ok up to pr3

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props.ma
index 7ec9240f425ce299434118e8227881e77a273c55..386f49f8de58c1471c4392b77d53546c9cb3522d 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props.ma
@@ -18,6 +18,8 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props".
 
 include "leq/defs.ma".
 
+include "aplus/props.ma".
+
 theorem ahead_inj_snd:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall 
 (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
@@ -86,10 +88,82 @@ k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
 A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 
 H1 a6 a5 H3))))))))) a1 a2 H)))).
 
-axiom leq_trans:
+theorem leq_trans:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
 (a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
-.
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 
+a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort 
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g 
+(ASort h2 n2) a3)).(let H2 \def (match H1 in leq return (\lambda (a: 
+A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to 
+((eq A a0 a3) \to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3 
+k0 H2) \Rightarrow (\lambda (H3: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda 
+(H4: (eq A (ASort h3 n3) a3)).((let H5 \def (f_equal A nat (\lambda (e: 
+A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n 
+| (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H3) in ((let H6 
+\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) 
+with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) 
+(ASort h2 n2) H3) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to 
+((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort 
+h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H7: (eq nat n0 
+n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A 
+(aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 
+n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) 
+(\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0)) 
+\to (leq g (ASort h1 n1) a))) (\lambda (H9: (eq A (aplus g (ASort h2 n2) k0) 
+(aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3 
+n3)) (\lambda (H10: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) 
+(ASort h2 n2) k k H0 (minus k0 k)) in (let H11 \def (eq_ind_r nat (plus 
+(minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g 
+(ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0 
+H10)))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0) 
+(aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11 H9))))) (\lambda 
+(H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0 
+k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus (minus k k0) k0) 
+(\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g (ASort h3 n3) n))) 
+H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A 
+(aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k) 
+H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0 (sym_eq nat h0 h2 H6))) 
+H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A 
+(AHead a0 a5) (ASort h2 n2))).(\lambda (H5: (eq A (AHead a4 a6) a3)).((let H6 
+\def (eq_ind A (AHead a0 a5) (\lambda (e: A).(match e in A return (\lambda 
+(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
+True])) I (ASort h2 n2) H4) in (False_ind ((eq A (AHead a4 a6) a3) \to ((leq 
+g a0 a4) \to ((leq g a5 a6) \to (leq g (ASort h1 n1) a3)))) H6)) H5 H2 
+H3)))]) in (H2 (refl_equal A (ASort h2 n2)) (refl_equal A a3))))))))))) 
+(\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: 
+((\forall (a5: A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: 
+A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: 
+A).((leq g a6 a7) \to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g 
+(AHead a4 a6) a0)).(let H5 \def (match H4 in leq return (\lambda (a: 
+A).(\lambda (a7: A).(\lambda (_: (leq ? a a7)).((eq A a (AHead a4 a6)) \to 
+((eq A a7 a0) \to (leq g (AHead a3 a5) a0)))))) with [(leq_sort h1 h2 n1 n2 k 
+H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n1) (AHead a4 a6))).(\lambda 
+(H7: (eq A (ASort h2 n2) a0)).((let H8 \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 a4 a6) H6) in 
+(False_ind ((eq A (ASort h2 n2) a0) \to ((eq A (aplus g (ASort h1 n1) k) 
+(aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 a5) a0))) H8)) H7 H5))) | 
+(leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda (H7: (eq A (AHead a7 a9) 
+(AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10) a0)).((let H9 \def 
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
+[(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a])) (AHead a7 a9) 
+(AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e: A).(match e in 
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _) 
+\Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4 (\lambda (a: 
+A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8) \to ((leq g 
+a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9 
+a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a8 a10) a0) \to ((leq g a4 
+a8) \to ((leq g a a10) \to (leq g (AHead a3 a5) a0))))) (\lambda (H12: (eq A 
+(AHead a8 a10) a0)).(eq_ind A (AHead a8 a10) (\lambda (a: A).((leq g a4 a8) 
+\to ((leq g a6 a10) \to (leq g (AHead a3 a5) a)))) (\lambda (H13: (leq g a4 
+a8)).(\lambda (H14: (leq g a6 a10)).(leq_head g a3 a8 (H1 a8 H13) a5 a10 (H3 
+a10 H14)))) a0 H12)) a9 (sym_eq A a9 a6 H11))) a7 (sym_eq A a7 a4 H10))) H9)) 
+H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A 
+a0))))))))))))) a1 a2 H)))).
