B [ ← N]

subgoal 2 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using ,)





econstructor. 
3 subgoals, subgoal 1 (ID 1645)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : Π (A), B
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ N : A
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ →e Γ'
  ============================
   Γ' ⊢ M : Π (?1644), B

subgoal 2 (ID 1646) is:
 Γ' ⊢ N : ?1644
subgoal 3 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using , ?1644 open,)



apply IHtyp1; eauto. 
2 subgoals, subgoal 1 (ID 1646)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : Π (A), B
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ N : A
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ →e Γ'
  ============================
   Γ' ⊢ N : A

subgoal 2 (ID 909) is:
 Γ' ⊢ M : B
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using , ?1644 using ,)



apply IHtyp2; eauto.
1 subgoals, subgoal 1 (ID 909)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   Γ' ⊢ M : B

(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using , ?1644 using ,)





econstructor. 
3 subgoals, subgoal 1 (ID 1665)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   ?1663 ≡ B

subgoal 2 (ID 1666) is:
 Γ' ⊢ M : ?1663
subgoal 3 (ID 1667) is:
 Γ' ⊢ B : !?1664
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using , ?1644 using , ?1663 open, ?1664 open,)



apply H. 
2 subgoals, subgoal 1 (ID 1666)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   Γ' ⊢ M : A

subgoal 2 (ID 1667) is:
 Γ' ⊢ B : !?1664
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using , ?1644 using , ?1663 using , ?1664 open,)



eapply IHtyp1; eauto. 
1 subgoals, subgoal 1 (ID 1667)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ →e Γ' -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ →e Γ'
  ============================
   Γ' ⊢ B : !?1664

(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using , ?1644 using , ?1663 using , ?1664 open,)



apply IHtyp2; eauto.
No more subgoals.
(dependent evars: ?1169 using , ?1390 using , ?1444 using , ?1445 using , ?1498 using , ?1499 using , ?1559 using , ?1560 using , ?1571 using , ?1591 using , ?1592 using , ?1593 using , ?1605 using , ?1623 using , ?1644 using , ?1663 using , ?1664 using ,)





Qed.
Beta_env_sound is defined















Same with expansion in the context.





Lemma Beta_env_sound_up :   forall Γ M T, Γ ⊢ M : T -> forall Γ', Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : T  .
1 subgoals, subgoal 1 (ID 1682)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T -> forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : T

(dependent evars:)





induction 1; intros.
6 subgoals, subgoal 1 (ID 1778)
  
  Γ : Env
  s : X.Sorts
  t : X.Sorts
  H : Y.Ax s t
  H0 : Γ ⊣
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ' →e Γ
  ============================
   Γ' ⊢ !s : !t

subgoal 2 (ID 1781) is:
 Γ' ⊢ #v : A
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)





constructor; trivial.
5 subgoals, subgoal 1 (ID 1781)
  
  Γ : Env
  A : Term
  v : nat
  H : Γ ⊣
  H0 : A ↓ v ⊂ Γ
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ' →e Γ
  ============================
   Γ' ⊢ #v : A

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)





revert A v H0 H H1. 
5 subgoals, subgoal 1 (ID 1798)
  
  Γ : Env
  Γ' : Env
  H2 : Γ' →e Γ
  ============================
   forall (A : Term) (v : nat), A ↓ v ⊂ Γ -> Γ ⊣ -> Γ' ⊣ -> Γ' ⊢ #v : A

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)



induction H2; intros. 
6 subgoals, subgoal 1 (ID 1820)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  A0 : Term
  v : nat
  H0 : A0 ↓ v ⊂ B :: Γ
  H1 : B :: Γ ⊣
  H2 : A :: Γ ⊣
  ============================
   A :: Γ ⊢ #v : A0

subgoal 2 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)



destruct H0 as (AA&  ? & ?). 
6 subgoals, subgoal 1 (ID 1836)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  A0 : Term
  v : nat
  AA : Term
  H0 : A0 = AA ↑ (S v)
  H3 : AA ↓ v ∈ B :: Γ
  H1 : B :: Γ ⊣
  H2 : A :: Γ ⊣
  ============================
   A :: Γ ⊢ #v : A0

subgoal 2 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)



inversion H3; subst; clear H3.
7 subgoals, subgoal 1 (ID 1925)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  H1 : B :: Γ ⊣
  H2 : A :: Γ ⊣
  ============================
   A :: Γ ⊢ #0 : B ↑ 1

subgoal 2 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)





inversion H1; subst; clear H1. 
7 subgoals, subgoal 1 (ID 1975)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  H2 : A :: Γ ⊣
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  ============================
   A :: Γ ⊢ #0 : B ↑ 1

subgoal 2 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)



inversion H2; subst; clear H2.
7 subgoals, subgoal 1 (ID 2024)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   A :: Γ ⊢ #0 : B ↑ 1

subgoal 2 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)





apply Cnv with ( A  ↑ 1 ) s. 
9 subgoals, subgoal 1 (ID 2025)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   A ↑ 1 ≡ B ↑ 1

subgoal 2 (ID 2026) is:
 A :: Γ ⊢ #0 : A ↑ 1
subgoal 3 (ID 2027) is:
 A :: Γ ⊢ B ↑ 1 : !s
subgoal 4 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 5 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 6 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 7 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 8 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 9 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)



intuition.
8 subgoals, subgoal 1 (ID 2026)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   A :: Γ ⊢ #0 : A ↑ 1

subgoal 2 (ID 2027) is:
 A :: Γ ⊢ B ↑ 1 : !s
subgoal 3 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 4 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 5 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 6 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 7 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 8 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)





constructor. 
9 subgoals, subgoal 1 (ID 2055)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   A :: Γ ⊣

subgoal 2 (ID 2056) is:
 A ↑ 1 ↓ 0 ⊂ A :: Γ
subgoal 3 (ID 2027) is:
 A :: Γ ⊢ B ↑ 1 : !s
subgoal 4 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 5 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 6 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 7 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 8 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 9 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars:)



econstructor; apply H1. 
8 subgoals, subgoal 1 (ID 2056)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   A ↑ 1 ↓ 0 ⊂ A :: Γ

subgoal 2 (ID 2027) is:
 A :: Γ ⊢ B ↑ 1 : !s
subgoal 3 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 4 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 5 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 6 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 7 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 8 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



exists A; intuition.
7 subgoals, subgoal 1 (ID 2027)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   A :: Γ ⊢ B ↑ 1 : !s

subgoal 2 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





change !s with !s ↑ 1. 
7 subgoals, subgoal 1 (ID 2077)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  s : X.Sorts
  H3 : Γ ⊢ B : !s
  s0 : X.Sorts
  H1 : Γ ⊢ A : !s0
  ============================
   A :: Γ ⊢ B ↑ 1 : !s ↑ 1

subgoal 2 (ID 1926) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



apply thinning with s0 ;trivial.
6 subgoals, subgoal 1 (ID 1926)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  AA : Term
  H1 : B :: Γ ⊣
  H2 : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   A :: Γ ⊢ #(S n) : AA ↑ (S (S n))

subgoal 2 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





constructor. 
7 subgoals, subgoal 1 (ID 2082)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  AA : Term
  H1 : B :: Γ ⊣
  H2 : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   A :: Γ ⊣

subgoal 2 (ID 2083) is:
 AA ↑ (S (S n)) ↓ S n ⊂ A :: Γ
subgoal 3 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



trivial. 
6 subgoals, subgoal 1 (ID 2083)
  
  A : Term
  B : Term
  Γ : list Term
  H : A → B
  AA : Term
  H1 : B :: Γ ⊣
  H2 : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ
  ============================
   AA ↑ (S (S n)) ↓ S n ⊂ A :: Γ

subgoal 2 (ID 1825) is:
 A :: Γ ⊢ #v : A0
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



exists AA; intuition.
5 subgoals, subgoal 1 (ID 1825)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  A0 : Term
  v : nat
  H0 : A0 ↓ v ⊂ A :: Γ'
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  ============================
   A :: Γ ⊢ #v : A0

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





destruct H0 as (AA&  ? & ?). 
5 subgoals, subgoal 1 (ID 2111)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  A0 : Term
  v : nat
  AA : Term
  H0 : A0 = AA ↑ (S v)
  H3 : AA ↓ v ∈ A :: Γ'
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  ============================
   A :: Γ ⊢ #v : A0

