(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/spare".
+include "LambdaDelta-1/theory.ma".
-include "theory.ma".
-
-definition nfs2:
- C \to (TList \to Prop)
-\def
- let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil
-\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))])
-in nfs2.
-
-theorem nf2_gen_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
-(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
-((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2)
-\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P:
-Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind
-Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead
-(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in
-(False_ind P H0))))))).
-
-theorem nf2_gen__aux:
- \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
-(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
+theorem lifts_inj:
+ \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
+nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
\def
- \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u:
-T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
-(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
-nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort
-n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
-d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n:
-nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u
-(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind
-T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
-(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall
-(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
-(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u:
-T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to
-(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1:
-(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
-t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
-_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
-(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1]))
-(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let
-H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat)))
-(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort
+ \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
+d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts
+h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_:
+nat).(\lambda (H: (eq TList TNil TNil)).H))) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList TNil
+(lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) (lifts h d t0)))).(let
+H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList return
+(\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
+\Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in (False_ind
+(eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: nat).(\forall (d:
+nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq TList t0
+ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: TList).(\forall (h:
+nat).(\forall (d: nat).((eq TList (lifts h d (TCons t t0)) (lifts h d t1))
+\to (eq TList (TCons t t0) t1))))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d t0)) TNil)).(let
+H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) (\lambda (ee:
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in (False_ind
+(eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda (t2:
+TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList (TCons
+(lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t t0)
+t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList (TCons
+(lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)))).(let H2
+\def (f_equal TList T (\lambda (e: TList).(match e in TList return (\lambda
+(_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
+(d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) \Rightarrow (TSort
n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
-| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0
-(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0:
-nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat)
-(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) |
-(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i |
-false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map
-f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus
-x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map
-(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort
-n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
-with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
-\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f:
-((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
-\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1)
-\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
-t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7
-\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0))
-H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t
-t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
-(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8
-P)))))) H3)) H2))))))))))) x)).
-
-theorem nf2_gen_abbr:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u
-t)) \to (\forall (P: Prop).P))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
-T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t)
-t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x
-in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t
-(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift
-(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O
-x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O
-x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
-_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S
-O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind
-Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u)
-t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda
-(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in
-(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O)
-O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c
-(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H
-(lift (S O) O x) H2) in (nf2_gen__aux Abbr x u O (H3 x (pr2_free c (THead
-(Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl
-x) u))) P))) H1))) H0))))))).
+| false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow (THead k (lref_map
+f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda (x: nat).(plus x
+h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0))
+(TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList
+TList (\lambda (e: TList).(match e in TList return (\lambda (_: TList).TList)
+with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) (ts0: TList) on
+ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | (TCons t3 ts1)
+\Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in lifts) h d t0) |
+(TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons
+(lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h
+d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2)))
+(f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1
+(lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
-theorem nf2_gen_void:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
-(lift (S O) O t))) \to (\forall (P: Prop).P))))
+theorem nfs2_tapp:
+ \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t))
+\to (land (nfs2 c ts) (nf2 c t)))))
\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
-T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
-Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__aux Void t u O
-(H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t (pr0_zeta Void
-not_void_abst t t (pr0_refl t) u))) P))))).
+ \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0:
+TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H:
+(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True
+(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I
+H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c
+(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c
+t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c
+(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2:
+(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let
+H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c
+t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj
+(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5)
+H6))) H4))))) H1)))))) ts))).
-theorem arity_nf2_inv_all:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
-a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat
-(\lambda (n: nat).(eq T t (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c
-ws))))))))))))
+theorem pc3_nf2_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
+t2) \to (pr3 c t1 t2)))))
\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
-A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
-(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
-ws))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (_: (nf2 c0 (TSort
-n))).(or3_intro1 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (TSort n)
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
-(\lambda (n0: nat).(eq T (TSort n) (TSort n0)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(TSort n) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda
-(i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst)
-v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
-T).(nfs2 c0 ws)))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort
-n0))) n (refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_:
-(((nf2 d u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads
-(Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0:
-nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 d (CHead d0 (Bind Abst)
-v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
-T).(nfs2 d ws)))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(nf2_gen_lref c0 d
-u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (TLRef i)
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_4 TList nat C T (\lambda
-(ws: TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TLRef
-i) (THeads (Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0:
-nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 c0 (CHead d0 (Bind Abst)
-v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
-T).(nfs2 c0 ws))))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst)
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_:
-(((nf2 d u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads
-(Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0:
-nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 d (CHead d0 (Bind Abst)
-v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
-T).(nfs2 d ws)))))))))).(\lambda (_: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T
