\lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
le_Sn_n in (False_ind P (H0 x H))))).
+theorem le_n_pred:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
+(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
+nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
+(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
+
theorem minus_le:
\forall (x: nat).(\forall (y: nat).(le (minus x y) x))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus
x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
+theorem lt_x_O:
+ \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
+\def
+ \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
+(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
+ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
+\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))).
+
theorem le_gen_S:
\forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
(le_plus_minus_sym h n (le_trans_plus_r d h n H))) in (le_S d (minus n h)
(le_minus d n h H))))))).
+theorem lt_x_pred_y:
+ \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
+n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
+(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
+(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x)
+n H0)))))) y)).
+
H=@
RT_BASEDIR=../../../
-OPTIONS=-bench
+OPTIONS=-bench -onepass
MMAKE=$(RT_BASEDIR)matitamake $(OPTIONS)
CLEAN=$(RT_BASEDIR)matitaclean $(OPTIONS)
MMAKEO=$(RT_BASEDIR)matitamake.opt $(OPTIONS)
alias id "I" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1/1)".
alias id "land" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1)".
alias id "le" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1)".
+alias id "le_ind" = "cic:/Coq/Init/Peano/le_ind.con".
alias id "le_lt_n_Sm" = "cic:/Coq/Arith/Lt/le_lt_n_Sm.con".
alias id "le_lt_or_eq" = "cic:/Coq/Arith/Lt/le_lt_or_eq.con".
alias id "le_n" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/1)".
alias id "le_plus_minus" = "cic:/Coq/Arith/Minus/le_plus_minus.con".
alias id "le_plus_minus_r" = "cic:/Coq/Arith/Minus/le_plus_minus_r.con".
alias id "le_plus_r" = "cic:/Coq/Arith/Plus/le_plus_r.con".
+alias id "le_pred_n" = "cic:/Coq/Arith/Le/le_pred_n.con".
alias id "le_S" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/2)".
alias id "le_S_n" = "cic:/Coq/Arith/Le/le_S_n.con".
alias id "le_Sn_n" = "cic:/Coq/Arith/Le/le_Sn_n.con".
set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/spare".
include "theory.ma".
+(*
+inductive pr: Set \def
+| pr_zero: pr
+| pr_succ: pr
+| pr_proj: nat \to pr
+| pr_comp: ((nat \to pr)) \to (pr \to pr)
+| pr_prec: pr \to (pr \to pr).
+definition pr_type:
+ Set
+\def
+ ((nat \to nat)) \to nat.
+
+definition prec_appl:
+ pr_type \to (pr_type \to (nat \to pr_type))
+\def
+ let rec prec_appl (f: pr_type) (g: pr_type) (n: nat) on n: pr_type \def
+(match n with [O \Rightarrow f | (S m) \Rightarrow (\lambda (ns: ((nat \to
+nat))).(g (\lambda (i: nat).(match i with [O \Rightarrow (prec_appl f g m ns)
+| (S n0) \Rightarrow (match n0 with [O \Rightarrow m | (S j) \Rightarrow (ns
+j)])]))))]) in prec_appl.
+
+definition pr_appl:
+ pr \to pr_type
+\def
+ let rec pr_appl (h: pr) on h: pr_type \def (match h with [pr_zero
+\Rightarrow (\lambda (_: ((nat \to nat))).O) | pr_succ \Rightarrow (\lambda
+(ns: ((nat \to nat))).(S (ns O))) | (pr_proj i) \Rightarrow (\lambda (ns:
+((nat \to nat))).(ns i)) | (pr_comp fs g) \Rightarrow (\lambda (ns: ((nat \to
+nat))).(pr_appl g (\lambda (i: nat).(pr_appl (fs i) ns)))) | (pr_prec f g)
+\Rightarrow (\lambda (ns: ((nat \to nat))).(prec_appl (pr_appl f) (pr_appl g)
+(ns O) (\lambda (i: nat).(ns (S i)))))]) in pr_appl.
+
+inductive pr_arity: pr \to (nat \to Prop) \def
+| pr_arity_zero: \forall (n: nat).(pr_arity pr_zero n)
+| pr_arity_succ: \forall (n: nat).((lt O n) \to (pr_arity pr_succ n))
+| pr_arity_proj: \forall (n: nat).(\forall (i: nat).((lt i n) \to (pr_arity
+(pr_proj i) n)))
+| pr_arity_comp: \forall (n: nat).(\forall (m: nat).(\forall (fs: ((nat \to
+pr))).(\forall (g: pr).((pr_arity g m) \to (((\forall (i: nat).((lt i m) \to
+(pr_arity (fs i) n)))) \to (pr_arity (pr_comp fs g) n))))))
+| pr_arity_prec: \forall (n: nat).(\forall (f: pr).(\forall (g: pr).((lt O n)
+\to ((pr_arity f (pred n)) \to ((pr_arity g (S n)) \to (pr_arity (pr_prec f
+g) n)))))).