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



inversion H3; subst; clear H3.
6 subgoals, subgoal 1 (ID 2200)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  ============================
   A :: Γ ⊢ #0 : A ↑ 1

subgoal 2 (ID 2201) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





constructor. 
7 subgoals, subgoal 1 (ID 2204)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  ============================
   A :: Γ ⊣

subgoal 2 (ID 2205) is:
 A ↑ 1 ↓ 0 ⊂ A :: Γ
subgoal 3 (ID 2201) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



trivial. 
6 subgoals, subgoal 1 (ID 2205)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  ============================
   A ↑ 1 ↓ 0 ⊂ A :: Γ

subgoal 2 (ID 2201) is:
 A :: Γ ⊢ #(S n) : AA ↑ (S (S n))
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



exists A; intuition.
5 subgoals, subgoal 1 (ID 2201)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  ============================
   A :: Γ ⊢ #(S n) : AA ↑ (S (S n))

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





change (S(S n)) with (1+S n). 
5 subgoals, subgoal 1 (ID 2221)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  ============================
   A :: Γ ⊢ #(S n) : AA ↑ (1 + S n)

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



change #(S n) with (#n  ↑ 1).
5 subgoals, subgoal 1 (ID 2223)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  ============================
   A :: Γ ⊢ #n ↑ 1 : AA ↑ (1 + S n)

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





rewrite <- lift_lift.
5 subgoals, subgoal 1 (ID 2224)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  H1 : A :: Γ ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  ============================
   A :: Γ ⊢ #n ↑ 1 : AA ↑ (S n) ↑ 1

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





inversion H1; subst; clear H1.
5 subgoals, subgoal 1 (ID 2273)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  ============================
   A :: Γ ⊢ #n ↑ 1 : AA ↑ (S n) ↑ 1

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)





apply thinning with s; trivial. 
5 subgoals, subgoal 1 (ID 2274)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  ============================
   Γ ⊢ #n : AA ↑ (S n)

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



apply IHBeta_env. 
7 subgoals, subgoal 1 (ID 2276)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  ============================
   AA ↑ (S n) ↓ n ⊂ Γ'

subgoal 2 (ID 2277) is:
 Γ' ⊣
subgoal 3 (ID 2278) is:
 Γ ⊣
subgoal 4 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 5 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 6 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 7 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



exists AA; intuition. 
6 subgoals, subgoal 1 (ID 2277)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  ============================
   Γ' ⊣

subgoal 2 (ID 2278) is:
 Γ ⊣
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



inversion H; subst; clear H. 
6 subgoals, subgoal 1 (ID 2333)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  n : nat
  H7 : AA ↓ n ∈ Γ'
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  s0 : X.Sorts
  H1 : Γ' ⊢ A : !s0
  ============================
   Γ' ⊣

subgoal 2 (ID 2278) is:
 Γ ⊣
subgoal 3 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using ,)



eauto.
5 subgoals, subgoal 1 (ID 2278)
  
  A : Term
  Γ : Env
  Γ' : Env
  H2 : Γ →e Γ'
  IHBeta_env : forall (A : Term) (v : nat),
               A ↓ v ⊂ Γ' -> Γ' ⊣ -> Γ ⊣ -> Γ ⊢ #v : A
  AA : Term
  H : A :: Γ' ⊣
  n : nat
  H7 : AA ↓ n ∈ Γ'
  s : X.Sorts
  H3 : Γ ⊢ A : !s
  ============================
   Γ ⊣

subgoal 2 (ID 1784) is:
 Γ' ⊢ Π (A), B : !u
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using ,)





eauto.
4 subgoals, subgoal 1 (ID 1784)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   Γ' ⊢ Π (A), B : !u

subgoal 2 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 3 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 4 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using ,)





econstructor. 
6 subgoals, subgoal 1 (ID 2451)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   Y.Rel ?2449 ?2450 u

subgoal 2 (ID 2452) is:
 Γ' ⊢ A : !?2449
subgoal 3 (ID 2453) is:
 A :: Γ' ⊢ B : !?2450
subgoal 4 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 open, ?2450 open,)



apply H. 
5 subgoals, subgoal 1 (ID 2452)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   Γ' ⊢ A : !s

subgoal 2 (ID 2453) is:
 A :: Γ' ⊢ B : !t
subgoal 3 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using ,)



apply IHtyp1; eauto. 
4 subgoals, subgoal 1 (ID 2453)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !t
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   A :: Γ' ⊢ B : !t

subgoal 2 (ID 1787) is:
 Γ' ⊢ λ [A], M : Π (A), B
subgoal 3 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 4 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using ,)



apply IHtyp2; eauto.
3 subgoals, subgoal 1 (ID 1787)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ' →e Γ
  ============================
   Γ' ⊢ λ [A], M : Π (A), B

subgoal 2 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 3 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using ,)





econstructor. 
6 subgoals, subgoal 1 (ID 2484)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ' →e Γ
  ============================
   Y.Rel ?2481 ?2482 ?2483

subgoal 2 (ID 2485) is:
 Γ' ⊢ A : !?2481
subgoal 3 (ID 2486) is:
 A :: Γ' ⊢ B : !?2482
subgoal 4 (ID 2487) is:
 A :: Γ' ⊢ M : B
subgoal 5 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 6 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 open, ?2482 open, ?2483 open,)



apply H. 
5 subgoals, subgoal 1 (ID 2485)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ' →e Γ
  ============================
   Γ' ⊢ A : !s1

subgoal 2 (ID 2486) is:
 A :: Γ' ⊢ B : !s2
subgoal 3 (ID 2487) is:
 A :: Γ' ⊢ M : B
subgoal 4 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 5 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using ,)



apply IHtyp1; eauto. 
4 subgoals, subgoal 1 (ID 2486)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ' →e Γ
  ============================
   A :: Γ' ⊢ B : !s2

subgoal 2 (ID 2487) is:
 A :: Γ' ⊢ M : B
subgoal 3 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 4 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using ,)



apply IHtyp2; eauto. 
3 subgoals, subgoal 1 (ID 2487)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ A : !s1
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ B : !s2
  IHtyp3 : forall Γ' : Env, Γ' ⊣ -> Γ' →e (A :: Γ) -> Γ' ⊢ M : B
  Γ' : Env
  H3 : Γ' ⊣
  H4 : Γ' →e Γ
  ============================
   A :: Γ' ⊢ M : B

subgoal 2 (ID 1790) is:
 Γ' ⊢ M · N : B [ ← N]
subgoal 3 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using ,)



apply IHtyp3; eauto.
2 subgoals, subgoal 1 (ID 1790)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : Π (A), B
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ N : A
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ' →e Γ
  ============================
   Γ' ⊢ M · N : B [ ← N]

subgoal 2 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using ,)





econstructor. 
3 subgoals, subgoal 1 (ID 2535)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : Π (A), B
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ N : A
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ' →e Γ
  ============================
   Γ' ⊢ M : Π (?2534), B

subgoal 2 (ID 2536) is:
 Γ' ⊢ N : ?2534
subgoal 3 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using , ?2534 open,)



apply IHtyp1; eauto. 
2 subgoals, subgoal 1 (ID 2536)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : Π (A), B
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ N : A
  Γ' : Env
  H1 : Γ' ⊣
  H2 : Γ' →e Γ
  ============================
   Γ' ⊢ N : A

subgoal 2 (ID 1793) is:
 Γ' ⊢ M : B
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using , ?2534 using ,)



apply IHtyp2; eauto.
1 subgoals, subgoal 1 (ID 1793)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   Γ' ⊢ M : B

(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using , ?2534 using ,)





econstructor. 
3 subgoals, subgoal 1 (ID 2555)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   ?2553 ≡ B

subgoal 2 (ID 2556) is:
 Γ' ⊢ M : ?2553
subgoal 3 (ID 2557) is:
 Γ' ⊢ B : !?2554
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using , ?2534 using , ?2553 open, ?2554 open,)



apply H. 
2 subgoals, subgoal 1 (ID 2556)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   Γ' ⊢ M : A

subgoal 2 (ID 2557) is:
 Γ' ⊢ B : !?2554
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using , ?2534 using , ?2553 using , ?2554 open,)



eapply IHtyp1; eauto. 
1 subgoals, subgoal 1 (ID 2557)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ M : A
  IHtyp2 : forall Γ' : Env, Γ' ⊣ -> Γ' →e Γ -> Γ' ⊢ B : !s
  Γ' : Env
  H2 : Γ' ⊣
  H3 : Γ' →e Γ
  ============================
   Γ' ⊢ B : !?2554

(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using , ?2534 using , ?2553 using , ?2554 open,)



apply IHtyp2; eauto.
No more subgoals.
(dependent evars: ?2059 using , ?2334 using , ?2335 using , ?2388 using , ?2389 using , ?2449 using , ?2450 using , ?2461 using , ?2481 using , ?2482 using , ?2483 using , ?2495 using , ?2513 using , ?2534 using , ?2553 using , ?2554 using ,)





Qed.
Beta_env_sound_up is defined















Second Step: reduction in the term.