-T (\lambda (w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(TLRef i) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
-(i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TLRef i) (THeads (Flat Appl)
-ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0: nat).(\lambda (d0:
-C).(\lambda (v: T).(getl i0 c0 (CHead d0 (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))
-(ex3_4_intro TList nat C T (\lambda (ws: TList).(\lambda (i0: nat).(\lambda
-(_: C).(\lambda (_: T).(eq T (TLRef i) (THeads (Flat Appl) ws (TLRef
-i0))))))) (\lambda (_: TList).(\lambda (i0: nat).(\lambda (d0: C).(\lambda
-(v: T).(getl i0 c0 (CHead d0 (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))
-TNil i d u (refl_equal T (TLRef i)) H0 I))))))))))) (\lambda (b: B).(\lambda
-(H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2
-T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 (Bind b) u) t0
-a2)).(\lambda (_: (((nf2 (CHead c0 (Bind b) u) t0) \to (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind b) u) w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind b) u) (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda
-(ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0
-(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
-nat).(\lambda (d: C).(\lambda (v: T).(getl i (CHead c0 (Bind b) u) (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 (CHead c0 (Bind b) u) ws)))))))))).(\lambda (H5:
-(nf2 c0 (THead (Bind b) u t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst))
-\to ((arity g (CHead c0 (Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u
-t0)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind
-b0) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
-w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_4
-TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda
-(_: T).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws (TLRef i)))))))
-(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
-c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))))) (\lambda (_: (not
-(eq B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0
-a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0
-H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr)
-u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
-w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_4
-TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda
-(_: T).(eq T (THead (Bind Abbr) u t0) (THeads (Flat Appl) ws (TLRef i)))))))
-(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
-c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))))) (\lambda (H6:
-(not (eq B Abst Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
-a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u t0))).(let H9 \def (match (H6
-(refl_equal B Abst)) in False return (\lambda (_: False).(or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T (THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))))) with []) in H9)))) (\lambda (_: (not
-(eq B Void Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0
-a2)).(\lambda (H8: (nf2 c0 (THead (Bind Void) u t0))).(let H9 \def
-(arity_gen_cvoid g (CHead c0 (Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void
-c0 u)) in (ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T
-T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u t0) (THead
-(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
-(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
-(n: nat).(eq T (THead (Bind Void) u t0) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x: T).(\lambda (H10: (eq T t0
-(lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead
-(Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T (lift (S O) O x)
-(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
-(THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
-(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u t1)
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind Void) u t1) (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))
-(nf2_gen_void c0 u x H11 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
-T (THead (Bind Void) u (lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
-Void) u (lift (S O) O x)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind
-Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))))) t0 H10)))) H9))))) b H0 H3
-H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
-(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T
-T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0:
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
-a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) (\lambda
-(w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind Abst) w)
-u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
-nat).(\lambda (d: C).(\lambda (v: T).(getl i (CHead c0 (Bind Abst) u) (CHead
-d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 (CHead c0 (Bind Abst) u) ws)))))))))).(\lambda (H4:
-(nf2 c0 (THead (Bind Abst) u t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in
-(and_ind (nf2 c0 u) (nf2 (CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T (THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7:
-(nf2 (CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind
-Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_2_intro T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind
-Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
-(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws
-(TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
-C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
-ws)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g c0 t0
-(AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
-(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
-C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda
-(_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0
-(CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda
-(_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (H4: (nf2 c0 (THead
-(Flat Appl) u t0))).(let H5 \def (nf2_gen_flat Appl c0 u t0 H4) in (and_ind
-(nf2 c0 u) (nf2 c0 t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
-(THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
-(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t0)
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
-(\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def (H3 H7) in
-(let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
-T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))
-(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u
-t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_4 TList
-nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_:
-T).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i)))))))
-(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
-c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H9: (ex3_2
-T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
-T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w:
-T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
-c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
-T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
-t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t0 (THead (Bind Abst)
-x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: (nf2 (CHead c0 (Bind Abst)
-x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat
-Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in (let H14 \def (eq_ind T t0
-(\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THead (Bind Abst) x0 x1)
-H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T
-T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead
-(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
-(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
-(n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead
-(Bind Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind
-Abst) x0 x1)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
-(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (THead
-(Bind Abst) x0 x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))))) t0 H10)))))))) H9)) (\lambda (H9: (ex
-nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind nat (\lambda (n:
-nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0:
-T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
-t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
-(\lambda (x: nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T
-t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in
-(let H12 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2
-(TSort x) H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))))) (let H13 \def (match (arity_gen_sort g
-c0 x (AHead a1 a2) H12) in leq return (\lambda (a0: A).(\lambda (a3:
-A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) \to ((eq A a3 (ASort
-O x)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
-Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x))
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x))
-(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
-nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
-(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
-c0 ws)))))))))))) with [(leq_sort h1 h2 n1 n2 k H13) \Rightarrow (\lambda
-(H14: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H15: (eq A (ASort h2 n2)
-(ASort O x))).((let H16 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
-in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
-_ _) \Rightarrow False])) I (AHead a1 a2) H14) in (False_ind ((eq A (ASort h2
-n2) (ASort O x)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
-Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x))
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x))
-(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
-nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
-(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
-c0 ws))))))))) H16)) H15 H13))) | (leq_head a0 a3 H13 a4 a5 H14) \Rightarrow
-(\lambda (H15: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H16: (eq A
-(AHead a3 a5) (ASort O x))).((let H17 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 |
-(AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H15) in ((let H18
-\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
-with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0
-a4) (AHead a1 a2) H15) in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to
-((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to
-(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u
-(TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n))))
-(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
-C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl)
-ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