+
+theorem pr_arity_le:
+ \forall (h: pr).(\forall (m: nat).((pr_arity h m) \to (\forall (n: nat).((le
+m n) \to (pr_arity h n)))))
+\def
+ \lambda (h: pr).(\lambda (m: nat).(\lambda (H: (pr_arity h m)).(pr_arity_ind
+(\lambda (p: pr).(\lambda (n: nat).(\forall (n0: nat).((le n n0) \to
+(pr_arity p n0))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: (le n
+n0)).(pr_arity_zero n0)))) (\lambda (n: nat).(\lambda (H0: (lt O n)).(\lambda
+(n0: nat).(\lambda (H1: (le n n0)).(pr_arity_succ n0 (lt_le_trans O n n0 H0
+H1)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (H0: (lt i n)).(\lambda
+(n0: nat).(\lambda (H1: (le n n0)).(pr_arity_proj n0 i (lt_le_trans i n n0 H0
+H1))))))) (\lambda (n: nat).(\lambda (m0: nat).(\lambda (fs: ((nat \to
+pr))).(\lambda (g: pr).(\lambda (H0: (pr_arity g m0)).(\lambda (_: ((\forall
+(n0: nat).((le m0 n0) \to (pr_arity g n0))))).(\lambda (_: ((\forall (i:
+nat).((lt i m0) \to (pr_arity (fs i) n))))).(\lambda (H3: ((\forall (i:
+nat).((lt i m0) \to (\forall (n0: nat).((le n n0) \to (pr_arity (fs i)
+n0))))))).(\lambda (n0: nat).(\lambda (H4: (le n n0)).(pr_arity_comp n0 m0 fs
+g H0 (\lambda (i: nat).(\lambda (H5: (lt i m0)).(H3 i H5 n0 H4))))))))))))))
+(\lambda (n: nat).(\lambda (f: pr).(\lambda (g: pr).(\lambda (H0: (lt O
+n)).(\lambda (_: (pr_arity f (pred n))).(\lambda (H2: ((\forall (n0:
+nat).((le (pred n) n0) \to (pr_arity f n0))))).(\lambda (_: (pr_arity g (S
+n))).(\lambda (H4: ((\forall (n0: nat).((le (S n) n0) \to (pr_arity g
+n0))))).(\lambda (n0: nat).(\lambda (H5: (le n n0)).(pr_arity_prec n0 f g
+(lt_le_trans O n n0 H0 H5) (H2 (pred n0) (le_n_pred n n0 H5)) (H4 (S n0)
+(le_n_S n n0 H5))))))))))))) h m H))).
+
+theorem pr_arity_appl:
+ \forall (h: pr).(\forall (n: nat).((pr_arity h n) \to (\forall (ns: ((nat
+\to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt i n) \to
+(eq nat (ns i) (ms i))))) \to (eq nat (pr_appl h ns) (pr_appl h ms)))))))
+\def
+ \lambda (h: pr).(\lambda (n: nat).(\lambda (H: (pr_arity h n)).(pr_arity_ind
+(\lambda (p: pr).(\lambda (n0: nat).(\forall (ns: ((nat \to nat))).(\forall
+(ms: ((nat \to nat))).(((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
+i))))) \to (eq nat (pr_appl p ns) (pr_appl p ms))))))) (\lambda (n0:
+nat).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
+(_: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(refl_equal
+nat O))))) (\lambda (n0: nat).(\lambda (H0: (lt O n0)).(\lambda (ns: ((nat
+\to nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H1: ((\forall (i:
+nat).((lt i n0) \to (eq nat (ns i) (ms i)))))).(f_equal nat nat S (ns O) (ms
+O) (H1 O H0))))))) (\lambda (n0: nat).(\lambda (i: nat).(\lambda (H0: (lt i
+n0)).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to nat))).(\lambda
+(H1: ((\forall (i0: nat).((lt i0 n0) \to (eq nat (ns i0) (ms i0)))))).(H1 i
+H0))))))) (\lambda (n0: nat).(\lambda (m: nat).(\lambda (fs: ((nat \to
+pr))).(\lambda (g: pr).(\lambda (_: (pr_arity g m)).(\lambda (H1: ((\forall
+(ns: ((nat \to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i: nat).((lt
+i m) \to (eq nat (ns i) (ms i))))) \to (eq nat (pr_appl g ns) (pr_appl g
+ms))))))).(\lambda (_: ((\forall (i: nat).((lt i m) \to (pr_arity (fs i)
+n0))))).(\lambda (H3: ((\forall (i: nat).((lt i m) \to (\forall (ns: ((nat
+\to nat))).(\forall (ms: ((nat \to nat))).(((\forall (i0: nat).((lt i0 n0)
+\to (eq nat (ns i0) (ms i0))))) \to (eq nat (pr_appl (fs i) ns) (pr_appl (fs
+i) ms))))))))).(\lambda (ns: ((nat \to nat))).(\lambda (ms: ((nat \to
+nat))).(\lambda (H4: ((\forall (i: nat).((lt i n0) \to (eq nat (ns i) (ms