Lemma Beta_sound : forall Γ M T, Γ ⊢ M : T -> forall N, M → N ->  Γ ⊢ N : T.
1 subgoals, subgoal 1 (ID 2572)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T -> forall N : Term, M → N -> Γ ⊢ N : T

(dependent evars:)





induction 1; intros.
6 subgoals, subgoal 1 (ID 2667)
  
  Γ : Env
  s : X.Sorts
  t : X.Sorts
  H : Y.Ax s t
  H0 : Γ ⊣
  N : Term
  H1 : !s → N
  ============================
   Γ ⊢ N : !t

subgoal 2 (ID 2669) is:
 Γ ⊢ N : A
subgoal 3 (ID 2671) is:
 Γ ⊢ N : !u
subgoal 4 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 5 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars:)





inversion H1.
5 subgoals, subgoal 1 (ID 2669)
  
  Γ : Env
  A : Term
  v : nat
  H : Γ ⊣
  H0 : A ↓ v ⊂ Γ
  N : Term
  H1 : #v → N
  ============================
   Γ ⊢ N : A

subgoal 2 (ID 2671) is:
 Γ ⊢ N : !u
subgoal 3 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars:)





inversion H1.
4 subgoals, subgoal 1 (ID 2671)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  N : Term
  H2 : Π (A), B → N
  ============================
   Γ ⊢ N : !u

subgoal 2 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars:)





inversion H2; subst; clear H2. 
5 subgoals, subgoal 1 (ID 3098)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  B' : Term
  H6 : B → B'
  ============================
   Γ ⊢ Π (A), B' : !u

subgoal 2 (ID 3099) is:
 Γ ⊢ Π (A'), B : !u
subgoal 3 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars:)



econstructor; eauto. 
4 subgoals, subgoal 1 (ID 3099)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  A' : Term
  H6 : A → A'
  ============================
   Γ ⊢ Π (A'), B : !u

subgoal 2 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using ,)



econstructor. 
6 subgoals, subgoal 1 (ID 3125)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  A' : Term
  H6 : A → A'
  ============================
   Y.Rel ?3123 ?3124 u

subgoal 2 (ID 3126) is:
 Γ ⊢ A' : !?3123
subgoal 3 (ID 3127) is:
 A' :: Γ ⊢ B : !?3124
subgoal 4 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 5 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 open, ?3124 open,)



apply H. 
5 subgoals, subgoal 1 (ID 3126)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  A' : Term
  H6 : A → A'
  ============================
   Γ ⊢ A' : !s

subgoal 2 (ID 3127) is:
 A' :: Γ ⊢ B : !t
subgoal 3 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using ,)



intuition.
4 subgoals, subgoal 1 (ID 3127)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  A' : Term
  H6 : A → A'
  ============================
   A' :: Γ ⊢ B : !t

subgoal 2 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using ,)





eapply Beta_env_sound. 
6 subgoals, subgoal 1 (ID 3144)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  A' : Term
  H6 : A → A'
  ============================
   ?3143 ⊢ B : !t

subgoal 2 (ID 3145) is:
 A' :: Γ ⊣
subgoal 3 (ID 3146) is:
 ?3143 →e (A' :: Γ)
subgoal 4 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 5 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 open,)



apply H1. 
5 subgoals, subgoal 1 (ID 3145)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  A' : Term
  H6 : A → A'
  ============================
   A' :: Γ ⊣

subgoal 2 (ID 3146) is:
 (A :: Γ) →e (A' :: Γ)
subgoal 3 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using ,)



eauto. 
4 subgoals, subgoal 1 (ID 3146)
  
  Γ : Env
  A : Term
  B : Term
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H : Y.Rel s t u
  H0 : Γ ⊢ A : !s
  H1 : A :: Γ ⊢ B : !t
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !t
  A' : Term
  H6 : A → A'
  ============================
   (A :: Γ) →e (A' :: Γ)

subgoal 2 (ID 2673) is:
 Γ ⊢ N : Π (A), B
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using ,)



intuition.
3 subgoals, subgoal 1 (ID 2673)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  N : Term
  H3 : λ [A], M → N
  ============================
   Γ ⊢ N : Π (A), B

subgoal 2 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using ,)





inversion H3; subst; clear H3. 
4 subgoals, subgoal 1 (ID 3349)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  M' : Term
  H7 : M → M'
  ============================
   Γ ⊢ λ [A], M' : Π (A), B

subgoal 2 (ID 3350) is:
 Γ ⊢ λ [A'], M : Π (A), B
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using ,)



eauto.
3 subgoals, subgoal 1 (ID 3350)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Γ ⊢ λ [A'], M : Π (A), B

subgoal 2 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using ,)





apply Cnv with ( Π (A'), B) s3. 
5 subgoals, subgoal 1 (ID 5203)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Π (A'), B ≡ Π (A), B

subgoal 2 (ID 5204) is:
 Γ ⊢ λ [A'], M : Π (A'), B
subgoal 3 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using ,)



eauto.
4 subgoals, subgoal 1 (ID 5204)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Γ ⊢ λ [A'], M : Π (A'), B

subgoal 2 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using ,)





econstructor. 
7 subgoals, subgoal 1 (ID 5454)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Y.Rel ?5451 ?5452 ?5453

subgoal 2 (ID 5455) is:
 Γ ⊢ A' : !?5451
subgoal 3 (ID 5456) is:
 A' :: Γ ⊢ B : !?5452
subgoal 4 (ID 5457) is:
 A' :: Γ ⊢ M : B
subgoal 5 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 6 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 7 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 open, ?5452 open, ?5453 open,)



apply H. 
6 subgoals, subgoal 1 (ID 5455)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Γ ⊢ A' : !s1

subgoal 2 (ID 5456) is:
 A' :: Γ ⊢ B : !s2
subgoal 3 (ID 5457) is:
 A' :: Γ ⊢ M : B
subgoal 4 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 5 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using ,)



eauto.
5 subgoals, subgoal 1 (ID 5456)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   A' :: Γ ⊢ B : !s2

subgoal 2 (ID 5457) is:
 A' :: Γ ⊢ M : B
subgoal 3 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using ,)





eapply Beta_env_sound. 
7 subgoals, subgoal 1 (ID 5465)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   ?5464 ⊢ B : !s2

subgoal 2 (ID 5466) is:
 A' :: Γ ⊣
subgoal 3 (ID 5467) is:
 ?5464 →e (A' :: Γ)
subgoal 4 (ID 5457) is:
 A' :: Γ ⊢ M : B
subgoal 5 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 6 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 7 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 open,)



apply H1. 
6 subgoals, subgoal 1 (ID 5466)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   A' :: Γ ⊣

subgoal 2 (ID 5467) is:
 (A :: Γ) →e (A' :: Γ)
subgoal 3 (ID 5457) is:
 A' :: Γ ⊢ M : B
subgoal 4 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 5 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using ,)



econstructor. 
6 subgoals, subgoal 1 (ID 5471)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Γ ⊢ A' : !?5470

subgoal 2 (ID 5467) is:
 (A :: Γ) →e (A' :: Γ)
subgoal 3 (ID 5457) is:
 A' :: Γ ⊢ M : B
subgoal 4 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 5 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 open,)



eauto. 
5 subgoals, subgoal 1 (ID 5467)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   (A :: Γ) →e (A' :: Γ)

subgoal 2 (ID 5457) is:
 A' :: Γ ⊢ M : B
subgoal 3 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using ,)



intuition.
4 subgoals, subgoal 1 (ID 5457)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   A' :: Γ ⊢ M : B

subgoal 2 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using ,)





eapply Beta_env_sound. 
6 subgoals, subgoal 1 (ID 5497)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   ?5496 ⊢ M : B

subgoal 2 (ID 5498) is:
 A' :: Γ ⊣
subgoal 3 (ID 5499) is:
 ?5496 →e (A' :: Γ)
subgoal 4 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 5 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 open,)



apply H2. 
5 subgoals, subgoal 1 (ID 5498)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   A' :: Γ ⊣

subgoal 2 (ID 5499) is:
 (A :: Γ) →e (A' :: Γ)
subgoal 3 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using ,)



econstructor. 
5 subgoals, subgoal 1 (ID 5503)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Γ ⊢ A' : !?5502

subgoal 2 (ID 5499) is:
 (A :: Γ) →e (A' :: Γ)
subgoal 3 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 4 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 open,)



eauto. 
4 subgoals, subgoal 1 (ID 5499)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   (A :: Γ) →e (A' :: Γ)

subgoal 2 (ID 5205) is:
 Γ ⊢ Π (A), B : !s3
subgoal 3 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using ,)



intuition.
3 subgoals, subgoal 1 (ID 5205)
  