-C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
-ws)))))))))))) (\lambda (H19: (eq A a4 a2)).(eq_ind A a2 (\lambda (a6:
-A).((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a1 a3) \to ((leq g a6 a5)
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl)
-u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n))))
-(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
-C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl)
-ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
-C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
-ws))))))))))) (\lambda (H20: (eq A (AHead a3 a5) (ASort O x))).(let H21 \def
-(eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort O x) H20) in (False_ind ((leq g a1 a3) \to ((leq g a2 a5)
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl)
-u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n))))
-(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
-C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl)
-ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
-C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))
-H21))) a4 (sym_eq A a4 a2 H19))) a0 (sym_eq A a0 a1 H18))) H17)) H16 H13
-H14)))]) in (H13 (refl_equal A (AHead a1 a2)) (refl_equal A (ASort O x)))) t0
-H10))))) H9)) (\lambda (H9: (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
-ws))))))).(ex3_4_ind TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x0: TList).(\lambda (x1:
-nat).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq T t0 (THeads (Flat
-Appl) x0 (TLRef x1)))).(\lambda (H11: (getl x1 c0 (CHead x2 (Bind Abst)
-x3))).(\lambda (H12: (nfs2 c0 x0)).(let H13 \def (eq_ind T t0 (\lambda (t1:
-T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1))
-H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1
-a2))) H2 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (eq_ind_r T (THeads (Flat
-Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-(THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat
-Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))))) (or3_intro2 (ex3_2 T T (\lambda (w:
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef
-x1))) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
-w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THeads (Flat Appl) x0
-(TLRef x1))) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
-(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (THeads
-(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda
-(_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0
-(CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda
-(_: C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_4_intro TList nat C T (\lambda
-(ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead
-(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws
-(TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
-C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))
-(TCons u x0) x1 x2 x3 (refl_equal T (THead (Flat Appl) u (THeads (Flat Appl)
-x0 (TLRef x1)))) H11 (conj (nf2 c0 u) (nfs2 c0 x0) H6 H12))) t0 H10))))))))))
-H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0:
-A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (_: (((nf2 c0 u) \to
-(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0:
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
-t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (H4: (nf2
-c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 u t0 H4 (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Cast) u t0) (THead (Bind
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
-nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Flat Cast) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws))))))))))))))))) (\lambda (c0: C).(\lambda
-(t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 t0 a1)).(\lambda (H1:
-(((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
-(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
-ws)))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2
-c0 t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda
-(_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))))
-(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
-c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))) (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda
-(_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))))
-(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
-c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H5: (ex3_2
-T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
-T).(nf2 (CHead c0 (Bind Abst) w) u))))).(ex3_2_ind T T (\lambda (w:
-T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
-c0 (Bind Abst) w) u))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T
-t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
-(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
-(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
-(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind Abst)
-x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind Abst)
-x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T
-T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w u))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
-T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T t1 (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))) (or3_intro0 (ex3_2 T
-T (\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
-nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-(THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_2_intro T T (\lambda (w:
-T).(\lambda (u: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u))))
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
-T).(nf2 (CHead c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst)
-x0 x1)) H7 H8)) t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq
-T t0 (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3
-(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w
-u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
-(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0
-(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x:
-nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T (TSort x) (\lambda (t1:
-T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind
-Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
-nat).(eq T t1 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
-(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t1 (THeads (Flat Appl) ws
-(TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
-C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
-TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))
-(or3_intro1 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead
-(Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
-T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
-nat).(eq T (TSort x) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TSort x)
-(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
-nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
-(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
-c0 ws)))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x
-(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_4 TList nat C T
-(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
-t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
-nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
-(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
-c0 ws))))))).(ex3_4_ind TList nat C T (\lambda (ws: TList).(\lambda (i:
-nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
-i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
-T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) (or3 (ex3_2 T T
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
-n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda
-(_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))))
-(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
-c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
-nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x0:
-TList).(\lambda (x1: nat).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H6: (eq
-T t0 (THeads (Flat Appl) x0 (TLRef x1)))).(\lambda (H7: (getl x1 c0 (CHead x2
-(Bind Abst) x3))).(\lambda (H8: (nfs2 c0 x0)).(eq_ind_r T (THeads (Flat Appl)
-x0 (TLRef x1)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u:
-T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2
-c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))))
-(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_4 TList nat C T (\lambda
-(ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t1
-(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
-nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
-(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
-c0 ws)))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T
-(THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w:
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
-c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl)
-x0 (TLRef x1)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THeads (Flat
-Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_4_intro TList nat C T (\lambda (ws:
-TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THeads (Flat
-Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
-(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
-C).(\lambda (_: T).(nfs2 c0 ws))))) x0 x1 x2 x3 (refl_equal T (THeads (Flat
-Appl) x0 (TLRef x1))) H7 H8)) t0 H6)))))))) H5)) H4))))))))))) c t a H))))).
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t:
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x:
+T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def
+(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t:
+T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
-theorem pc3_gen_sort_abst:
- \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c
-(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
+theorem pc3_pr3_conf:
+ \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall
+(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda
-(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0
-\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0:
-T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c
-(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def
-(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
-T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u
-x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
-u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n)))
-(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind
-T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P
-H8)))))))) H3))))) H0))))))).
+ \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t
+t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c
+t2 t H0) t1 H)))))).