 
 theorem leq_ahead_false:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) 

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/fwd.ma
new file mode 100644 (file)
index 0000000..6c0e84a
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/fwd.ma
@@ -0,0 +1,1471 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/pr3/fwd".
+
+include "pr3/props.ma".
+
+include "pr2/fwd.ma".
+
+theorem pr3_gen_sort:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to 
+(eq T x (TSort n)))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort 
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (eq T x (TSort n)) 
+(\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort n))))) (\lambda 
+(t: T).(\lambda (H1: (eq T t (TSort n))).H1)) (\lambda (t2: T).(\lambda (t1: 
+T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 
+t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 (TSort n))))).(\lambda 
+(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t 
+t2)) H1 (TSort n) H4) in (H3 (pr2_gen_sort c t2 n H5)))))))))) y x H0))) 
+H)))).
+
+theorem pr3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c 
+y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Abst) u1 t)) \to 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead 
+(Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x0 t2))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: 
+T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind Abst) x0 x1)) \to 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t2))))))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H1: (eq T t (THead (Bind Abst) x0 x1))).(ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) x0 x1 H1 
+(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) 
+u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 
+t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) \to (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 
+t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead 
+(Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) 
+H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 t2 H5) 
+in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t2 (THead 
+(Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 x2)).(\lambda (H9: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H10 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Abst) x4 x5)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x5 t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 
+\def (H10 x2 x3 (refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead 
+(Bind Abst) x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) 
+x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: 
+T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) 
+H11)))))))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_cast:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c 
+t1 x))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c 
+t1 x)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: 
+T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t 
+t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y 
+(THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2)))) (pr3 c x0 x))))) 
+(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: 
+T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 
+t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 x1))).(eq_ind_r T (THead (Flat 
+Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c 
+x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead 
+(Flat Cast) x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c 
+x1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 
+t2))) x0 x1 (refl_equal T (THead (Flat Cast) x0 x1)) (pr3_refl c x0) 
+(pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: 
+(pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: 
+((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat Cast) x0 x1)) \to 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4))))))).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Flat Cast) x0 x1))).(let 
+H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 
+x1) H4) in (let H6 \def (pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr2 c x1 t5)))) (pr2 c x1 t2) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T 
+t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: 
+(pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: 
+T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) x2 x3) H8) 
+in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 x3))) in 
+(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T 
+t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: 
+(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat 
+Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead 
+(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 
+(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) 
+(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 
+t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c 
+x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 
+H2))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
+(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
+(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 
+t2))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) 
+(\lambda (t: T).(pr3 c t x)) (\forall (e: C).((drop h d c e) \to (ex2 T 
+(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 t2))))) 
+(\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq 
+T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda 
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) (pr3_ind 
+c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h d x0)) 
+\to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T t0 
+(lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: 
+T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: 
+C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h 
+d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 
+(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: 
+T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda 
+(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: 
+(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 
+(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in 
+(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e 
+x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: 
+T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d 
+x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift 
+h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 
+(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda 
+(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T 
+(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 
+H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x 
+H0)))) H)))))).
+
+theorem pr3_gen_lref:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to 
+(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T x (lift (S n) O v))))))))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef 
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (or (eq T x (TLRef 
+n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c 
+(CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: 
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T x 
+(lift (S n) O v))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c 
+(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 (TLRef 
+n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c 
+(CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: 
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t0 
+(lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (H1: (eq T t (TLRef 
+n))).(or_introl (eq T t (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) H1))) (\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda 
+(H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or (eq T t3 
+(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: 
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 
+(lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef n))).(let H5 \def 
+(eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) in (let H6 \def 
+(pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef n)) (ex2_2 C T (\lambda 
+(d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T t3 (TLRef n)) 
+(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead 
+d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
+v))))))) (\lambda (H7: (eq T t2 (TLRef n))).(H3 H7)) (\lambda (H7: (ex2_2 C T 
+(\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda 
+(_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda 
+(d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) 
+(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead 
+d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
+v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 
+(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def 
+(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let 
+H11 \def (pr3_gen_lift c x1 t3 (S n) O H10 x0 (getl_drop Abbr c x0 x1 n H8)) 
+in (ex2_ind T (\lambda (t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: 
+T).(pr3 x0 x1 t4)) (or (eq T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) 
+(\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))))) (\lambda 
+(x2: T).(\lambda (H12: (eq T t3 (lift (S n) O x2))).(\lambda (H13: (pr3 x0 x1 
+x2)).(or_intror (eq T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v)))))) (ex3_3_intro C T T 
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind 
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) 
+x0 x1 x2 H8 H13 H12))))) H11))))))) H7)) H6)))))))))) y x H0))) H)))).