+i)))))).(H1 (\lambda (i: nat).(pr_appl (fs i) ns)) (\lambda (i: nat).(pr_appl
+(fs i) ms)) (\lambda (i: nat).(\lambda (H5: (lt i m)).(H3 i H5 ns ms
+H4))))))))))))))) (\lambda (n0: nat).(\lambda (f: pr).(\lambda (g:
+pr).(\lambda (H0: (lt O n0)).(\lambda (_: (pr_arity f (pred n0))).(\lambda
+(H2: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
+nat))).(((\forall (i: nat).((lt i (pred n0)) \to (eq nat (ns i) (ms i)))))
+\to (eq nat (pr_appl f ns) (pr_appl f ms))))))).(\lambda (_: (pr_arity g (S
+n0))).(\lambda (H4: ((\forall (ns: ((nat \to nat))).(\forall (ms: ((nat \to
+nat))).(((\forall (i: nat).((lt i (S n0)) \to (eq nat (ns i) (ms i))))) \to
+(eq nat (pr_appl g ns) (pr_appl g ms))))))).(\lambda (ns: ((nat \to
+nat))).(\lambda (ms: ((nat \to nat))).(\lambda (H5: ((\forall (i: nat).((lt i
+n0) \to (eq nat (ns i) (ms i)))))).(eq_ind nat (ns O) (\lambda (n1: nat).(eq
+nat (prec_appl (pr_appl f) (pr_appl g) (ns O) (\lambda (i: nat).(ns (S i))))
+(prec_appl (pr_appl f) (pr_appl g) n1 (\lambda (i: nat).(ms (S i)))))) (let
+n1 \def (ns O) in (nat_ind (\lambda (n2: nat).(eq nat (prec_appl (pr_appl f)
+(pr_appl g) n2 (\lambda (i: nat).(ns (S i)))) (prec_appl (pr_appl f) (pr_appl
+g) n2 (\lambda (i: nat).(ms (S i)))))) (H2 (\lambda (i: nat).(ns (S i)))
+(\lambda (i: nat).(ms (S i))) (\lambda (i: nat).(\lambda (H6: (lt i (pred
+n0))).(H5 (S i) (lt_x_pred_y i n0 H6))))) (\lambda (n2: nat).(\lambda (IHn0:
+(eq nat (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ns (S i))))
+(prec_appl (pr_appl f) (pr_appl g) n2 (\lambda (i: nat).(ms (S i)))))).(H4
+(\lambda (i: nat).(match i with [O \Rightarrow (prec_appl (pr_appl f)
+(pr_appl g) n2 (\lambda (i0: nat).(ns (S i0)))) | (S n3) \Rightarrow (match
+n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))])])) (\lambda (i:
+nat).(match i with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
+(\lambda (i0: nat).(ms (S i0)))) | (S n3) \Rightarrow (match n3 with [O
+\Rightarrow n2 | (S j) \Rightarrow (ms (S j))])])) (\lambda (i: nat).(\lambda
+(H6: (lt i (S n0))).(nat_ind (\lambda (n3: nat).((lt n3 (S n0)) \to (eq nat
+(match n3 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2 (\lambda
+(i0: nat).(ns (S i0)))) | (S n4) \Rightarrow (match n4 with [O \Rightarrow n2
+| (S j) \Rightarrow (ns (S j))])]) (match n3 with [O \Rightarrow (prec_appl
+(pr_appl f) (pr_appl g) n2 (\lambda (i0: nat).(ms (S i0)))) | (S n4)
+\Rightarrow (match n4 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S
+j))])])))) (\lambda (_: (lt O (S n0))).IHn0) (\lambda (i0: nat).(\lambda (_:
+(((lt i0 (S n0)) \to (eq nat (match i0 with [O \Rightarrow (prec_appl
+(pr_appl f) (pr_appl g) n2 (\lambda (i1: nat).(ns (S i1)))) | (S n3)
+\Rightarrow (match n3 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S
+j))])]) (match i0 with [O \Rightarrow (prec_appl (pr_appl f) (pr_appl g) n2
+(\lambda (i1: nat).(ms (S i1)))) | (S n3) \Rightarrow (match n3 with [O
+\Rightarrow n2 | (S j) \Rightarrow (ms (S j))])]))))).(\lambda (H7: (lt (S
+i0) (S n0))).(let H_y \def (H5 i0 (lt_S_n i0 n0 H7)) in (nat_ind (\lambda
+(n3: nat).((eq nat (ns n3) (ms n3)) \to (eq nat (match n3 with [O \Rightarrow
+n2 | (S j) \Rightarrow (ns (S j))]) (match n3 with [O \Rightarrow n2 | (S j)
+\Rightarrow (ms (S j))])))) (\lambda (_: (eq nat (ns O) (ms O))).(refl_equal
+nat n2)) (\lambda (i1: nat).(\lambda (_: (((eq nat (ns i1) (ms i1)) \to (eq
+nat (match i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ns (S j))]) (match
+i1 with [O \Rightarrow n2 | (S j) \Rightarrow (ms (S j))]))))).(\lambda (H8:
+(eq nat (ns (S i1)) (ms (S i1)))).H8))) i0 H_y))))) i H6)))))) n1)) (ms O)
+(H5 O H0))))))))))))) h n H))).