  Γ : Env
  A : Term
  B : Term
  M : Term
  s1 : X.Sorts
  s2 : X.Sorts
  s3 : X.Sorts
  H : Y.Rel s1 s2 s3
  H0 : Γ ⊢ A : !s1
  H1 : A :: Γ ⊢ B : !s2
  H2 : A :: Γ ⊢ M : B
  IHtyp1 : forall N : Term, A → N -> Γ ⊢ N : !s1
  IHtyp2 : forall N : Term, B → N -> A :: Γ ⊢ N : !s2
  IHtyp3 : forall N : Term, M → N -> A :: Γ ⊢ N : B
  A' : Term
  H7 : A → A'
  ============================
   Γ ⊢ Π (A), B : !s3

subgoal 2 (ID 2675) is:
 Γ ⊢ N0 : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using ,)





econstructor; eauto.
2 subgoals, subgoal 1 (ID 2675)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N0 : Term
  H1 : M · N → N0
  ============================
   Γ ⊢ N0 : B [ ← N]

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)





inversion H1; subst; clear H1.
4 subgoals, subgoal 1 (ID 5751)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : Γ ⊢ λ [A0], M0 : Π (A), B
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  ============================
   Γ ⊢ M0 [ ← N] : B [ ← N]

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)





assert (exists u, Γ ⊢ Π(A),B : !u). 
5 subgoals, subgoal 1 (ID 5755)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : Γ ⊢ λ [A0], M0 : Π (A), B
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5756) is:
 Γ ⊢ M0 [ ← N] : B [ ← N]
subgoal 3 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 4 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)



apply TypeCorrect in H.
5 subgoals, subgoal 1 (ID 5758)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : (exists s : X.Sorts, Π (A), B = !s) \/
      (exists s : X.Sorts, Γ ⊢ Π (A), B : !s)
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5756) is:
 Γ ⊢ M0 [ ← N] : B [ ← N]
subgoal 3 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 4 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)





destruct H. 
6 subgoals, subgoal 1 (ID 5764)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : exists s : X.Sorts, Π (A), B = !s
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5766) is:
 exists u : X.Sorts, Γ ⊢ Π (A), B : !u
subgoal 3 (ID 5756) is:
 Γ ⊢ M0 [ ← N] : B [ ← N]
subgoal 4 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 5 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)



destruct H; discriminate. 
5 subgoals, subgoal 1 (ID 5766)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : exists s : X.Sorts, Γ ⊢ Π (A), B : !s
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5756) is:
 Γ ⊢ M0 [ ← N] : B [ ← N]
subgoal 3 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 4 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)



trivial.
4 subgoals, subgoal 1 (ID 5756)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : Γ ⊢ λ [A0], M0 : Π (A), B
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  H1 : exists u : X.Sorts, Γ ⊢ Π (A), B : !u
  ============================
   Γ ⊢ M0 [ ← N] : B [ ← N]

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)





destruct H1. 
4 subgoals, subgoal 1 (ID 5781)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : Γ ⊢ λ [A0], M0 : Π (A), B
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  H1 : Γ ⊢ Π (A), B : !x
  ============================
   Γ ⊢ M0 [ ← N] : B [ ← N]

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)



apply gen_pi in H1 as (s & t & u & h); decompose [and] h; clear h.
4 subgoals, subgoal 1 (ID 5813)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  H : Γ ⊢ λ [A0], M0 : Π (A), B
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  ============================
   Γ ⊢ M0 [ ← N] : B [ ← N]

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)





apply gen_la in H  as (u1 & u2 & u3 & D & ? & ? & ? & ? & ?).
4 subgoals, subgoal 1 (ID 5847)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H : Π (A), B ≡ Π (A0), D
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  ============================
   Γ ⊢ M0 [ ← N] : B [ ← N]

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)





apply PiInj in H as (? & ?).
4 subgoals, subgoal 1 (ID 5853)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   Γ ⊢ M0 [ ← N] : B [ ← N]

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using ,)





eapply Cnv with (s := t). 
6 subgoals, subgoal 1 (ID 5855)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5854 ≡ B [ ← N]

subgoal 2 (ID 5856) is:
 Γ ⊢ M0 [ ← N] : ?5854
subgoal 3 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 4 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 5 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 open,)



eapply Betac_subst2. 
6 subgoals, subgoal 1 (ID 5859)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5858 ≡ B

subgoal 2 (ID 5856) is:
 Γ ⊢ M0 [ ← N] : ?5858 [ ← N]
subgoal 3 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 4 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 5 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 open,)



apply Betac_sym. 
6 subgoals, subgoal 1 (ID 5860)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   B ≡ ?5858

subgoal 2 (ID 5856) is:
 Γ ⊢ M0 [ ← N] : ?5858 [ ← N]
subgoal 3 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 4 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 5 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 open,)



apply H9.
5 subgoals, subgoal 1 (ID 5856)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   Γ ⊢ M0 [ ← N] : D [ ← N]

subgoal 2 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 3 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 4 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using ,)





eapply substitution. 
8 subgoals, subgoal 1 (ID 5873)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5872 ⊢ M0 : D

subgoal 2 (ID 5876) is:
 ?5874 ⊢ N : ?5875
subgoal 3 (ID 5877) is:
 sub_in_env ?5874 N ?5875 0 ?5872 Γ
subgoal 4 (ID 5878) is:
 ?5872 ⊣
subgoal 5 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 6 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 7 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 8 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 open, ?5874 open, ?5875 open,)



apply H7. 
7 subgoals, subgoal 1 (ID 5876)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5874 ⊢ N : ?5875

subgoal 2 (ID 5877) is:
 sub_in_env ?5874 N ?5875 0 (A0 :: Γ) Γ
subgoal 3 (ID 5878) is:
 A0 :: Γ ⊣
subgoal 4 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 5 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 6 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 7 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 open, ?5875 open,)



eapply Cnv. 
9 subgoals, subgoal 1 (ID 5881)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5879 ≡ ?5875

subgoal 2 (ID 5882) is:
 ?5874 ⊢ N : ?5879
subgoal 3 (ID 5883) is:
 ?5874 ⊢ ?5875 : !?5880
subgoal 4 (ID 5877) is:
 sub_in_env ?5874 N ?5875 0 (A0 :: Γ) Γ
subgoal 5 (ID 5878) is:
 A0 :: Γ ⊣
subgoal 6 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 7 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 8 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 9 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 open, ?5875 open, ?5879 open, ?5880 open,)



apply H. 
8 subgoals, subgoal 1 (ID 5882)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5874 ⊢ N : A

subgoal 2 (ID 5883) is:
 ?5874 ⊢ A0 : !?5880
subgoal 3 (ID 5877) is:
 sub_in_env ?5874 N A0 0 (A0 :: Γ) Γ
subgoal 4 (ID 5878) is:
 A0 :: Γ ⊣
subgoal 5 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 6 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 7 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 8 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 open, ?5875 using , ?5879 using , ?5880 open,)



apply H0. 
7 subgoals, subgoal 1 (ID 5883)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   Γ ⊢ A0 : !?5880

subgoal 2 (ID 5877) is:
 sub_in_env Γ N A0 0 (A0 :: Γ) Γ
subgoal 3 (ID 5878) is:
 A0 :: Γ ⊣
subgoal 4 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 5 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 6 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 7 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 open,)



apply H6. 
6 subgoals, subgoal 1 (ID 5877)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   sub_in_env Γ N A0 0 (A0 :: Γ) Γ

subgoal 2 (ID 5878) is:
 A0 :: Γ ⊣
subgoal 3 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 4 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 5 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using ,)



constructor.
5 subgoals, subgoal 1 (ID 5878)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   A0 :: Γ ⊣

subgoal 2 (ID 5857) is:
 Γ ⊢ B [ ← N] : !t
subgoal 3 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 4 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using ,)





eauto. 
4 subgoals, subgoal 1 (ID 5857)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   Γ ⊢ B [ ← N] : !t

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using ,)



change !t with (!t  [ ← N] ). 
4 subgoals, subgoal 1 (ID 5896)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   Γ ⊢ B [ ← N] : !t [ ← N]