-theorem ty3_gen_abst_abst:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
-(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2
-T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst)
-u) t1 t2))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u
-t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T
-(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u)
-t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2)
-x)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c
-(THead (Bind Abst) u t3) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_:
-T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
-(CHead c (Bind Abst) u) t2 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda
-(t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0)))) (ex2 T (\lambda (w: T).(ty3
-g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind
-Abst) u x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c
-(Bind Abst) u) t2 x0)).(\lambda (_: (ty3 g (CHead c (Bind Abst) u) x0
-x2)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c
-(THead (Bind Abst) u t3) (THead (Bind Abst) u t2))))) (\lambda (_:
-T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda
-(_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t3)))) (\lambda (t3:
-T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0))))
-(ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind
-Abst) u) t1 t2))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
-(H5: (pc3 c (THead (Bind Abst) u x3) (THead (Bind Abst) u t2))).(\lambda (H6:
-(ty3 g c u x4)).(\lambda (H7: (ty3 g (CHead c (Bind Abst) u) t1 x3)).(\lambda
-(_: (ty3 g (CHead c (Bind Abst) u) x3 x5)).(and_ind (pc3 c u u) (\forall (b:
-B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x3 t2))) (ex2 T (\lambda (w:
-T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)))
-(\lambda (_: (pc3 c u u)).(\lambda (H10: ((\forall (b: B).(\forall (u0:
-T).(pc3 (CHead c (Bind b) u0) x3 t2))))).(ex_intro2 T (\lambda (w: T).(ty3 g
-c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x4 H6
-(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x3 H7 (H10 Abst u)))))
-(pc3_gen_abst c u u x3 t2 H5))))))))) (ty3_gen_bind g Abst c u t1 (THead
-(Bind Abst) u t2) H))))))))) (ty3_gen_bind g Abst c u t2 x H0))))
-(ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2) H))))))).
-
-theorem ty3_typecheck:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t
-v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H:
-(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u:
-T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g
-c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) v
-(ty3_cast g c t v H x H0)))) (ty3_correct g c t v H)))))).
-
-inductive sort: T \to Prop \def
-| sort_sort: \forall (n: nat).(sort (TSort n))
-| sort_abst: \forall (u: T).((sort u) \to (\forall (t: T).((sort t) \to (sort
-(THead (Bind Abst) u t))))).
+axiom pc3_gen_appls_sort_abst:
+ \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall
+(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u))
+\to False)))))
+.
-theorem sort_nf2:
- \forall (t: T).((sort t) \to (\forall (c: C).(nf2 c t)))
-\def
- \lambda (t: T).(\lambda (H: (sort t)).(sort_ind (\lambda (t0: T).(\forall
-(c: C).(nf2 c t0))) (\lambda (n: nat).(\lambda (c: C).(nf2_sort c n)))
-(\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1: ((\forall (c: C).(nf2 c
-u)))).(\lambda (t0: T).(\lambda (_: (sort t0)).(\lambda (H3: ((\forall (c:
-C).(nf2 c t0)))).(\lambda (c: C).(let H_y \def (H3 (CHead c (Bind Abst) u))
-in (nf2_abst c u (H1 c) Abst u t0 H_y))))))))) t H)).
+axiom pc3_gen_appls_lref_abst:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall
+(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to
+False))))))))
+.
-theorem sort_pc3:
- \forall (t1: T).((sort t1) \to (\forall (t2: T).((sort t2) \to (\forall (c:
-C).((pc3 c t1 t2) \to (eq T t1 t2))))))
-\def
- \lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda (t: T).(\forall
-(t2: T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t t2))))))
-(\lambda (n: nat).(\lambda (t2: T).(\lambda (H0: (sort t2)).(sort_ind
-(\lambda (t: T).(\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n)
-t)))) (\lambda (n0: nat).(\lambda (c: C).(\lambda (H1: (pc3 c (TSort n)
-(TSort n0))).(eq_ind nat n (\lambda (n1: nat).(eq T (TSort n) (TSort n1)))
-(refl_equal T (TSort n)) n0 (pc3_gen_sort c n n0 H1))))) (\lambda (u:
-T).(\lambda (_: (sort u)).(\lambda (_: ((\forall (c: C).((pc3 c (TSort n) u)
-\to (eq T (TSort n) u))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda
-(_: ((\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n) t))))).(\lambda
-(c: C).(\lambda (H5: (pc3 c (TSort n) (THead (Bind Abst) u
-t))).(pc3_gen_sort_abst c u t n H5 (eq T (TSort n) (THead (Bind Abst) u