+
+theorem pr_arity_comp_proj_zero:
+ \forall (n: nat).(pr_arity (pr_comp pr_proj pr_zero) n)
+\def
+ \lambda (n: nat).(pr_arity_comp n n pr_proj pr_zero (pr_arity_zero n)
+(\lambda (i: nat).(\lambda (H: (lt i n)).(pr_arity_proj n i H)))).
+
+*)
inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))).
+inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
+| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))).
+
inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
| or3_intro0: P0 \to (or3 P0 P1 P2)
| or3_intro1: P1 \to (or3 P0 P1 P2)
include "preamble.ma".
-(* object blt not inlined *)
+(* object blt not inlined *)
inline procedural "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/le_Sx_x.con".
+inline procedural "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/le_n_pred.con".
+
inline procedural "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/minus_le.con".
inline procedural
inline procedural
"cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/le_trans_plus_r.con".
+inline procedural "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/lt_x_O.con".
+
inline procedural "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/le_gen_S.con".
inline procedural
inline procedural "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/le_S_minus.con".
+inline procedural "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith/lt_x_pred_y.con".
+
include "preamble.ma".
+
(* object PList not inlined *)
(* object papp not inlined *)
-
include "../Base-1/definitions.ma".
-alias id "le_ind" = "cic:/Coq/Init/Peano/le_ind.con".
-
default "equality"
cic:/Coq/Init/Logic/eq.ind
cic:/matita/LAMBDA-TYPES/Base-1/preamble/sym_eq.con
include "preamble.ma".
+
(* object and3 not inlined *)
+(* object and4 not inlined *)
+
+
(* object or3 not inlined *)
(* object ex6_7 not inlined *)
-
(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans
g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
+theorem arity_repellent:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1:
+A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c
+(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P:
+Prop).P)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1:
+A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2:
+A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g
+a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst)
+w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind
+A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
+A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4:
+A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g
+x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def
+(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y
+(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind
+Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))).
+
theorem arity_appls_cast:
\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs:
TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to
include "nf2/defs.ma".
+include "ex2/defs.ma".
+
include "csubc/defs.ma".
include "pc1/defs.ma".
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex2/defs".
+
+include "C/defs.ma".
+
+definition ex2_c:
+ C
+\def
+ CSort O.
+
+definition ex2_t:
+ T
+\def
+ THead (Flat Appl) (TSort O) (TSort O).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ex2/props".
+
+include "ex2/defs.ma".
+
+include "nf2/defs.ma".
+
+include "pr2/fwd.ma".
+
+include "arity/fwd.ma".
+
+theorem ex2_nf2:
+ nf2 ex2_c ex2_t
+\def
+ \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O)
+(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2
+H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
+O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
+(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
+O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat
+Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O)
+(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2
+(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O)
+x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1
+(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O)
+(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t:
+T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort
+(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O))
+(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal
+T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1:
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
+(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T
+(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead
+(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2
+x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def
+(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O)
+(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O)
+x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7
+\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in
+(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr)
+(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort
+O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O)
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O)
+(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2)
+(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
+(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq
+T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0)
+x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O)
+(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead
+(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in
+(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O
+H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
+(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O))
+t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1
+x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2
+H8))))))))))))))) H1)) H0))).
+
+theorem ex2_arity:
+ \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat
+Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def
+(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda
+(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O)
+(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O)
+(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let
+H3 \def (match (arity_gen_sort g (CSort O) O (AHead x a) H2) in leq return
+(\lambda (a0: A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A a0
+(AHead x a)) \to ((eq A a1 (ASort O O)) \to P))))) with [(leq_sort h1 h2 n1
+n2 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1) (AHead x a))).(\lambda
+(H5: (eq A (ASort h2 n2) (ASort O O))).((let H6 \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 x a) H4) in
+(False_ind ((eq A (ASort h2 n2) (ASort O O)) \to ((eq A (aplus g (ASort h1
+n1) k) (aplus g (ASort h2 n2) k)) \to P)) H6)) H5 H3))) | (leq_head a1 a2 H3
+a3 a4 H4) \Rightarrow (\lambda (H5: (eq A (AHead a1 a3) (AHead x
+a))).(\lambda (H6: (eq A (AHead a2 a4) (ASort O O))).((let H7 \def (f_equal A
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a3 | (AHead _ a0) \Rightarrow a0])) (AHead a1 a3) (AHead x a) H5)
+in ((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0]))
+(AHead a1 a3) (AHead x a) H5) in (eq_ind A x (\lambda (a0: A).((eq A a3 a)
+\to ((eq A (AHead a2 a4) (ASort O O)) \to ((leq g a0 a2) \to ((leq g a3 a4)
+\to P))))) (\lambda (H9: (eq A a3 a)).(eq_ind A a (\lambda (a0: A).((eq A
+(AHead a2 a4) (ASort O O)) \to ((leq g x a2) \to ((leq g a0 a4) \to P))))
+(\lambda (H10: (eq A (AHead a2 a4) (ASort O O))).(let H11 \def (eq_ind A
+(AHead a2 a4) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O
+O) H10) in (False_ind ((leq g x a2) \to ((leq g a a4) \to P)) H11))) a3
+(sym_eq A a3 a H9))) a1 (sym_eq A a1 x H8))) H7)) H6 H3 H4)))]) in (H3
+(refl_equal A (AHead x a)) (refl_equal A (ASort O O))))))) H0))))).