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using ,)



eapply substitution. 
7 subgoals, subgoal 1 (ID 5909)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5908 ⊢ B : !t

subgoal 2 (ID 5912) is:
 ?5910 ⊢ N : ?5911
subgoal 3 (ID 5913) is:
 sub_in_env ?5910 N ?5911 0 ?5908 Γ
subgoal 4 (ID 5914) is:
 ?5908 ⊣
subgoal 5 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 6 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 7 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 open, ?5910 open, ?5911 open,)



apply H5. 
6 subgoals, subgoal 1 (ID 5912)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   ?5910 ⊢ N : ?5911

subgoal 2 (ID 5913) is:
 sub_in_env ?5910 N ?5911 0 (A :: Γ) Γ
subgoal 3 (ID 5914) is:
 A :: Γ ⊣
subgoal 4 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 5 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 6 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 open, ?5911 open,)



apply H0. 
5 subgoals, subgoal 1 (ID 5913)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   sub_in_env Γ N A 0 (A :: Γ) Γ

subgoal 2 (ID 5914) is:
 A :: Γ ⊣
subgoal 3 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 4 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using ,)



constructor.
4 subgoals, subgoal 1 (ID 5914)
  
  Γ : Env
  N : Term
  A : Term
  B : Term
  H0 : Γ ⊢ N : A
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  A0 : Term
  M0 : Term
  IHtyp1 : forall N : Term, λ [A0], M0 → N -> Γ ⊢ N : Π (A), B
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H5 : A :: Γ ⊢ B : !t
  u1 : X.Sorts
  u2 : X.Sorts
  u3 : X.Sorts
  D : Term
  H4 : Y.Rel u1 u2 u3
  H6 : Γ ⊢ A0 : !u1
  H7 : A0 :: Γ ⊢ M0 : D
  H8 : A0 :: Γ ⊢ D : !u2
  H : A ≡ A0
  H9 : B ≡ D
  ============================
   A :: Γ ⊣

subgoal 2 (ID 5752) is:
 Γ ⊢ M' · N : B [ ← N]
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using ,)





eauto.
3 subgoals, subgoal 1 (ID 5752)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  M' : Term
  H5 : M → M'
  ============================
   Γ ⊢ M' · N : B [ ← N]

subgoal 2 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using ,)








econstructor. 
4 subgoals, subgoal 1 (ID 5932)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  M' : Term
  H5 : M → M'
  ============================
   Γ ⊢ M' : Π (?5931), B

subgoal 2 (ID 5933) is:
 Γ ⊢ N : ?5931
subgoal 3 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 open,)



eauto. 
3 subgoals, subgoal 1 (ID 5933)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  M' : Term
  H5 : M → M'
  ============================
   Γ ⊢ N : A

subgoal 2 (ID 5753) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)



trivial.
2 subgoals, subgoal 1 (ID 5753)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  ============================
   Γ ⊢ M · N' : B [ ← N]

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)





assert (exists u, Γ ⊢ Π(A),B : !u). 
3 subgoals, subgoal 1 (ID 5941)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5942) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)



apply TypeCorrect in H.
3 subgoals, subgoal 1 (ID 5944)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : (exists s : X.Sorts, Π (A), B = !s) \/
      (exists s : X.Sorts, Γ ⊢ Π (A), B : !s)
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5942) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)





destruct H. 
4 subgoals, subgoal 1 (ID 5950)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : exists s : X.Sorts, Π (A), B = !s
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5952) is:
 exists u : X.Sorts, Γ ⊢ Π (A), B : !u
subgoal 3 (ID 5942) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)



destruct H; discriminate. 
3 subgoals, subgoal 1 (ID 5952)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : exists s : X.Sorts, Γ ⊢ Π (A), B : !s
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  ============================
   exists u : X.Sorts, Γ ⊢ Π (A), B : !u

subgoal 2 (ID 5942) is:
 Γ ⊢ M · N' : B [ ← N]
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)



trivial.
2 subgoals, subgoal 1 (ID 5942)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  H1 : exists u : X.Sorts, Γ ⊢ Π (A), B : !u
  ============================
   Γ ⊢ M · N' : B [ ← N]

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)





destruct H1. 
2 subgoals, subgoal 1 (ID 5967)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  H1 : Γ ⊢ Π (A), B : !x
  ============================
   Γ ⊢ M · N' : B [ ← N]

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)



apply gen_pi in H1 as (s & t & u & h); decompose [and] h; clear h.
2 subgoals, subgoal 1 (ID 5999)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   Γ ⊢ M · N' : B [ ← N]

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)





apply Cnv with (B [← N']) t. 
4 subgoals, subgoal 1 (ID 6000)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   B [ ← N'] ≡ B [ ← N]

subgoal 2 (ID 6001) is:
 Γ ⊢ M · N' : B [ ← N']
subgoal 3 (ID 6002) is:
 Γ ⊢ B [ ← N] : !t
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)



apply Betac_subst. 
4 subgoals, subgoal 1 (ID 6003)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   N' ≡ N

subgoal 2 (ID 6001) is:
 Γ ⊢ M · N' : B [ ← N']
subgoal 3 (ID 6002) is:
 Γ ⊢ B [ ← N] : !t
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)



auto.
3 subgoals, subgoal 1 (ID 6001)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   Γ ⊢ M · N' : B [ ← N']

subgoal 2 (ID 6002) is:
 Γ ⊢ B [ ← N] : !t
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using ,)





econstructor. 
4 subgoals, subgoal 1 (ID 6022)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   Γ ⊢ M : Π (?6021), B

subgoal 2 (ID 6023) is:
 Γ ⊢ N' : ?6021
subgoal 3 (ID 6002) is:
 Γ ⊢ B [ ← N] : !t
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 open,)



apply H. 
3 subgoals, subgoal 1 (ID 6023)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   Γ ⊢ N' : A

subgoal 2 (ID 6002) is:
 Γ ⊢ B [ ← N] : !t
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using ,)



eauto.
2 subgoals, subgoal 1 (ID 6002)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   Γ ⊢ B [ ← N] : !t

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using ,)





change !t with (!t  [ ← N] ). 
2 subgoals, subgoal 1 (ID 6031)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   Γ ⊢ B [ ← N] : !t [ ← N]

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using ,)



eapply substitution. 
5 subgoals, subgoal 1 (ID 6044)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   ?6043 ⊢ B : !t

subgoal 2 (ID 6047) is:
 ?6045 ⊢ N : ?6046
subgoal 3 (ID 6048) is:
 sub_in_env ?6045 N ?6046 0 ?6043 Γ
subgoal 4 (ID 6049) is:
 ?6043 ⊣
subgoal 5 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using , ?6036 using ?6043 , ?6037 using ?6045 , ?6038 using ?6046 , ?6043 open, ?6045 open, ?6046 open,)



apply H6. 
4 subgoals, subgoal 1 (ID 6047)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   ?6045 ⊢ N : ?6046

subgoal 2 (ID 6048) is:
 sub_in_env ?6045 N ?6046 0 (A :: Γ) Γ
subgoal 3 (ID 6049) is:
 A :: Γ ⊣
subgoal 4 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using , ?6036 using ?6043 , ?6037 using ?6045 , ?6038 using ?6046 , ?6043 using , ?6045 open, ?6046 open,)



apply H0. 
3 subgoals, subgoal 1 (ID 6048)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   sub_in_env Γ N A 0 (A :: Γ) Γ

subgoal 2 (ID 6049) is:
 A :: Γ ⊣
subgoal 3 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using , ?6036 using ?6043 , ?6037 using ?6045 , ?6038 using ?6046 , ?6043 using , ?6045 using , ?6046 using ,)



constructor.
2 subgoals, subgoal 1 (ID 6049)
  
  Γ : Env
  M : Term
  N : Term
  A : Term
  B : Term
  H : Γ ⊢ M : Π (A), B
  H0 : Γ ⊢ N : A
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : Π (A), B
  IHtyp2 : forall N0 : Term, N → N0 -> Γ ⊢ N0 : A
  N' : Term
  H5 : N → N'
  x : X.Sorts
  s : X.Sorts
  t : X.Sorts
  u : X.Sorts
  H1 : !x ≡ !u
  H3 : Y.Rel s t u
  H2 : Γ ⊢ A : !s
  H6 : A :: Γ ⊢ B : !t
  ============================
   A :: Γ ⊣

subgoal 2 (ID 2677) is:
 Γ ⊢ N : B
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using , ?6036 using ?6043 , ?6037 using ?6045 , ?6038 using ?6046 , ?6043 using , ?6045 using , ?6046 using ,)





eauto.
1 subgoals, subgoal 1 (ID 2677)
  