-t))))))))))) t2 H0)))) (\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1:
-((\forall (t2: T).((sort t2) \to (\forall (c: C).((pc3 c u t2) \to (eq T u
-t2))))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3: ((\forall (t2:
-T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t
-t2))))))).(\lambda (t2: T).(\lambda (H4: (sort t2)).(sort_ind (\lambda (t0:
-T).(\forall (c: C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind
-Abst) u t) t0)))) (\lambda (n: nat).(\lambda (c: C).(\lambda (H5: (pc3 c
-(THead (Bind Abst) u t) (TSort n))).(pc3_gen_sort_abst c u t n (pc3_s c
-(TSort n) (THead (Bind Abst) u t) H5) (eq T (THead (Bind Abst) u t) (TSort
-n)))))) (\lambda (u0: T).(\lambda (H5: (sort u0)).(\lambda (_: ((\forall (c:
-C).((pc3 c (THead (Bind Abst) u t) u0) \to (eq T (THead (Bind Abst) u t)
-u0))))).(\lambda (t0: T).(\lambda (H7: (sort t0)).(\lambda (_: ((\forall (c:
-C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind Abst) u t)
-t0))))).(\lambda (c: C).(\lambda (H9: (pc3 c (THead (Bind Abst) u t) (THead
-(Bind Abst) u0 t0))).(and_ind (pc3 c u u0) (\forall (b: B).(\forall (u1:
-T).(pc3 (CHead c (Bind b) u1) t t0))) (eq T (THead (Bind Abst) u t) (THead
-(Bind Abst) u0 t0)) (\lambda (H10: (pc3 c u u0)).(\lambda (H11: ((\forall (b:
-B).(\forall (u1: T).(pc3 (CHead c (Bind b) u1) t t0))))).(let H_y \def (H11
-Abbr u) in (let H_y0 \def (H1 u0 H5 c H10) in (let H_y1 \def (H3 t0 H7 (CHead
-c (Bind Abbr) u) H_y) in (let H12 \def (eq_ind_r T t0 (\lambda (t3: T).(pc3
-(CHead c (Bind Abbr) u) t t3)) H_y t H_y1) in (let H13 \def (eq_ind_r T t0
-(\lambda (t3: T).(sort t3)) H7 t H_y1) in (eq_ind T t (\lambda (t3: T).(eq T
-(THead (Bind Abst) u t) (THead (Bind Abst) u0 t3))) (let H14 \def (eq_ind_r T
-u0 (\lambda (t3: T).(pc3 c u t3)) H10 u H_y0) in (let H15 \def (eq_ind_r T u0
-(\lambda (t3: T).(sort t3)) H5 u H_y0) in (eq_ind T u (\lambda (t3: T).(eq T
-(THead (Bind Abst) u t) (THead (Bind Abst) t3 t))) (refl_equal T (THead (Bind
-Abst) u t)) u0 H_y0))) t0 H_y1)))))))) (pc3_gen_abst c u u0 t t0 H9))))))))))
-t2 H4))))))))) t1 H)).
+axiom pc3_gen_appls_lref_sort:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws:
+TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads
+(Flat Appl) ws (TSort n))) \to False))))))))
+.
-theorem sort_correct:
- \forall (g: G).(\forall (t1: T).((sort t1) \to (\forall (c: C).(ex3 T
-(\lambda (t2: T).(tau0 g c t1 t2)) (\lambda (t2: T).(ty3 g c t1 t2)) (\lambda
-(t2: T).(sort t2))))))
-\def
- \lambda (g: G).(\lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda
-(t: T).(\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2:
-T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2))))) (\lambda (n: nat).(\lambda
-(c: C).(ex3_intro T (\lambda (t2: T).(tau0 g c (TSort n) t2)) (\lambda (t2:
-T).(ty3 g c (TSort n) t2)) (\lambda (t2: T).(sort t2)) (TSort (next g n))
-(tau0_sort g c n) (ty3_sort g c n) (sort_sort (next g n))))) (\lambda (u:
-T).(\lambda (H0: (sort u)).(\lambda (H1: ((\forall (c: C).(ex3 T (\lambda
-(t2: T).(tau0 g c u t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2:
-T).(sort t2)))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3:
-((\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2:
-T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2)))))).(\lambda (c: C).(let H_x
-\def (H1 c) in (let H4 \def H_x in (ex3_ind T (\lambda (t2: T).(tau0 g c u
-t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2: T).(sort t2)) (ex3 T
-(\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(ty3
-g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) (\lambda (x0:
-T).(\lambda (_: (tau0 g c u x0)).(\lambda (H6: (ty3 g c u x0)).(\lambda (_:
-(sort x0)).(let H_x0 \def (H3 (CHead c (Bind Abst) u)) in (let H8 \def H_x0
-in (ex3_ind T (\lambda (t2: T).(tau0 g (CHead c (Bind Abst) u) t t2))
-(\lambda (t2: T).(ty3 g (CHead c (Bind Abst) u) t t2)) (\lambda (t2: T).(sort
-t2)) (ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda
-(t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2)))
-(\lambda (x1: T).(\lambda (H9: (tau0 g (CHead c (Bind Abst) u) t
-x1)).(\lambda (H10: (ty3 g (CHead c (Bind Abst) u) t x1)).(\lambda (H11:
-(sort x1)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) x1 t0))
-(ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2:
-T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2)))
-(\lambda (x: T).(\lambda (H12: (ty3 g (CHead c (Bind Abst) u) x1
-x)).(ex3_intro T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2))
-(\lambda (t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort
-t2)) (THead (Bind Abst) u x1) (tau0_bind g Abst c u t x1 H9) (ty3_bind g c u
-x0 H6 Abst t x1 H10 x H12) (sort_abst u H0 x1 H11)))) (ty3_correct g (CHead c
-(Bind Abst) u) t x1 H10)))))) H8))))))) H4)))))))))) t1 H))).
+inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
+| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c
+TNil u)))
+| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts:
+TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
-definition pchurch_context:
- T \to (T \to T)
+theorem tys3_gen_nil:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T
+(\lambda (u0: T).(ty3 g c u u0))))))
\def
- \lambda (t: T).(\lambda (u: T).(THead (Bind Abst) t (THead (Bind Abst)
-(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) u))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil
+u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_:
+TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda
+(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq
+TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0:
+T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList
+TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda
+(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts:
+TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to
+(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t
+ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee:
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind
+(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
-definition pnat:
- T \to T
+theorem tys3_gen_cons:
+ \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall
+(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts
+u)))))))
\def
- \lambda (t: T).(pchurch_context t (lift (S (S O)) O t)).
+ \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts)
+(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u)
+(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind
+g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to
+(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1:
+T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
+return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
+\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0)
+(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1:
+(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0
+u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0)
+(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t
+ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList
+return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _)
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal
+TList TList (\lambda (e: TList).(match e in TList return (\lambda (_:
+TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1]))
+(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def
+(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land
+(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList
+ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind
+T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3
+g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
-definition church_body:
- nat \to T
+theorem ty3_gen_appl_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))))))))
\def
- let rec church_body (n: nat) on n: T \def (match n with [O \Rightarrow
-(TLRef (S O)) | (S n0) \Rightarrow (THead (Flat Appl) (church_body n0) (TLRef
-O))]) in church_body.
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
+x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g
+c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in
+(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0
+x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl)
+w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))
+(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda
+(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def
+(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t:
+T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead
+(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t:
+T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3
+g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t)))))
+(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c
+(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind
+Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c
+(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c
+(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def
+(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c
+x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
+b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10
+(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13
+(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda
+(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u:
+T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c
+(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead
+(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6))
+(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w
+Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead
+(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind
+Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5
+x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2
+(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3))))))))
+(ty3_gen_appl g c w v x H))))))).
-definition pchurch:
- T \to (nat \to T)
+theorem ty3_inv_lref_nf2_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c
+(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to
+((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
\def
- \lambda (t: T).(\lambda (n: nat).(pchurch_context t (church_body n))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda
+(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t
+u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c
+u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda
+(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2:
+T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift
+(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2
+c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to
+((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0
+t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda
+(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10
+\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11
+\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to
+(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0:
+T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y
+\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2
+H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq
+T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2:
+T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m))
+u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in
+(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5)))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7:
+(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
+O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
+n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))))))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7:
+(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
+O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
+n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0
+(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y
+d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2
+(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq
+T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i)
+O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0:
+T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda
+(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S
+i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
+i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u)
+t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0
+(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
+u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda
+(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0
+u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda
+(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
+(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2)
+\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
+u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
+(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to
+(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq
+T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl)
+w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead
+(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0:
+T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
+t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0
+t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda
+(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to
+((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast)
+t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0
+(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2
+t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T
+u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
-theorem pnat_props__lift_SSO_O:
- \forall (t: T).(eq T (lift (S (S O)) O t) (lift (S O) O (lift (S O) O t)))
+theorem ty3_inv_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c
+(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0:
+T).(eq T u (lift (S i) O u0))))))))))
\def
- \lambda (t: T).(eq_ind_r T (lift (plus (S O) (S O)) O t) (\lambda (t0:
-T).(eq T (lift (S (S O)) O t) t0)) (refl_equal T (lift (plus (S O) (S O)) O