+
+theorem pr3_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 
+(\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t2)))))) 
+(pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 (\lambda 
+(t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2)))))) (pr3 
+(CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: 
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind 
+Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O 
+t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: 
+(eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind Void) x0 x1) 
+(\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O 
+t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead 
+(Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
+(lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr3_refl c x0) 
+(\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) u) x1))))) t 
+H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 
+t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let 
+H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 
+x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) 
+O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind 
+Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O 
+t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 
+(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c 
+(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 
+x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 
+(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead 
+c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: 
+(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind 
+Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift 
+(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) 
+(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S 
+O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) 
+O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))) (\lambda 
+(y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y 
+(THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t 
+t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) O x))))) (unintro T u1 
+(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Abbr) t x0)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 (CHead c 
+(Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: T).(\lambda 
+(t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind Abbr) x0 x1)) 
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind 
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 
+x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 
+x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) 
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
+T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 
+t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 
+(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
+b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead 
+c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z 
+t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 
+c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
+(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 
+(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: 
+T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 
+(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 
+x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) 
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c 
+x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T 
+(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 
+x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 
+(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 
+H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) 
+O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: 
+T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 
+x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) 
+x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall 
+(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 
+x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 
+x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) 
+(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 
+x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 
+x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O 
+t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T 
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) 
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
+(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 
+x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) 
+x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall 
+(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 
+x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 
+x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 
+H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind 
+Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 
+(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) 
+(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 
+(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing 
+(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 
+(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) 
+H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing 
+(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) 
+(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c 
+(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y 
+(THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t t2)))) (ex4_4 T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro 
+T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) 
+x))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 c x0 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))) (pr3_ind 
+c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
+(THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) 
+(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t 
+(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda 
+(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) 
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat 
+Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
+x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda 
+(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 
+t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat 
+Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 
+t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (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 x1 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 
+x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t 
+(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 
+(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 
+x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T 
+(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 
+x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5)))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) 
+x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T 
+(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 
+x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: 
+(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda 
+(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro 
+T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: 
+T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
+t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 
+(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: 
+T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 
+c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 
+(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 
+x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) 
+x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro 
+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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) 
+x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 
+(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) 
+(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq 
+T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 
+x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 
+x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t 
+(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind 
+Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) 
+x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (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 x1 
+(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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (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 x1 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T 
+x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 
+x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) 
+x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 
+(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) 
+x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let 
+H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro 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).(pr3 c 
+(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) 
+H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
+x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso.ma
new file mode 100644 (file)
index 0000000..fabf661
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso.ma
@@ -0,0 +1,182 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/pr3/iso".
+
+include "pr3/fwd.ma".
+
+include "iso/fwd.ma".
+
+theorem pr3_iso_appls_cast:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 
+\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: 
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
+(THeads (Flat Appl) vs t) u2))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0 
+(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2)))))) 
+(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda 
+(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1 
+\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t 
+t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t 
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0 
+x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def 
+(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall 
+(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat 
+Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head (Flat Cast) v x0 t 
+x1) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: (pr3 
+c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: 
+((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) 
+\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall 
+(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2: 
+T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead 
+(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P: 
+Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead 
+(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat 
+Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 c (THeads (Flat 
+Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) 
+(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 
+(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl) 
+x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) 
+t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2: 
+T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v 
+t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in 
+(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat 
+Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head (Flat Appl) t0 x0 
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) x1) (pr3 c (THead (Flat Appl) 
+t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3)) 
+(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
+t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) 
+(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
+(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6: 
+(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0 
+x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
+u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) 
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads 
+(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6 
+(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead 
+(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast 
+Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c 
+(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) 
+(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind 
+Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c 
+t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
+(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).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) 
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (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).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat 
+Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) 
+(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda 
+(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
+Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat 
+Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) 
+c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 
+(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead 
+(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads 
+(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda 
+(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0 
+Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2 
+c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead 
+(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead 
+(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
+x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2 
+(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) 
+x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift 
+(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead 
+c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind 
+x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7) 
+(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
+(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
+O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
+(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
+(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
+x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/pr1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/pr1.ma
new file mode 100644 (file)
index 0000000..820b4fe
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/pr1.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/pr3/pr1".
+
+include "pr3/defs.ma".
+
+include "pr1/defs.ma".
+
+theorem pr3_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: 
+T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 
+t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: 
+C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/pr3.ma
new file mode 100644 (file)
index 0000000..1969c14
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/pr3.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/pr3/pr3".