+
((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 (asucc g a0))).(eq_ind A
(asucc g a0) (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7) \to
P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g
-a0))).(leq_ahead_false g a a0 H13 P))) a6 (sym_eq A a6 (asucc g a0) H12))) a4
-(sym_eq A a4 a H11))) H10))) a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3 (AHead
-a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2))
-(refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
+a0))).(leq_ahead_false_1 g a a0 H13 P))) a6 (sym_eq A a6 (asucc g a0) H12)))
+a4 (sym_eq A a4 a H11))) H10))) a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3
+(AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a
+a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
theorem leq_asucc_false:
\forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A
a0))))))))))))) a1 a2 H)))).
-theorem leq_ahead_false:
+theorem leq_ahead_false_1:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
\to (\forall (P: Prop).P))))
\def
H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a
a0))))))))))) a1)).
+theorem leq_ahead_false_2:
+ \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
+A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
+n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
+a1 (ASort O n0)) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda
+(a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead a1 (ASort
+O n0))) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
+H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead a1 (ASort O
+n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O n0))).((let H4 \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 (ASort O n0)) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O n0))
+\to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H4))
+H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
+(AHead a0 a4) (AHead a1 (ASort O n0)))).(\lambda (H4: (eq A (AHead a3 a5)
+(ASort O n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
+\Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort O n0)) H3) in ((let H6 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
+(AHead a1 (ASort O n0)) H3) in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort
+O n0)) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g
+a4 a5) \to P))))) (\lambda (H7: (eq A a4 (ASort O n0))).(eq_ind A (ASort O
+n0) (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a1 a3) \to
+((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O n0))).(let
+H9 \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 n0) H8) in (False_ind ((leq g a1 a3) \to ((leq g (ASort O
+n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort O n0) H7))) a0 (sym_eq A a0 a1
+H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead a1 (ASort O n0)))
+(refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
+a1 (ASort (S n1) n0)) (ASort (S n1) n0))).(let H1 \def (match H0 in leq
+return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
+(AHead a1 (ASort (S n1) n0))) \to ((eq A a0 (ASort (S n1) n0)) \to P)))))
+with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1
+n2) (AHead a1 (ASort (S n1) n0)))).(\lambda (H3: (eq A (ASort h2 n3) (ASort
+(S n1) n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e
+in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
+_ _) \Rightarrow False])) I (AHead a1 (ASort (S n1) n0)) H2) in (False_ind
+((eq A (ASort h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k)
+(aplus g (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
+H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead a1 (ASort (S n1)
+n0)))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
+(AHead a1 (ASort (S n1) n0)) H3) in ((let H6 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
+(AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort (S n1) n0)) H3)
+in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort (S n1) n0)) \to ((eq A
+(AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to
+P))))) (\lambda (H7: (eq A a4 (ASort (S n1) n0))).(eq_ind A (ASort (S n1) n0)
+(\lambda (a: A).((eq A (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a1 a3)
+\to ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1)
+n0))).(let H9 \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 (S n1) n0) H8) in (False_ind ((leq g a1 a3)
+\to ((leq g (ASort (S n1) n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort (S n1)
+n0) H7))) a0 (sym_eq A a0 a1 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A
+(AHead a1 (ASort (S n1) n0))) (refl_equal A (ASort (S n1) n0))))))) n H))))))
+(\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) \to
+(\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1:
+A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1:
+A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P:
+Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
+A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a1 (AHead a a0))) \to ((eq A
+a4 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
+(\lambda (H3: (eq A (ASort h1 n1) (AHead a1 (AHead a a0)))).(\lambda (H4: (eq
+A (ASort h2 n2) (AHead a a0))).((let H5 \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 (AHead a a0))
+H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g
+(ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head
+a3 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead a1
+(AHead a a0)))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3
+a5) (AHead a1 (AHead a a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 |
+(AHead a7 _) \Rightarrow a7])) (AHead a3 a5) (AHead a1 (AHead a a0)) H4) in
+(eq_ind A a1 (\lambda (a7: A).((eq A a5 (AHead a a0)) \to ((eq A (AHead a4
+a6) (AHead a a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda
+(H8: (eq A a5 (AHead a a0))).(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A
+(AHead a4 a6) (AHead a a0)) \to ((leq g a1 a4) \to ((leq g a7 a6) \to P))))
+(\lambda (H9: (eq A (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A
+(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0)
+H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
+\Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7:
+A).((eq A a6 a0) \to ((leq g a1 a7) \to ((leq g (AHead a a0) a6) \to P))))
+(\lambda (H12: (eq A a6 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g a1 a) \to
+((leq g (AHead a a0) a7) \to P))) (\lambda (_: (leq g a1 a)).(\lambda (H14:
+(leq g (AHead a a0) a0)).(H0 a H14 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq
+A a4 a H11))) H10))) a5 (sym_eq A a5 (AHead a a0) H8))) a3 (sym_eq A a3 a1
+H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a1 (AHead a a0)))
+(refl_equal A (AHead a a0))))))))))) a2)).