-t)) (lift (S O) O (lift (S O) O t)) (lift_free t (S O) (S O) O O (le_O_n
-(plus O (S O))) (le_n O))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1:
+(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
-theorem pnat_props__lift_SO_SO:
- \forall (t: T).(eq T (lift (S O) (S O) (lift (S O) O t)) (lift (S O) O (lift
-(S O) O t)))
+theorem ty3_inv_appls_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1:
+T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to
+((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S
+i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u))
+u1))))))))))
\def
- \lambda (t: T).(eq_ind nat (plus (S O) O) (\lambda (n: nat).(eq T (lift (S
-O) n (lift (S O) O t)) (lift (S O) O (lift (S O) O t)))) (eq_ind_r T (lift (S
-O) O (lift (S O) O t)) (\lambda (t0: T).(eq T t0 (lift (S O) O (lift (S O) O
-t)))) (refl_equal T (lift (S O) O (lift (S O) O t))) (lift (S O) (plus (S O)
-O) (lift (S O) O t)) (lift_d t (S O) (S O) O O (le_n O))) (S O) (refl_equal
-nat (S O))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t:
+TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t
+(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u:
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t
+(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H:
+(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c
+u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in
+(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u:
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1)))
+(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def
+(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r
+T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i)
+O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda
+(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u)
+(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2))))))))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall
+(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef
+i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
+(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u))
+u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c
+(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t
+(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T
+T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind
+Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_:
+T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst)
+u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u:
+T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u)))
+u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat
+Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t
+x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def
+(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
+x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3
+c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1)))
+(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0)
+x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def
+(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c
+(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i)
+O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O
+u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
+(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O
+x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead
+(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
+(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S
+i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c
+(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t
+Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
-theorem pnat_ty3:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
-u) \to (\forall (n: nat).(ty3 g c (pchurch t n) (pnat t)))))))
+theorem ty3_inv_lref_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c
+(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i
+j)))))))
\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c t u)).(ex_ind T (\lambda (t0: T).(ty3 g c u t0)) (\forall (n:
-nat).(ty3 g c (THead (Bind Abst) t (THead (Bind Abst) (THead (Bind Abst)
-(lift (S O) O t) (lift (S (S O)) O t)) (church_body n))) (THead (Bind Abst) t
-(THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))
-(lift (S (S O)) O t))))) (\lambda (x: T).(\lambda (H0: (ty3 g c u
-x)).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(ty3 g c (THead (Bind Abst)
-t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
-t)) (church_body n0))) (THead (Bind Abst) t (THead (Bind Abst) (THead (Bind
-Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)))))
-(ty3_bind g c t u H Abst (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O
-t) (lift (S (S O)) O t)) (TLRef (S O))) (THead (Bind Abst) (THead (Bind Abst)
-(lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)) (ty3_bind g
-(CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
-t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g
-(CHead c (Bind Abst) t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H
-(CHead c (Bind Abst) t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c)
-t)) Abst (lift (S (S O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead
-(CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S
-Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O)
-(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
-(drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x)
-(ty3_lift g c u x H0 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O
-t)) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift
-(S O) O t)) c t (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead
-c (Bind Abst) t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t)))))
-Abst (TLRef (S O)) (lift (S (S O)) O t) (ty3_abst g (S O) (CHead (CHead c
-(Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
-O)) O t))) c t (getl_head (Bind Abst) O (CHead c (Bind Abst) t) (CHead c
-(Bind Abst) t) (getl_refl Abst c t) (THead (Bind Abst) (lift (S O) O t) (lift
-(S (S O)) O t))) u H) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead
-c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
-O)) O t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind
-Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O)
-(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
-(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
-(S (S O)) O t)))))) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t)
-(lift (S (S O)) O t)) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
-t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Bind
-Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
-t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead c (Bind Abst)
-t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst (lift (S (S
-O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
-t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead (CHead c
-(Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop (Bind Abst)
-O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl (CHead c (Bind
-Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g c u x H0
-(CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O))
-(drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t
-(S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst)
-t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (lift (S (S
-O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
-t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) O
-(S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) (drop_drop
-(Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl
-(CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O))
-O t))))) (lift (S (S O)) O x) (ty3_lift g c u x H0 (CHead (CHead c (Bind
-Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
-t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst)
-(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O)
-(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
-(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
-(S (S O)) O t))))))) (\lambda (n0: nat).(\lambda (H1: (ty3 g c (THead (Bind
-Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
-O)) O t)) (church_body n0))) (THead (Bind Abst) t (THead (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O
-t))))).(let H_x \def (ty3_gen_abst_abst g c t (THead (Bind Abst) (THead (Bind
-Abst) (lift (S O) O t) (lift (S (S O)) O t)) (church_body n0)) (THead (Bind
-Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S
-O)) O t)) H1) in (let H2 \def H_x in (ex2_ind T (\lambda (w: T).(ty3 g c t
-w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) t) (THead (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (church_body n0)) (THead
-(Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift
-(S (S O)) O t)))) (ty3 g c (THead (Bind Abst) t (THead (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Flat Appl)
-(church_body n0) (TLRef O)))) (THead (Bind Abst) t (THead (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t))))
-(\lambda (x0: T).(\lambda (_: (ty3 g c t x0)).(\lambda (H4: (ty3 g (CHead c
-(Bind Abst) t) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
-(S (S O)) O t)) (church_body n0)) (THead (Bind Abst) (THead (Bind Abst) (lift
-(S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)))).(let H_x0 \def
-(ty3_gen_abst_abst g (CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O
-t) (lift (S (S O)) O t)) (church_body n0) (lift (S (S O)) O t) H4) in (let H5