  Γ : Env
  M : Term
  A : Term
  B : Term
  s : X.Sorts
  H : A ≡ B
  H0 : Γ ⊢ M : A
  H1 : Γ ⊢ B : !s
  IHtyp1 : forall N : Term, M → N -> Γ ⊢ N : A
  IHtyp2 : forall N : Term, B → N -> Γ ⊢ N : !s
  N : Term
  H2 : M → N
  ============================
   Γ ⊢ N : B

(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using , ?6036 using ?6043 , ?6037 using ?6045 , ?6038 using ?6046 , ?6043 using , ?6045 using , ?6046 using , ?6051 using ,)





eauto.
No more subgoals.
(dependent evars: ?3103 using , ?3104 using , ?3123 using , ?3124 using , ?3143 using , ?3147 using , ?3351 using , ?3352 using , ?3377 using , ?3378 using , ?3379 using , ?5170 using , ?5171 using , ?5451 using , ?5452 using , ?5453 using , ?5464 using , ?5470 using , ?5496 using , ?5502 using , ?5531 using , ?5532 using , ?5854 using ?5858 , ?5858 using , ?5865 using ?5872 , ?5866 using ?5874 , ?5867 using ?5875 , ?5872 using , ?5874 using , ?5875 using , ?5879 using , ?5880 using , ?5885 using , ?5901 using ?5908 , ?5902 using ?5910 , ?5903 using ?5911 , ?5908 using , ?5910 using , ?5911 using , ?5916 using , ?5931 using , ?6021 using , ?6036 using ?6043 , ?6037 using ?6045 , ?6038 using ?6046 , ?6043 using , ?6045 using , ?6046 using , ?6051 using , ?6061 using , ?6062 using ,)





Qed.
Beta_sound is defined















With both lemmas, we can now prove the full SR.





Lemma SubjectRed : forall Γ M T, Γ ⊢ M : T-> 

  forall N , M →→ N -> Γ ⊢ N : T.
1 subgoals, subgoal 1 (ID 6087)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T -> forall N : Term, M →→ N -> Γ ⊢ N : T

(dependent evars:)





intros. 
1 subgoals, subgoal 1 (ID 6093)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  N : Term
  H0 : M →→ N
  ============================
   Γ ⊢ N : T

(dependent evars:)



induction H0.
3 subgoals, subgoal 1 (ID 6110)
  
  Γ : Env
  T : Term
  M : Term
  H : Γ ⊢ M : T
  ============================
   Γ ⊢ M : T

subgoal 2 (ID 6115) is:
 Γ ⊢ N : T
subgoal 3 (ID 6125) is:
 Γ ⊢ P : T
(dependent evars:)





trivial. 
2 subgoals, subgoal 1 (ID 6115)
  
  Γ : Env
  T : Term
  M : Term
  H : Γ ⊢ M : T
  N : Term
  H0 : M → N
  ============================
   Γ ⊢ N : T

subgoal 2 (ID 6125) is:
 Γ ⊢ P : T
(dependent evars:)



eapply Beta_sound.
3 subgoals, subgoal 1 (ID 6127)
  
  Γ : Env
  T : Term
  M : Term
  H : Γ ⊢ M : T
  N : Term
  H0 : M → N
  ============================
   Γ ⊢ ?6126 : T

subgoal 2 (ID 6128) is:
 ?6126 → N
subgoal 3 (ID 6125) is:
 Γ ⊢ P : T
(dependent evars: ?6126 open,)





apply H. 
2 subgoals, subgoal 1 (ID 6128)
  
  Γ : Env
  T : Term
  M : Term
  H : Γ ⊢ M : T
  N : Term
  H0 : M → N
  ============================
   M → N

subgoal 2 (ID 6125) is:
 Γ ⊢ P : T
(dependent evars: ?6126 using ,)



trivial. 
1 subgoals, subgoal 1 (ID 6125)
  
  Γ : Env
  T : Term
  M : Term
  H : Γ ⊢ M : T
  N : Term
  P : Term
  H0_ : M →→ N
  H0_0 : N →→ P
  IHBetas1 : Γ ⊢ M : T -> Γ ⊢ N : T
  IHBetas2 : Γ ⊢ N : T -> Γ ⊢ P : T
  ============================
   Γ ⊢ P : T

(dependent evars: ?6126 using ,)



intuition.
No more subgoals.
(dependent evars: ?6126 using ,)





Qed.
SubjectRed is defined















Reduction in the type is valid.





Lemma Betas_typ_sound : forall Γ M T, Γ ⊢ M : T -> 

 forall T', T →→ T' -> Γ ⊢ M : T'.
1 subgoals, subgoal 1 (ID 6139)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T -> forall T' : Term, T →→ T' -> Γ ⊢ M : T'

(dependent evars:)





intros.
1 subgoals, subgoal 1 (ID 6145)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  T' : Term
  H0 : T →→ T'
  ============================
   Γ ⊢ M : T'

(dependent evars:)





assert ((exists s, T = !s )\/ exists s,  Γ ⊢ T : !s ) by ( apply TypeCorrect in H; intuition).
1 subgoals, subgoal 1 (ID 6150)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  T' : Term
  H0 : T →→ T'
  H1 : (exists s : X.Sorts, T = !s) \/ (exists s : X.Sorts, Γ ⊢ T : !s)
  ============================
   Γ ⊢ M : T'

(dependent evars:)





destruct H1 as [[s Hs]|[s Hs]].
2 subgoals, subgoal 1 (ID 6180)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  T' : Term
  H0 : T →→ T'
  s : X.Sorts
  Hs : T = !s
  ============================
   Γ ⊢ M : T'

subgoal 2 (ID 6185) is:
 Γ ⊢ M : T'
(dependent evars:)





rewrite Hs in *. 
2 subgoals, subgoal 1 (ID 6189)
  
  Γ : Env
  M : Term
  T : Term
  s : X.Sorts
  H : Γ ⊢ M : !s
  T' : Term
  H0 : !s →→ T'
  Hs : T = !s
  ============================
   Γ ⊢ M : T'

subgoal 2 (ID 6185) is:
 Γ ⊢ M : T'
(dependent evars:)



apply Betas_S in H0. 
2 subgoals, subgoal 1 (ID 6191)
  
  Γ : Env
  M : Term
  T : Term
  s : X.Sorts
  H : Γ ⊢ M : !s
  T' : Term
  H0 : T' = !s
  Hs : T = !s
  ============================
   Γ ⊢ M : T'

subgoal 2 (ID 6185) is:
 Γ ⊢ M : T'
(dependent evars:)



rewrite H0; trivial.
1 subgoals, subgoal 1 (ID 6185)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  T' : Term
  H0 : T →→ T'
  s : X.Sorts
  Hs : Γ ⊢ T : !s
  ============================
   Γ ⊢ M : T'

(dependent evars:)





apply Cnv with T s; intuition.
1 subgoals, subgoal 1 (ID 6195)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  T' : Term
  H0 : T →→ T'
  s : X.Sorts
  Hs : Γ ⊢ T : !s
  ============================
   Γ ⊢ T' : !s

(dependent evars:)





apply SubjectRed with T; intuition.
No more subgoals.
(dependent evars:)





Qed.
Betas_typ_sound is defined















Wellformation of a context after reduction.