+
H=@
RT_BASEDIR=../../../
-OPTIONS=-bench
+OPTIONS=-bench -onepass
MMAKE=$(RT_BASEDIR)matitamake $(OPTIONS)
CLEAN=$(RT_BASEDIR)matitaclean $(OPTIONS)
MMAKEO=$(RT_BASEDIR)matitamake.opt $(OPTIONS)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/nf2/arity".
+
+include "nf2/fwd.ma".
+
+include "arity/subst0.ma".
+
+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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort
+n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i0:
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0:
+nat).(nf2 d (TLRef i0))))))))).(\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_2
+TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads
+(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
+ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
+i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (_: (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_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u
+(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef
+i0))))))))).(\lambda (H3: (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_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
+(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
+i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
+(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
+i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))))).(\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2
+(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead
+c0 (Bind b) u) (TLRef i))))))))).(\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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
+(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\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_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
+(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))) 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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
+T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) 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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\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_2 TList
+nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind
+Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind
+Abst) u) (TLRef i))))))))).(\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
+T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))))).(\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\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_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
+ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
+T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat
+Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) 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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat
+Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))))) 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_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
+(TSort x)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))))) 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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat
+Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))))))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\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_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x))
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))))) 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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0:
+TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0
+(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef
+x1))).(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_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
+t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))) (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_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
+(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads
+(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda
+(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u
+(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11)
+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_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\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_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))))))))))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))))).(\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i:
+nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i)))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) (\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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
+Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c0 (TLRef i))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) (\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_2 TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x
+(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
+ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda
+(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0
+(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef
+x1))).(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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i))))))) (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_2 TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
+(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
+x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))) x0 x1 (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).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1
t2)))).
+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.
+
include "pr2/clen.ma".
+include "subst0/dec.ma".
+
include "T/props.ma".
theorem nf2_gen_lref:
(pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u)))
P))))).
+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_flat:
\forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
+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)))))
+\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
+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))))))).
+
+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))))
+\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))))).
+
x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3))))))
H2)))))))))).
+theorem nf2_abst_shift:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c
+(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
+\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2
+(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda
+(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2
+H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2
+c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
+b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T
+(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst)
+u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2
+H3)))))) H2)))))))).
+
theorem nf2_appl_lref:
\forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c
(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1
x2 x0 h d H11)))))))) H7))))) 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)))))
+\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))))))).
+
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)))))
-\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
-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))))))).
-
-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))))
-\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))))).
-
-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))))))))))))
-\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))))).
-
-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)))))
-\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))))))).
-
-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
(\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)).
+in (nf2_abst_shift c u (H1 c) t0 H_y))))))))) t H)).
theorem sort_pc3:
\forall (t1: T).((sort t1) \to (\forall (t2: T).((sort t2) \to (\forall (c:
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))).
-definition pchurch_context:
- T \to (T \to T)
-\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))).
-
-definition pnat:
- T \to T
-\def
- \lambda (t: T).(pchurch_context t (lift (S (S O)) O t)).
-
-definition church_body:
- nat \to 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.
-
-definition pchurch:
- T \to (nat \to T)
-\def
- \lambda (t: T).(\lambda (n: nat).(pchurch_context t (church_body n))).
-
-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)))
-\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))).
-
-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)))
-\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))).
-
-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)))))))
-\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))
-(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 (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
-t) (lift (S O) O (lift (S O) O t)))) (TLRef O) t0)) (ty3_abst g O (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)))) (CHead c (Bind Abst) t) (THead (Bind Abst) (lift
-(S O) O t) (lift (S O) O (lift (S O) O t))) (getl_refl Abst (CHead c (Bind
-Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O
-t)))) x1 H9) (THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O)
-(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
-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)))) (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)))
-(lift (S O) O (lift (S O) O (lift (S O) O t))) (pr0_refl (lift (S O) O (lift
-(S O) O (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
-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)) (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
-(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)))))))))) H5)))))) H2))))) n)))) (ty3_correct g c t u H)))))).
-
include "pr3/wcpr0.ma".
+include "ex2/props.ma".
+
include "ex1/props.ma".
include "ty3/tau0.ma".
+include "ty3/nf2.ma".
+
include "ty3/dec.ma".
set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/arity".
-include "ty3/defs.ma".
+include "ty3/pr3_props.ma".
include "arity/pr3.ma".
c v (asucc g (AHead x3 x4)) H13 (asucc g x3) H11) P))))))) H9)))))
H6))))))))))) (ty3_gen_bind g Abst c v t u H1)))))))))).