-\def H_x0 in (ex2_ind T (\lambda (w: T).(ty3 g (CHead c (Bind Abst) t) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) w)) (\lambda (_: T).(ty3 g
-(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
-t) (lift (S (S O)) O t))) (church_body n0) (lift (S (S O)) O t))) (ty3 g c
-(THead (Bind Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t)
-(lift (S (S O)) O t)) (THead (Flat Appl) (church_body n0) (TLRef O)))) (THead
-(Bind Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S
-(S O)) O t)) (lift (S (S O)) O t)))) (\lambda (x1: T).(\lambda (H6: (ty3 g
-(CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
-t)) x1)).(\lambda (H7: (ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst)
-(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (church_body n0)
-(lift (S (S O)) O t))).(ty3_bind g c t u H Abst (THead (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Flat Appl)
-(church_body n0) (TLRef O))) (THead (Bind Abst) (THead (Bind Abst) (lift (S
-O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)) (ty3_bind g (CHead c
-(Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))
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-(lift (S (S O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c
-(Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead
-(CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop
-(Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl
-(CHead c (Bind Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g
-c u x H0 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S
-O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t))
-c t (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind
-Abst) t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (THead
-(Flat Appl) (church_body n0) (TLRef O)) (lift (S (S O)) O t) (ex_ind T
-(\lambda (t0: T).(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (lift (S (S O)) O t) t0))
-(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S
-O) O t) (lift (S (S O)) O t))) (THead (Flat Appl) (church_body n0) (TLRef O))
-(lift (S (S O)) O t)) (\lambda (x2: T).(\lambda (H8: (ty3 g (CHead (CHead c
-(Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
-O)) O t))) (lift (S (S O)) O t) x2)).(ty3_conv g (CHead (CHead c (Bind Abst)
-t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)))
-(lift (S (S O)) O t) x2 H8 (THead (Flat Appl) (church_body n0) (TLRef O))
-(THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S (S O)) O t)
-(lift (S O) (S O) (lift (S O) O (lift (S O) O t))))) (ty3_appl g (CHead
-(CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
-(S (S O)) O t))) (church_body n0) (lift (S (S O)) O t) H7 (TLRef O) (lift (S
-O) (S O) (lift (S O) O (lift (S O) O t))) (eq_ind_r T (lift (S O) O (lift (S
-O) O t)) (\lambda (t0: T).(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst)
-(THead (Bind Abst) (lift (S O) O t) t0)) (TLRef O) (THead (Bind Abst) t0
-(lift (S O) (S O) (lift (S O) O (lift (S O) O t)))))) (let H9 \def (eq_ind T
-(lift (S (S O)) O t) (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t) (THead
-(Bind Abst) (lift (S O) O t) t0) x1)) H6 (lift (S O) O (lift (S O) O t))
-(pnat_props__lift_SSO_O t)) in (eq_ind T (lift (S O) O (THead (Bind Abst)
-(lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (\lambda (t0: T).(ty3 g
-(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
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-(CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
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-(S O) O t) (lift (S O) O (lift (S O) O t))) (getl_refl Abst (CHead c (Bind
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-(S O) (lift (S O) O (lift (S O) O t)))) (lift_bind Abst (lift (S O) O t)
-(lift (S O) O (lift (S O) O t)) (S O) O))) (lift (S (S O)) O t)
-(pnat_props__lift_SSO_O t))) (eq_ind_r T (lift (S O) O (lift (S O) O (lift (S
-O) O t))) (\lambda (t0: T).(pc3 (CHead (CHead c (Bind Abst) t) (Bind Abst)
-(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (THead (Flat Appl)
-(church_body n0) (THead (Bind Abst) (lift (S (S O)) O t) t0)) (lift (S (S O))
-O t))) (eq_ind_r T (lift (S O) O (lift (S O) O t)) (\lambda (t0: T).(pc3
-(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
-t) t0)) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) t0 (lift (S O)
-O (lift (S O) O (lift (S O) O t))))) t0)) (pc3_pr3_r (CHead (CHead c (Bind
-Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift
-(S O) O t)))) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S
-O) O (lift (S O) O t)) (lift (S O) O (lift (S O) O (lift (S O) O t))))) (lift
-(S O) O (lift (S O) O t)) (pr3_t (THead (Bind Abbr) (church_body n0) (lift (S
-O) O (lift (S O) O (lift (S O) O t)))) (THead (Flat Appl) (church_body n0)
-(THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O) O (lift (S O)
-O (lift (S O) O t))))) (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (pr3_pr2
-(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
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-(THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O) O (lift (S O)
-O (lift (S O) O t))))) (THead (Bind Abbr) (church_body n0) (lift (S O) O
-(lift (S O) O (lift (S O) O t)))) (pr2_free (CHead (CHead c (Bind Abst) t)
-(Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O
-t)))) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S O) O
-(lift (S O) O t)) (lift (S O) O (lift (S O) O (lift (S O) O t))))) (THead
-(Bind Abbr) (church_body n0) (lift (S O) O (lift (S O) O (lift (S O) O t))))
-(pr0_beta (lift (S O) O (lift (S O) O t)) (church_body n0) (church_body n0)
-(pr0_refl (church_body n0)) (lift (S O) O (lift (S O) O (lift (S O) O t)))
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-(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
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-t)) (pr2_free (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst)
-(lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (THead (Bind Abbr)
-(church_body n0) (lift (S O) O (lift (S O) O (lift (S O) O t)))) (lift (S O)
-O (lift (S O) O t)) (pr0_zeta Abbr not_abbr_abst (lift (S O) O (lift (S O) O
-t)) (lift (S O) O (lift (S O) O t)) (pr0_refl (lift (S O) O (lift (S O) O
-t))) (church_body n0)))))) (lift (S (S O)) O t) (pnat_props__lift_SSO_O t))
-(lift (S O) (S O) (lift (S O) O (lift (S O) O t))) (pnat_props__lift_SO_SO
-(lift (S O) O t)))))) (ty3_correct g (CHead (CHead c (Bind Abst) t) (Bind
-Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (church_body
-n0) (lift (S (S O)) O t) H7)) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead
-(CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
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-(Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S
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-(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
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-(lift (S (S O)) O t)) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
-t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Bind
-Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
-t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead c (Bind Abst)
-t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst (lift (S (S
-O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
-t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead (CHead c
-(Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop (Bind Abst)
-O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl (CHead c (Bind
-Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g c u x H0
-(CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O))
-(drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t
-(S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst)
-t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (lift (S (S
-O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
-t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) O
-(S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
-(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) (drop_drop
-(Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl
-(CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O))
-O t))))) (lift (S (S O)) O x) (ty3_lift g c u x H0 (CHead (CHead c (Bind
-Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
-t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst)
-(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O)
-(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
-(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
-(S (S O)) O t)))))))))) H5)))))) H2))))) n)))) (ty3_correct g c t u H)))))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda
+(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda
+(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0
+H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift
+(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S
+i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
+in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x
+(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
+(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
+j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
+(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S
+i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i)
+j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4)))))
+H2))))))))).
projects / helm.git / blobdiff