+
+include "pr3/props.ma".
+
+include "pr2/pr2.ma".
+
+theorem pr3_strip:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
+(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t 
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t 
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
+t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda 
+(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 
+t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: 
+T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda 
+(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: 
+T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c 
+t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c 
+x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) 
+t0 t1 H)))).
+
+theorem pr3_confluence:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
+(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t 
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t 
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
+t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda 
+(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda 
+(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 
+t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c 
+t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 
+t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) 
+(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 
+c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 
+H)))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/props.ma
new file mode 100644 (file)
index 0000000..d4b3d64
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/props.ma
@@ -0,0 +1,379 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/pr3/props".
+
+include "pr3/defs.ma".
+
+include "pr2/props.ma".
+
+theorem clear_pr3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to 
+(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1 
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t: 
+T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: 
+T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 
+t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
+
+theorem pr3_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
+
+theorem pr3_t:
+ \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall 
+(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0 
+t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3 
+c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall 
+(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: 
+(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+theorem pr3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
+T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t: 
+T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 
+t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u 
+t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c 
+t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
+
+theorem pr3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall 
+(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda 
+(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda 
+(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c 
+(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing 
+c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) 
+(H2 k t)))))))))) u1 u2 H)))).
+
+theorem pr3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u) 
+(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0)))) 
+(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: 
+(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u 
+t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) 
+(THead k u t4) H2))))))) t1 t2 H)))))).
+
+theorem pr3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
+(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c 
+u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
+
+theorem pr3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
+(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c 
+u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pr3_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v: 
+T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f: 
+F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead 
+c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c 
+(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
+
+theorem pr3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: 
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 
+(CHead c k u2) t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 
+(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1 
+t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 
+(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 
+t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) 
+\to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3 
+(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda 
+(t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 
+t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 
+(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) 
+H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda 
+(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq 
+T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t 
+t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i 
+u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 
+t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) 
+(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3 
+(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda 
+(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
+((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i 
+(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
+(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) 
+(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1 
+t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
+u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((getl O 
+(CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t1 t2))) 
+(\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind 
+Abbr) u))).(let H15 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow 
+c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c 
+(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind 
+Abbr) u) H14))) in ((let H16 \def (f_equal C B (\lambda (e: C).(match e in C 
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) 
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
+(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H17 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d 
+(Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) 
+u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in 
+(\lambda (H18: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind 
+T u (\lambda (t4: T).(subst0 O t4 t3 t2)) H13 u2 H17) in (eq_ind B Abbr 
+(\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t4: 
+T).(subst0 O u1 t3 t4)) (\lambda (t4: T).(pr0 t4 t2)) (pr3 (CHead c (Bind 
+Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H21: (subst0 O u1 t3 x)).(\lambda 
+(H22: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) u1) x t1 (pr2_delta (CHead c 
+(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21) t2 (pr3_pr2 
+(CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c (Bind Abbr) u1) x t2 
+H22)))))) (pr0_subst0_back u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15)))) 
+(\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat f) u2) (CHead d (Bind 
+Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta 
+c d u O (getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) 
+(clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) 
+u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f u1)))) k H12))) 
+(\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind 
+Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 
+t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) 
+u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k0: K).((getl 
+(S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k0 
+u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k0 
+u1) t1 t2)))) \to (pr3 (CHead c k0 u1) t1 t2)))) (\lambda (b: B).(\lambda 
+(H14: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda 
+(_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 
+u t3 t2) \to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b) 
+u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 
+c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) u2 
+i0 H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0) 
+(CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0 
+(CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to 
+(pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 
+(pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c 
+(CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi))))) 
+i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 
+(CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) 
+(refl_equal T t1) (refl_equal T t2)))))))))).