+theorem ty3_repellent:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1:
+T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead
+c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P:
+Prop).P)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1:
+T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2:
+T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O
+u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0:
+T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x:
+T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3
+\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop
+(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3
+(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2
+t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O)
+O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5
+(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind
+Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda
+(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g
+(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1:
+A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity
+g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def
+(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in
+(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda
+(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity
+g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g
+x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g
+c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1)
+(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2)
+H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w)
+t (lift (S O) O u2) H0))))))))))).
+
theorem ty3_acyclic:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2".
+
+include "ty3/arity.ma".
+
+include "nf2/arity.ma".
+
+theorem ty3_nf2_inv_all:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
+t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat
+(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H)
+in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
+(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w)
+u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2:
+(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g
+c t x H2 H0)))) H1)))))))).
+
+theorem ty3_nf2_inv_sort:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t
+(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort
+n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda
+(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def
+(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (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_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))
+(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
+m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
+i)))))) (\lambda (H2: (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))))).(ex3_2_ind 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))) (or (ex2 nat (\lambda
+(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0
+x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0)
+x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H
+(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
+(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex4_3_ind T T T (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2)
+(TSort m))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c x0
+t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind
+Abst) x0) x1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g
+(CHead c (Bind Abst) x0) t2 t1)))) (or (ex2 nat (\lambda (n: nat).(eq T
+(THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
+(Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i:
+nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H7: (pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_:
+(ty3 g c x0 x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1
+x2)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x2 x4)).(pc3_gen_sort_abst
+c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat
+(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n:
+nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))))) (ty3_gen_bind g Abst c
+x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n:
+nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n)))
+(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat
+m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
+i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def
+(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in
+(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T
+t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x)
+(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort
+n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda
+(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x)
+(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
+(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
+(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef
+i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda
+(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x))
+(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g
+c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda
+(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n:
+nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0:
+TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef
+x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat
+Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1))
+(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda
+(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda
+(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n:
+nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
+(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda
+(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
+x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))
+x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3)))))))
+H2)) H1)))))))).
+
include "ty3/fwd.ma".
+include "pc3/fwd.ma".
+
theorem ty3_lift:
\forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e
t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c
(pc3 c0 t2 t4)).(\lambda (_: (ty3 g c0 t0 t2)).H5)) (ty3_gen_cast g c0 t0 t2
t4 H4)))))))))))) c u 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)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/LOGIC/PNF/defs".
+
+(* NORMAL FORM PREDICATE FOR PARALLEL REDUCTION
+*)
+
+include "PRed/defs.ma".
+
+inductive PNF: Proof \to Prop \def
+ | pnf: \forall p. (\forall q. p => q \to p = q) \to PNF p
+.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/LOGIC/PRed/defs".
+
+(* SINGLE STEP PARALLEL REDUCTION
+ For cut elimination
+*)
+
+include "datatypes/Proof.ma".
+
+inductive PRed: Proof \to Proof \to Prop \def
+ | pred_lref: \forall i. PRed (lref i) (lref i)
+ | pred_parx: \forall h. PRed (parx h) (parx h)
+ | pred_impw: \forall p1,p2. PRed p1 p2 \to PRed (impw p1) (impw p2)
+ | pred_impr: \forall p1,p2. PRed p1 p2 \to PRed (impr p1) (impr p2)
+ | pred_impi: \forall p1,p2. PRed p1 p2 \to \forall q1,q2. PRed q1 q2 \to
+ \forall r1,r2. PRed r1 r2 \to
+ PRed (impi p1 q1 r1) (impi p2 q2 r2)
+ | pred_scut: \forall p1,p2. PRed p1 p2 \to \forall q1,q2. PRed q1 q2 \to
+ PRed (scut p1 q1) (scut p2 q2)
+.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation
+ "single step parallel reduction in B->"
+ 'parred x y = (cic:/matita/LOGIC/PRed/defs/PRed.ind#xpointer(1/1) x y)
+.
notation "hvbox(\lnot a)"
non associative with precedence 40
for @{ 'not $a }.
+
+notation "hvbox(a break => b)"
+ non associative with precedence 45
+for @{ 'parred $a $b }.
(* $Id$ *)
-open Printf
-
type thingsToInitilaize =
ConfigurationFile | Db | Environment | Getter | Makelib | CmdLine | Registry
List.iter (fun key, value -> Helm_registry.set ~key ~value)
let fill_registry init_status =
+ wants [ ConfigurationFile ] init_status;
if not (already_configured [ Registry ] init_status) then begin
set_registry_values registry_defaults;
Registry :: init_status
init_status
let load_configuration init_status =
- wants [ Registry ] init_status;
if not (already_configured [ConfigurationFile] init_status) then
begin
Helm_registry.load_from !conffile;
List.iter
(fun (name, s) -> Hashtbl.replace usages name s)
[ "matitac",
- sprintf "MatitaC v%s
+ Printf.sprintf "MatitaC v%s
Usage: matitac [ OPTION ... ] FILE
Options:"
BuildTimeConf.version;
"gragrep",
- sprintf "Grafite Grep v%s
+ Printf.sprintf "Grafite Grep v%s
Usage: gragrep [ -r ] PATH
Options:"
BuildTimeConf.version;
"matitaprover",
- sprintf "Matita's prover v%s
+ Printf.sprintf "Matita's prover v%s
Usage: matitaprover [ -tptppath ] FILE.p
Options:"
BuildTimeConf.version;
"matita",
- sprintf "Matita v%s
+ Printf.sprintf "Matita v%s
Usage: matita [ OPTION ... ] [ FILE ... ]
Options:"
BuildTimeConf.version;
"cicbrowser",
- sprintf
+ Printf.sprintf
"CIC Browser v%s
Usage: cicbrowser [ URL | WHELP QUERY ]
Options:"
BuildTimeConf.version;
"matitadep",
- sprintf "MatitaDep v%s
+ Printf.sprintf "MatitaDep v%s
Usage: matitadep [ OPTION ... ] FILE ...
