--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/T/defs.ma".
+
+theorem terms_props__bind_dec:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq
+B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b:
+B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl
+(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B
+Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P:
+Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def
+(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow
+False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B
+Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda
+(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda
+(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall
+(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall
+(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def
+(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
+False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B
+Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B
+Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
+Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P
+H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B
+Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void
+Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P:
+Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
+Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B
+Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
+Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee:
+B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
+Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P
+H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P:
+Prop).P)) (refl_equal B Void)) b2)) b1).
+
+theorem bind_dec_not:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
+\def
+ \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2)
+in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P:
+Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
+b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
+(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1
+b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
+
+theorem terms_props__flat_dec:
+ \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq
+F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f:
+F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl
+(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F
+Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P:
+Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def
+(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop)
+with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in
+(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F
+Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast
+Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast
+Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match
+ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast
+\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast
+Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2))
+f1).
+
+theorem terms_props__kind_dec:
+ \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq
+K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind
+(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P:
+Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in
+(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P:
+Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to
+(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1:
+B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P:
+Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to
+(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B
+b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq
+K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b)
+(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e:
+K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
+(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B
+b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in
+(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b)
+(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H:
+(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b)
+(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind
+P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or
+(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda
+(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to
+(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P:
+Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def
+(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F
+f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat
+f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind
+F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1))
+\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat
+f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0))
+(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K
+(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P))
+(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def
+(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with
+[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1)
+in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall
+(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1).
+
+theorem term_dec:
+ \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall
+(P: Prop).P))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq
+T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2:
+T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to
+(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in
+(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
+Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to
+(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda
+(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to
+(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort
+n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0))
+(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T
+(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P))
+(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _)
+\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0
+(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in
+(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T
+(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P))
+(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0))))))
+(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T
+(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or
+(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P:
+Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n)
+(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n)
+(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n:
+nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T
+(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T
+(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P))
+(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0))))))
+(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind
+(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n)
+(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda
+(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n)
+(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P))))
+(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to
+(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq
+nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0))
+((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T
+(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n]))
+(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1:
+nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal
+nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T
+(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0:
+T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P:
+Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n)
+(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n)
+(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t
+t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall
+(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P:
+Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t
+t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort
+n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort
+n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T
+(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P:
+Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P:
+Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in
+(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq
+T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P:
+Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq
+T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in
+(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P:
+Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0)
+(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let
+H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
+(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t
+(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t
+t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in
+(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P:
+Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0)
+(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let
+H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
+(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0
+(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t
+t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def
+(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq
+K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0))
+((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda
+(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0)
+(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P:
+Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t
+t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0)))
+k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror
+(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0))
+\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t
+t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
+t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to
+(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9)))
+t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror
+(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4))
+\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t
+t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
+t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in
+(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq
+T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r
+T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5)
+\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P)))))
+H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P:
+Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k
+t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead
+k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
+(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t
+| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0)
+(THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3
+t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def
+(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k
+t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r
+T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in
+(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5)
+((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12
+(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
+
+theorem binder_dec:
+ \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
+T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall
+(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b:
+B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w
+u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind
+b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n)
+(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
+(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w
+u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n:
+nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
+T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P:
+Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq
+T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T
+(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0)))))))))
+(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
+u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead
+(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3
+B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind
+b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
+(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda
+(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b)
+w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0
+t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b:
+B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda
+(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0:
+B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to
+(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T
+(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w
+u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
+(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T
+(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1)
+(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u:
+T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P:
+Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u:
+T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal
+T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda
+(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0
+(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u:
+T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda
+(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda
+(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P:
+Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w:
+T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))))))
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1)
+(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
+(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead
+(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0
+t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
+\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1)
+in (False_ind P H2))))))))))))) k)) t).
+
+theorem abst_dec:
+ \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead
+(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to
+(\forall (P: Prop).P)))))
+\def
+ \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda
+(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead
+(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda
+(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v
+t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall
+(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind
+Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n:
+nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead
+(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t))
+\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n)
+(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0:
+T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind
+Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_:
+((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v
+t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P:
+Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind
+Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind
+Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t
+t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead
+(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind
+Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1:
+T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T
+(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let
+H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t
+v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind
+Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind
+Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda
+(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq
+T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq
+T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P:
+Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0)
+(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0)
+(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda
+(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0
+(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v)
+\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead
+(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead
+(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))
+(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind
+Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t |
+(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst)
+t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind
+Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8
+P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall
+(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead
+(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind
+Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T
+(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
+\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _)
+\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
+\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_:
+(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4)))))))
+H1))))))))) u).
+
projects / helm.git / blobdiff