+
+theorem pr3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 
+u2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u1) 
+\to ((eq T t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 
+(CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))))))) with 
+[(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: 
+(eq T t1 u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T 
+t1 u1) \to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: 
+T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 
+t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T 
+t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
+K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda 
+(H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3: 
+T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 
+(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t3 
+t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H7)))))) t2 (sym_eq T t2 u2 H5))) t1 
+(sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u 
+i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq 
+T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1 
+u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 
+t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
+K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))))) 
+(\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to 
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) 
+\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 
+t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t 
+u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to 
+((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: 
+T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 
+t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
+(pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda 
+(t0: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t3 t0)).(let H12 
+\def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: 
+T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t3) 
+\to ((eq T t5 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c1 t4 
+t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14: 
+(eq T t4 t3)).(\lambda (H15: (eq T t5 t0)).(eq_ind C (CHead c k u2) (\lambda 
+(_: C).((eq T t4 t3) \to ((eq T t5 t0) \to ((pr0 t4 t5) \to (pr3 (CHead c k 
+u1) t3 t0))))) (\lambda (H16: (eq T t4 t3)).(eq_ind T t3 (\lambda (t6: 
+T).((eq T t5 t0) \to ((pr0 t6 t5) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda 
+(H17: (eq T t5 t0)).(eq_ind T t0 (\lambda (t6: T).((pr0 t3 t6) \to (pr3 
+(CHead c k u1) t3 t0))) (\lambda (H18: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1) 
+t3 t0 (pr2_free (CHead c k u1) t3 t0 H18))) t5 (sym_eq T t5 t0 H17))) t4 
+(sym_eq T t4 t3 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) | 
+(pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C 
+c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t3)).(\lambda (H17: (eq T t6 
+t0)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t3) \to ((eq T t6 
+t0) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0 
+i0 u0 t5 t6) \to (pr3 (CHead c k u1) t3 t0))))))) (\lambda (H18: (eq T t4 
+t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t6 t0) \to ((getl i0 (CHead c k u2) 
+(CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to 
+(pr3 (CHead c k u1) t3 t0)))))) (\lambda (H19: (eq T t6 t0)).(eq_ind T t0 
+(\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to 
+((pr0 t3 t5) \to ((subst0 i0 u0 t5 t7) \to (pr3 (CHead c k u1) t3 t0))))) 
+(\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda 
+(H21: (pr0 t3 t5)).(\lambda (H22: (subst0 i0 u0 t5 t0)).((match i0 in nat 
+return (\lambda (n: nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) 
+\to ((subst0 n u0 t5 t0) \to (pr3 (CHead c k u1) t3 t0)))) with [O 
+\Rightarrow (\lambda (H23: (getl O (CHead c k u2) (CHead d0 (Bind Abbr) 
+u0))).(\lambda (H24: (subst0 O u0 t5 t0)).((match k in K return (\lambda (k0: 
+K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k0 
+u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda (H25: (clear (CHead c (Bind 
+b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 | 
+(CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) 
+u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \def 
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K 
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) 
+(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind 
+Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) 
+u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31 
+\def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t0)) H24 u2 H28) in 
+(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t3 t0)) (ex2_ind 
+T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i 
+O)) u t7 t0)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x: T).(\lambda 
+(H32: (subst0 O t2 t5 x)).(\lambda (H33: (subst0 (S (plus i O)) u x t0)).(let 
+H34 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O 
+i))) in (let H35 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n 
+u x t0)) H33 (S i) H34) in (ex2_ind T (\lambda (t7: T).(subst0 O u1 t5 t7)) 
+(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda 
+(x0: T).(\lambda (H36: (subst0 O u1 t5 x0)).(\lambda (H37: (pr0 x0 
+x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) 
+u1) c u1 O (getl_refl Abbr c u1) t3 t5 H21 x0 H36) t0 (pr3_pr2 (CHead c (Bind 
+Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) 
+(getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 (clear_bind Abbr c u1) 
+(CHead d (Bind Abbr) u) i H8) x0 x H37 t0 H35)))))) (pr0_subst0_back t2 t5 x 
+O H32 u1 H9))))))) (subst0_subst0 t5 t0 u2 O H31 t2 u i H10)) b H29))))) 
+H27)) H26))) | (Flat f) \Rightarrow (\lambda (H25: (clear (CHead c (Flat f) 
+u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 
+(pr2_cflat c t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 (Bind Abbr) 
+u0) c (drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25)) t3 
+t5 H21 t0 H24) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) 
+H23)))) | (S n) \Rightarrow (\lambda (H23: (getl (S n) (CHead c k u2) (CHead 
+d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 (S n) u0 t5 t0)).((match k in K 
+return (\lambda (k0: K).((getl (S n) (CHead c k0 u2) (CHead d0 (Bind Abbr) 
+u0)) \to (pr3 (CHead c k0 u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda 
+(H25: (getl (S n) (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 
+(CHead c (Bind b) u1) t3 t0 (pr2_delta (CHead c (Bind b) u1) d0 u0 (S n) 
+(getl_head (Bind b) n c (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c 
+(CHead d0 (Bind Abbr) u0) u2 n H25) u1) t3 t5 H21 t0 H24))) | (Flat f) 
+\Rightarrow (\lambda (H25: (getl (S n) (CHead c (Flat f) u2) (CHead d0 (Bind 
+Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0 
+(pr2_delta c d0 u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d0 (Bind 
+Abbr) u0) u2 n H25) t3 t5 H21 t0 H24) f u1)))]) H23)))]) H20 H22)))) t6 
+(sym_eq T t6 t0 H19))) t4 (sym_eq T t4 t3 H18))) c1 (sym_eq C c1 (CHead c k 
+u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead c k u2)) 
+(refl_equal T t3) (refl_equal T t0)))))))))) t (sym_eq T t u2 H7))) t1 
+(sym_eq T t1 u1 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 
+(refl_equal C c) (refl_equal T u1) (refl_equal T u2)))))).