Options:"
BuildTimeConf.version;
"matitaclean",
- sprintf "MatitaClean v%s
+ Printf.sprintf "MatitaClean v%s
Usage: matitaclean all
matitaclean [ (FILE | URI) ... ]
Options:"
BuildTimeConf.version;
"matitamake",
- sprintf "MatitaMake v%s
+ Printf.sprintf "MatitaMake v%s
Usage: matitamake [ OPTION ... ] (init | clean | list | destroy | build)
init
Parameters: name (the name of the development, required)
BuildTimeConf.version;
]
let default_usage =
- sprintf "Matita v%s\nUsage: matita [ ARG ]\nOptions:" BuildTimeConf.version
+ Printf.sprintf
+ "Matita v%s\nUsage: matita [ ARG ]\nOptions:" BuildTimeConf.version
let usage () =
let basename = Filename.basename Sys.argv.(0) in
Arg.Unit (fun () -> Helm_registry.set_bool "matita.debug" true),
("Do not catch top-level exception "
^ "(useful for backtrace inspection)");
+ "-onepass",
+ Arg.Unit (fun () -> GrafiteDisambiguator.only_one_pass := true),
+ "Enable only one disambiguation pass";
]
else []
in
print_endline (usage ());
exit 1
+let conf_components =
+ [ parse_cmdline; load_configuration; fill_registry ]
+
+let other_components =
+ [ initialize_makelib; initialize_db; initialize_environment ]
+
let initialize_all () =
status :=
List.fold_left (fun s f -> f s) !status
- [ fill_registry; parse_cmdline; load_configuration; initialize_makelib;
- initialize_db; initialize_environment ]
+ (conf_components @ other_components)
(* initialize_notation
(initialize_environment
(initialize_db
(load_configuration
(parse_cmdline !status))))) *)
-let load_configuration_file () =
- status := load_configuration !status
-
-let parse_cmdline () =
- status := parse_cmdline !status
+let parse_cmdline_and_configuration_file () =
+ status := List.fold_left (fun s f -> f s) !status conf_components
-let fill_registry () =
- status := fill_registry !status
;;
Inversion_principle.init ()
val initialize_all: unit -> unit
(** {2 per-components initialization} *)
-val fill_registry: unit -> unit (** fill registry with default values *)
-val parse_cmdline: unit -> unit (** parse cmdline setting registry keys *)
-val load_configuration_file: unit -> unit
+val parse_cmdline_and_configuration_file: unit -> unit
(** {2 Utilities} *)
let resolve alias current_buri =
let buri = buri alias in
if buri <> current_buri then Some buri else None in
- MatitaInit.fill_registry ();
let dot_file = ref "" in
MatitaInit.add_cmdline_spec
["-dot", Arg.Set_string dot_file,
"<file> Save dependency graph in dot format to the given file"];
- MatitaInit.parse_cmdline ();
- MatitaInit.load_configuration_file ();
+ MatitaInit.parse_cmdline_and_configuration_file ();
let include_paths =
Helm_registry.get_list Helm_registry.string "matita.includes" in
let args = Helm_registry.get_list Helm_registry.string "matita.args" in
module MK = MatitamakeLib ;;
let main () =
- MatitaInit.fill_registry ();
- MatitaInit.parse_cmdline ();
- MatitaInit.load_configuration_file ();
+ MatitaInit.parse_cmdline_and_configuration_file ();
MK.initialize ();
let usage = ref (fun () -> ()) in
let dev_of_name name =
;;
let main () =
- MatitaInit.fill_registry ();
let tptppath = ref "./" in
let timeout = ref 600 in
MatitaInit.add_cmdline_spec
"Where to find the Axioms/ and Problems/ directory";
"-timeout", Arg.Int (fun x -> timeout := x),
"Timeout in seconds"];
- MatitaInit.parse_cmdline ();
- MatitaInit.load_configuration_file ();
+ MatitaInit.parse_cmdline_and_configuration_file ();
Helm_registry.set_bool "matita.nodisk" true;
HLog.set_log_callback (fun _ _ -> ());
let args = Helm_registry.get_list Helm_registry.string "matita.args" in
projects / helm.git / commitdiff