Lemma Beta_env_sound2 : forall Γ Γ', Γ ⊣ -> Γ →e Γ' -> Γ' ⊣.
1 subgoals, subgoal 1 (ID 6231)
  
  ============================
   forall Γ Γ' : Env, Γ ⊣ -> Γ →e Γ' -> Γ' ⊣

(dependent evars:)





intros. 
1 subgoals, subgoal 1 (ID 6235)
  
  Γ : Env
  Γ' : Env
  H : Γ ⊣
  H0 : Γ →e Γ'
  ============================
   Γ' ⊣

(dependent evars:)



induction H0; intuition.
2 subgoals, subgoal 1 (ID 6251)
  
  A : Term
  Γ : list Term
  H : A :: Γ ⊣
  B : Term
  H0 : A → B
  ============================
   B :: Γ ⊣

subgoal 2 (ID 6258) is:
 A :: Γ' ⊣
(dependent evars:)





inversion H; subst; clear H.
2 subgoals, subgoal 1 (ID 6320)
  
  A : Term
  Γ : list Term
  B : Term
  H0 : A → B
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   B :: Γ ⊣

subgoal 2 (ID 6258) is:
 A :: Γ' ⊣
(dependent evars:)





econstructor. 
2 subgoals, subgoal 1 (ID 6324)
  
  A : Term
  Γ : list Term
  B : Term
  H0 : A → B
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   Γ ⊢ B : !?6323

subgoal 2 (ID 6258) is:
 A :: Γ' ⊣
(dependent evars: ?6323 open,)



eapply Beta_sound. 
3 subgoals, subgoal 1 (ID 6326)
  
  A : Term
  Γ : list Term
  B : Term
  H0 : A → B
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   Γ ⊢ ?6325 : !?6323

subgoal 2 (ID 6327) is:
 ?6325 → B
subgoal 3 (ID 6258) is:
 A :: Γ' ⊣
(dependent evars: ?6323 open, ?6325 open,)



apply H2. 
2 subgoals, subgoal 1 (ID 6327)
  
  A : Term
  Γ : list Term
  B : Term
  H0 : A → B
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   A → B

subgoal 2 (ID 6258) is:
 A :: Γ' ⊣
(dependent evars: ?6323 using , ?6325 using ,)



trivial.
1 subgoals, subgoal 1 (ID 6258)
  
  A : Term
  Γ : Env
  H : A :: Γ ⊣
  Γ' : Env
  H0 : Γ →e Γ'
  IHBeta_env : Γ ⊣ -> Γ' ⊣
  ============================
   A :: Γ' ⊣

(dependent evars: ?6323 using , ?6325 using ,)





inversion H; subst; clear H.
1 subgoals, subgoal 1 (ID 6376)
  
  A : Term
  Γ : Env
  Γ' : Env
  H0 : Γ →e Γ'
  IHBeta_env : Γ ⊣ -> Γ' ⊣
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   A :: Γ' ⊣

(dependent evars: ?6323 using , ?6325 using ,)





econstructor. 
1 subgoals, subgoal 1 (ID 6380)
  
  A : Term
  Γ : Env
  Γ' : Env
  H0 : Γ →e Γ'
  IHBeta_env : Γ ⊣ -> Γ' ⊣
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   Γ' ⊢ A : !?6379

(dependent evars: ?6323 using , ?6325 using , ?6379 open,)



eapply Beta_env_sound. 
3 subgoals, subgoal 1 (ID 6382)
  
  A : Term
  Γ : Env
  Γ' : Env
  H0 : Γ →e Γ'
  IHBeta_env : Γ ⊣ -> Γ' ⊣
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   ?6381 ⊢ A : !?6379

subgoal 2 (ID 6383) is:
 Γ' ⊣
subgoal 3 (ID 6384) is:
 ?6381 →e Γ'
(dependent evars: ?6323 using , ?6325 using , ?6379 open, ?6381 open,)



apply H2. 
2 subgoals, subgoal 1 (ID 6383)
  
  A : Term
  Γ : Env
  Γ' : Env
  H0 : Γ →e Γ'
  IHBeta_env : Γ ⊣ -> Γ' ⊣
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   Γ' ⊣

subgoal 2 (ID 6384) is:
 Γ →e Γ'
(dependent evars: ?6323 using , ?6325 using , ?6379 using , ?6381 using ,)



apply IHBeta_env.
2 subgoals, subgoal 1 (ID 6385)
  
  A : Term
  Γ : Env
  Γ' : Env
  H0 : Γ →e Γ'
  IHBeta_env : Γ ⊣ -> Γ' ⊣
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   Γ ⊣

subgoal 2 (ID 6384) is:
 Γ →e Γ'
(dependent evars: ?6323 using , ?6325 using , ?6379 using , ?6381 using ,)





eauto. 
1 subgoals, subgoal 1 (ID 6384)
  
  A : Term
  Γ : Env
  Γ' : Env
  H0 : Γ →e Γ'
  IHBeta_env : Γ ⊣ -> Γ' ⊣
  s : X.Sorts
  H2 : Γ ⊢ A : !s
  ============================
   Γ →e Γ'

(dependent evars: ?6323 using , ?6325 using , ?6379 using , ?6381 using , ?6386 using , ?6387 using ,)



trivial.
No more subgoals.
(dependent evars: ?6323 using , ?6325 using , ?6379 using , ?6381 using , ?6386 using , ?6387 using ,)





Qed.
Beta_env_sound2 is defined








Lemma Betas_env_sound2 : forall Γ Γ', Γ ⊣ -> Γ →→e Γ' -> Γ' ⊣.
1 subgoals, subgoal 1 (ID 6442)
  
  ============================
   forall Γ Γ' : Env, Γ ⊣ -> Γ →→e Γ' -> Γ' ⊣

(dependent evars:)





intros.
1 subgoals, subgoal 1 (ID 6446)
  
  Γ : Env
  Γ' : Env
  H : Γ ⊣
  H0 : Γ →→e Γ'
  ============================
   Γ' ⊣

(dependent evars:)





induction H0; intuition.
1 subgoals, subgoal 1 (ID 6468)
  
  Γ : Env
  H : Γ ⊣
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ' ⊣

(dependent evars:)





eapply Beta_env_sound2. 
2 subgoals, subgoal 1 (ID 6491)
  
  Γ : Env
  H : Γ ⊣
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   ?6490 ⊣

subgoal 2 (ID 6492) is:
 ?6490 →e Γ'
(dependent evars: ?6490 open,)



apply H. 
1 subgoals, subgoal 1 (ID 6492)
  
  Γ : Env
  H : Γ ⊣
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ →e Γ'

(dependent evars: ?6490 using ,)



trivial.
No more subgoals.
(dependent evars: ?6490 using ,)





Qed.
Betas_env_sound2 is defined








Lemma Betas_env_sound :   forall Γ M T, Γ ⊢ M : T-> forall Γ', Γ →→e Γ' -> Γ' ⊢ M : T.
1 subgoals, subgoal 1 (ID 6497)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T -> forall Γ' : Env, Γ →→e Γ' -> Γ' ⊢ M : T

(dependent evars:)





intros.
1 subgoals, subgoal 1 (ID 6503)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ →→e Γ'
  ============================
   Γ' ⊢ M : T

(dependent evars:)





induction H0; intuition.
1 subgoals, subgoal 1 (ID 6525)
  
  M : Term
  T : Term
  Γ : Env
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ' ⊢ M : T

(dependent evars:)





eapply Beta_env_sound. 
3 subgoals, subgoal 1 (ID 6550)
  
  M : Term
  T : Term
  Γ : Env
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   ?6549 ⊢ M : T

subgoal 2 (ID 6551) is:
 Γ' ⊣
subgoal 3 (ID 6552) is:
 ?6549 →e Γ'
(dependent evars: ?6549 open,)



apply H. 
2 subgoals, subgoal 1 (ID 6551)
  
  M : Term
  T : Term
  Γ : Env
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ' ⊣

subgoal 2 (ID 6552) is:
 Γ →e Γ'
(dependent evars: ?6549 using ,)



apply Beta_env_sound2 with Γ. 
3 subgoals, subgoal 1 (ID 6553)
  
  M : Term
  T : Term
  Γ : Env
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ ⊣

subgoal 2 (ID 6554) is:
 Γ →e Γ'
subgoal 3 (ID 6552) is:
 Γ →e Γ'
(dependent evars: ?6549 using ,)



apply wf_typ in H. 
3 subgoals, subgoal 1 (ID 6556)
  
  M : Term
  T : Term
  Γ : Env
  H : Γ ⊣
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ ⊣

subgoal 2 (ID 6554) is:
 Γ →e Γ'
subgoal 3 (ID 6552) is:
 Γ →e Γ'
(dependent evars: ?6549 using ,)



trivial.
2 subgoals, subgoal 1 (ID 6554)
  
  M : Term
  T : Term
  Γ : Env
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ →e Γ'

subgoal 2 (ID 6552) is:
 Γ →e Γ'
(dependent evars: ?6549 using ,)





trivial. 
1 subgoals, subgoal 1 (ID 6552)
  
  M : Term
  T : Term
  Γ : Env
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ →e Γ'
  ============================
   Γ →e Γ'

(dependent evars: ?6549 using ,)



trivial.
No more subgoals.
(dependent evars: ?6549 using ,)





Qed.
Betas_env_sound is defined








Lemma Betas_env_sound_up :   forall Γ M T, Γ ⊢ M : T -> forall Γ', Γ' ⊣ -> Γ' →→e Γ -> Γ' ⊢ M : T.
1 subgoals, subgoal 1 (ID 6561)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T -> forall Γ' : Env, Γ' ⊣ -> Γ' →→e Γ -> Γ' ⊢ M : T