+
+theorem pr3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to 
+(pr3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) 
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3 
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c 
+u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: 
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2) 
+\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 
+u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
+u1 H3)))))))))) t1 t2 H)))))).
+
+theorem pr3_pr3_pr3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall 
+(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1 
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
+K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda 
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 
+t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 
+(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: 
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 
+t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
+
+theorem pr3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift 
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1 
+t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h 
+d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda 
+(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0 
+t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d 
+t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 
+H0)))))))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/subst1.ma
new file mode 100644 (file)
index 0000000..c1f85ae
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/subst1.ma
@@ -0,0 +1,91 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/pr3/subst1".
+
+include "pr3/defs.ma".
+
+include "pr2/subst1.ma".
+
+theorem pr3_subst1:
+ \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) 
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c 
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
+T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 
+w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: 
+T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 
+w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: 
+T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v 
+t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i 
+v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T 
+(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T 
+(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) 
+(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 
+x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 
+w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 
+w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v 
+t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 
+i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c 
+e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
+
+theorem pr3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T 
+(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a 
+x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
+(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 
+d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) 
+(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
+(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: 
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 
+a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d 
+x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) 
+(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: 
+T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
+(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: 
+T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 
+x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
+(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: 
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 
+a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d 
+x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda 
+(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
+x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 
+d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda 
+(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) 
+(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: 
+T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) 
+d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d 
+u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x 
+x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 
+e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/wcpr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/wcpr0.ma
new file mode 100644 (file)
index 0000000..949cc41
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/wcpr0.ma
@@ -0,0 +1,79 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/pr3/wcpr0".
+
+include "pr3/props.ma".
+
+include "wcpr0/getl.ma".
+
+theorem pr3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: 
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 
+t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: 
+T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) 
+(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: 
+T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead 
+c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 
+(CHead c0 k u1) (let H7 \def (match H4 in pr2 return (\lambda (c: C).(\lambda 
+(t: T).(\lambda (t5: T).(\lambda (_: (pr2 c t t5)).((eq C c (CHead c3 k u1)) 
+\to ((eq T t t3) \to ((eq T t5 t0) \to (pr3 (CHead c0 k u1) t3 t0)))))))) 
+with [(pr2_free c t5 t6 H7) \Rightarrow (\lambda (H8: (eq C c (CHead c3 k 
+u1))).(\lambda (H9: (eq T t5 t3)).(\lambda (H10: (eq T t6 t0)).(eq_ind C 
+(CHead c3 k u1) (\lambda (_: C).((eq T t5 t3) \to ((eq T t6 t0) \to ((pr0 t5 
+t6) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H11: (eq T t5 t3)).(eq_ind 
+T t3 (\lambda (t: T).((eq T t6 t0) \to ((pr0 t t6) \to (pr3 (CHead c0 k u1) 
+t3 t0)))) (\lambda (H12: (eq T t6 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 
+t) \to (pr3 (CHead c0 k u1) t3 t0))) (\lambda (H13: (pr0 t3 t0)).(pr3_pr2 
+(CHead c0 k u1) t3 t0 (pr2_free (CHead c0 k u1) t3 t0 H13))) t6 (sym_eq T t6 
+t0 H12))) t5 (sym_eq T t5 t3 H11))) c (sym_eq C c (CHead c3 k u1) H8) H9 H10 
+H7)))) | (pr2_delta c d u i H7 t5 t6 H8 t H9) \Rightarrow (\lambda (H10: (eq 
+C c (CHead c3 k u1))).(\lambda (H11: (eq T t5 t3)).(\lambda (H12: (eq T t 
+t0)).(eq_ind C (CHead c3 k u1) (\lambda (c4: C).((eq T t5 t3) \to ((eq T t 
+t0) \to ((getl i c4 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i 
+u t6 t) \to (pr3 (CHead c0 k u1) t3 t0))))))) (\lambda (H13: (eq T t5 
+t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t t0) \to ((getl i (CHead c3 k u1) 
+(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (pr3 
+(CHead c0 k u1) t3 t0)))))) (\lambda (H14: (eq T t t0)).(eq_ind T t0 (\lambda 
+(t7: T).((getl i (CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t3 t6) 
+\to ((subst0 i u t6 t7) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H15: 
+(getl i (CHead c3 k u1) (CHead d (Bind Abbr) u))).(\lambda (H16: (pr0 t3 
+t6)).(\lambda (H17: (subst0 i u t6 t0)).(ex3_2_ind C T (\lambda (e2: 
+C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: 
+T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H18: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H20: (pr0 x1 u)).(ex2_ind T 
+(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t0)) (pr3 
+(CHead c0 k u1) t3 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x1 t6 
+x)).(\lambda (H22: (pr0 x t0)).(pr3_sing (CHead c0 k u1) x t3 (pr2_delta 
+(CHead c0 k u1) x0 x1 i H18 t3 t6 H16 x H21) t0 (pr3_pr2 (CHead c0 k u1) x t0 
+(pr2_free (CHead c0 k u1) x t0 H22)))))) (pr0_subst0_back u t6 t0 i H17 x1 
+H20))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 
+H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H15))))) t (sym_eq T t t0 H14))) 
+t5 (sym_eq T t5 t3 H13))) c (sym_eq C c (CHead c3 k u1) H10) H11 H12 H7 H8 
+H9))))]) in (H7 (refl_equal C (CHead c3 k u1)) (refl_equal T t3) (refl_equal 
+T t0))) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 
+u2 H2)))))))))))))) c2 c1 H))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
index 673c65b476a000e42f5147c896edba3c05ea7aa8..f479a8baa531607138c9041ab662ab4ea56c9e8c 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
@@ -232,6 +232,12 @@ include "pr1/props.ma".
 
 include "pr1/pr1.ma".
 
+include "wcpr0/defs.ma".
+
+include "wcpr0/fwd.ma".
+
+include "wcpr0/getl.ma".
+
 include "pr2/defs.ma".
 
 include "pr2/fwd.ma".
@@ -246,3 +252,17 @@ include "pr2/subst1.ma".
 
 include "pr3/defs.ma".
 
+include "pr3/props.ma".
+
+include "pr3/fwd.ma".
+
+include "pr3/wcpr0.ma".
+
+include "pr3/pr1.ma".
+
+include "pr3/pr3.ma".
+
+include "pr3/subst1.ma".
+
+include "pr3/iso.ma".
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/defs.ma
new file mode 100644 (file)
index 0000000..5cdec7d
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/defs.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/wcpr0/defs".
+
+include "pr0/defs.ma".
+
+include "C/defs.ma".
+
+inductive wcpr0: C \to (C \to Prop) \def
+| wcpr0_refl: \forall (c: C).(wcpr0 c c)
+| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall 
+(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k 
+u1) (CHead c2 k u2)))))))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/fwd.ma
new file mode 100644 (file)
index 0000000..25d8c29
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/fwd.ma
@@ -0,0 +1,102 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/wcpr0/fwd".
+
+include "wcpr0/defs.ma".
+
+theorem wcpr0_gen_sort:
+ \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort 
+n))))
+\def
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let H0 
+\def (match H in wcpr0 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
+(wcpr0 c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort 
+n))))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort 
+n))).(\lambda (H1: (eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0 
+x) \to (eq C x (CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C 
+(CSort n) (\lambda (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x 
+H2)) c (sym_eq C c (CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k) 
+\Rightarrow (\lambda (H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq 
+C (CHead c2 k u2) x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e: 
+C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
+False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in (False_ind ((eq 
+C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort 
+n))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C 
+x))))).