(dependent evars:)





intros. 
1 subgoals, subgoal 1 (ID 6568)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  Γ' : Env
  H0 : Γ' ⊣
  H1 : Γ' →→e Γ
  ============================
   Γ' ⊢ M : T

(dependent evars:)



revert M T H H0 .
1 subgoals, subgoal 1 (ID 6570)
  
  Γ : Env
  Γ' : Env
  H1 : Γ' →→e Γ
  ============================
   forall M T : Term, Γ ⊢ M : T -> Γ' ⊣ -> Γ' ⊢ M : T

(dependent evars:)





induction H1; intros.
3 subgoals, subgoal 1 (ID 6598)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  H0 : Γ ⊣
  ============================
   Γ ⊢ M : T

subgoal 2 (ID 6602) is:
 Γ ⊢ M : T
subgoal 3 (ID 6606) is:
 Γ ⊢ M : T
(dependent evars:)





trivial.
2 subgoals, subgoal 1 (ID 6602)
  
  Γ : Env
  Γ' : Env
  H : Γ →e Γ'
  M : Term
  T : Term
  H0 : Γ' ⊢ M : T
  H1 : Γ ⊣
  ============================
   Γ ⊢ M : T

subgoal 2 (ID 6606) is:
 Γ ⊢ M : T
(dependent evars:)





eapply Beta_env_sound_up. 
4 subgoals, subgoal 1 (ID 6608)
  
  Γ : Env
  Γ' : Env
  H : Γ →e Γ'
  M : Term
  T : Term
  H0 : Γ' ⊢ M : T
  H1 : Γ ⊣
  ============================
   ?6607 ⊢ M : T

subgoal 2 (ID 6609) is:
 Γ ⊣
subgoal 3 (ID 6610) is:
 Γ →e ?6607
subgoal 4 (ID 6606) is:
 Γ ⊢ M : T
(dependent evars: ?6607 open,)



apply H0. 
3 subgoals, subgoal 1 (ID 6609)
  
  Γ : Env
  Γ' : Env
  H : Γ →e Γ'
  M : Term
  T : Term
  H0 : Γ' ⊢ M : T
  H1 : Γ ⊣
  ============================
   Γ ⊣

subgoal 2 (ID 6610) is:
 Γ →e Γ'
subgoal 3 (ID 6606) is:
 Γ ⊢ M : T
(dependent evars: ?6607 using ,)



trivial. 
2 subgoals, subgoal 1 (ID 6610)
  
  Γ : Env
  Γ' : Env
  H : Γ →e Γ'
  M : Term
  T : Term
  H0 : Γ' ⊢ M : T
  H1 : Γ ⊣
  ============================
   Γ →e Γ'

subgoal 2 (ID 6606) is:
 Γ ⊢ M : T
(dependent evars: ?6607 using ,)



trivial.
1 subgoals, subgoal 1 (ID 6606)
  
  Γ : Env
  Γ' : Env
  Γ'' : Env
  H1_ : Γ →→e Γ'
  H1_0 : Γ' →→e Γ''
  IHBetas_env1 : forall M T : Term, Γ' ⊢ M : T -> Γ ⊣ -> Γ ⊢ M : T
  IHBetas_env2 : forall M T : Term, Γ'' ⊢ M : T -> Γ' ⊣ -> Γ' ⊢ M : T
  M : Term
  T : Term
  H : Γ'' ⊢ M : T
  H0 : Γ ⊣
  ============================
   Γ ⊢ M : T

(dependent evars: ?6607 using ,)





apply IHBetas_env1. 
2 subgoals, subgoal 1 (ID 6611)
  
  Γ : Env
  Γ' : Env
  Γ'' : Env
  H1_ : Γ →→e Γ'
  H1_0 : Γ' →→e Γ''
  IHBetas_env1 : forall M T : Term, Γ' ⊢ M : T -> Γ ⊣ -> Γ ⊢ M : T
  IHBetas_env2 : forall M T : Term, Γ'' ⊢ M : T -> Γ' ⊣ -> Γ' ⊢ M : T
  M : Term
  T : Term
  H : Γ'' ⊢ M : T
  H0 : Γ ⊣
  ============================
   Γ' ⊢ M : T

subgoal 2 (ID 6612) is:
 Γ ⊣
(dependent evars: ?6607 using ,)



apply IHBetas_env2. 
3 subgoals, subgoal 1 (ID 6613)
  
  Γ : Env
  Γ' : Env
  Γ'' : Env
  H1_ : Γ →→e Γ'
  H1_0 : Γ' →→e Γ''
  IHBetas_env1 : forall M T : Term, Γ' ⊢ M : T -> Γ ⊣ -> Γ ⊢ M : T
  IHBetas_env2 : forall M T : Term, Γ'' ⊢ M : T -> Γ' ⊣ -> Γ' ⊢ M : T
  M : Term
  T : Term
  H : Γ'' ⊢ M : T
  H0 : Γ ⊣
  ============================
   Γ'' ⊢ M : T

subgoal 2 (ID 6614) is:
 Γ' ⊣
subgoal 3 (ID 6612) is:
 Γ ⊣
(dependent evars: ?6607 using ,)



trivial. 
2 subgoals, subgoal 1 (ID 6614)
  
  Γ : Env
  Γ' : Env
  Γ'' : Env
  H1_ : Γ →→e Γ'
  H1_0 : Γ' →→e Γ''
  IHBetas_env1 : forall M T : Term, Γ' ⊢ M : T -> Γ ⊣ -> Γ ⊢ M : T
  IHBetas_env2 : forall M T : Term, Γ'' ⊢ M : T -> Γ' ⊣ -> Γ' ⊢ M : T
  M : Term
  T : Term
  H : Γ'' ⊢ M : T
  H0 : Γ ⊣
  ============================
   Γ' ⊣

subgoal 2 (ID 6612) is:
 Γ ⊣
(dependent evars: ?6607 using ,)



apply Betas_env_sound2 with Γ; trivial.
1 subgoals, subgoal 1 (ID 6612)
  
  Γ : Env
  Γ' : Env
  Γ'' : Env
  H1_ : Γ →→e Γ'
  H1_0 : Γ' →→e Γ''
  IHBetas_env1 : forall M T : Term, Γ' ⊢ M : T -> Γ ⊣ -> Γ ⊢ M : T
  IHBetas_env2 : forall M T : Term, Γ'' ⊢ M : T -> Γ' ⊣ -> Γ' ⊢ M : T
  M : Term
  T : Term
  H : Γ'' ⊢ M : T
  H0 : Γ ⊣
  ============================
   Γ ⊣

(dependent evars: ?6607 using ,)





trivial.
No more subgoals.
(dependent evars: ?6607 using ,)





Qed.
Betas_env_sound_up is defined















In its complete form: SubjectReduction. 





Lemma Subject_Reduction :  forall Γ M T, Γ ⊢ M : T ->

  forall Γ' N T', M →→ N -> Γ →→e Γ' ->  T →→ T' -> Γ' ⊢ N : T'.
1 subgoals, subgoal 1 (ID 6623)
  
  ============================
   forall (Γ : Env) (M T : Term),
   Γ ⊢ M : T ->
   forall (Γ' : Env) (N T' : Term),
   M →→ N -> Γ →→e Γ' -> T →→ T' -> Γ' ⊢ N : T'

(dependent evars:)





intros.
1 subgoals, subgoal 1 (ID 6633)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  Γ' : Env
  N : Term
  T' : Term
  H0 : M →→ N
  H1 : Γ →→e Γ'
  H2 : T →→ T'
  ============================
   Γ' ⊢ N : T'

(dependent evars:)





apply Betas_typ_sound with T; intuition.
1 subgoals, subgoal 1 (ID 6634)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  Γ' : Env
  N : Term
  T' : Term
  H0 : M →→ N
  H1 : Γ →→e Γ'
  H2 : T →→ T'
  ============================
   Γ' ⊢ N : T

(dependent evars:)





apply Betas_env_sound with Γ; intuition.
1 subgoals, subgoal 1 (ID 6647)
  
  Γ : Env
  M : Term
  T : Term
  H : Γ ⊢ M : T
  Γ' : Env
  N : Term
  T' : Term
  H0 : M →→ N
  H1 : Γ →→e Γ'
  H2 : T →→ T'
  ============================
   Γ ⊢ N : T

(dependent evars:)





apply SubjectRed with M; intuition.
No more subgoals.
(dependent evars:)





Qed.
Subject_Reduction is defined








End ut_sr_mod.
Module Type ut_sr_mod is defined



















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