+
+theorem wcpr0_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 
+(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
+(H: (wcpr0 (CHead c1 k u1) x)).(let H0 \def (match H in wcpr0 return (\lambda 
+(c: C).(\lambda (c0: C).(\lambda (_: (wcpr0 c c0)).((eq C c (CHead c1 k u1)) 
+\to ((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))) with 
+[(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda 
+(H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 x) \to 
+(or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq 
+C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) 
+(\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))) (\lambda (H2: (eq C (CHead 
+c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead 
+c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C c0 (CHead c2 k 
+u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C (CHead c1 k u1) (CHead 
+c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C (CHead c1 k u1) 
+(CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda 
+(_: C).(\lambda (u2: T).(pr0 u1 u2)))) (refl_equal C (CHead c1 k u1))) x H2)) 
+c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0) 
+\Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda 
+(H3: (eq C (CHead c2 k0 u2) x)).((let H4 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let 
+H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) 
+with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 
+k0 u0) (CHead c1 k u1) H2) in ((let H6 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | 
+(CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in 
+(eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead 
+c2 k0 u2) x) \to ((wcpr0 c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k 
+u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) 
+(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u1 u3))))))))))) (\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda 
+(k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to 
+((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: 
+C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: 
+T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))) 
+(\lambda (H8: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k 
+u2) x) \to ((wcpr0 c1 c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1)) 
+(ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) 
+(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u1 u3))))))))) (\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C 
+(CHead c2 k u2) (\lambda (c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C 
+c (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c 
+(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1 
+c2)).(\lambda (H11: (pr0 u1 u2)).(or_intror (eq C (CHead c2 k u2) (CHead c1 k 
+u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) 
+(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: 
+C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: 
+C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
+u3))) c2 u2 (refl_equal C (CHead c2 k u2)) H10 H11)))) x H9)) u0 (sym_eq T u0 
+u1 H8))) k0 (sym_eq K k0 k H7))) c0 (sym_eq C c0 c1 H6))) H5)) H4)) H3 H0 
+H1)))]) in (H0 (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))).
+

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/getl.ma
new file mode 100644 (file)
index 0000000..ba017b9
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/wcpr0/getl.ma
@@ -0,0 +1,470 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Level-1/LambdaDelta/wcpr0/getl".
+
+include "wcpr0/defs.ma".
+
+include "getl/props.ma".
+
+theorem wcpr0_drop:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c0 k u1) (CHead 
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
+c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c0 k u1) 
+(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c0 k u1) (CHead 
+e1 k0 u0) H3) in eq return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c 
+(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O 
+(CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))))) with [refl_equal 
+\Rightarrow (\lambda (H4: (eq C (CHead c0 k u1) (CHead e1 k0 u0))).(let H5 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k 
+u1) (CHead e1 k0 u0) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
+e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 
+_) \Rightarrow k1])) (CHead c0 k u1) (CHead e1 k0 u0) H4) in ((let H7 \def 
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u1) 
+(CHead e1 k0 u0) H4) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T 
+u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k 
+u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))))) (\lambda (H8: (eq K k 
+k0)).(eq_ind K k0 (\lambda (k1: K).((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop O O (CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u0 u3)))))) (\lambda (H9: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C 
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))))) (let H10 \def (eq_ind T u1 (\lambda (t: 
+T).(pr0 t u2)) H2 u0 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(wcpr0 
+c c3)) H0 e1 H7) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop 
+O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) c3 u2 
+(drop_refl (CHead c3 k0 u2)) H11 H10))) u1 (sym_eq T u1 u0 H9))) k (sym_eq K 
+k k0 H8))) c0 (sym_eq C c0 e1 H7))) H6)) H5)))]) in (H4 (refl_equal C (CHead 
+e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c0 k0 u1) (CHead e1 k1 
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 
+u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c0 k0 u1) (CHead 
+e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O 
+(CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: 
+B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
+(k0: K).((drop n O (CHead c0 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u2) (CHead e2 
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: 
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Bind b) u1) (CHead 
+e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c3 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c3 (CHead 
+x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 
+x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
+c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop 
+(Bind b) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 n e1 u0 k0 
+(drop_gen_drop (Bind b) c0 (CHead e1 k0 u0) u1 n H4)))))))))) (\lambda (f: 
+F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
+(k0: K).((drop n O (CHead c0 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u2) (CHead e2 
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: 
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Flat f) u1) (CHead 
+e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(drop (S n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c3 
+(CHead x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 
+x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
+c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop 
+(Flat f) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 (S n) e1 u0 k0 
+(drop_gen_drop (Flat f) c0 (CHead e1 k0 u0) u1 n H4)))))))))) k) h)))))))))) 
+c1 c2 H))).
+
+theorem wcpr0_drop_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 
+u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u2) (CHead 
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
+c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u2) 
+(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c3 k u2) (CHead 
+e1 k0 u0) H3) in eq return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c 
+(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O 
+(CHead c0 k u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))))) with [refl_equal 
+\Rightarrow (\lambda (H4: (eq C (CHead c3 k u2) (CHead e1 k0 u0))).(let H5 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k 
+u2) (CHead e1 k0 u0) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
+e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 
+_) \Rightarrow k1])) (CHead c3 k u2) (CHead e1 k0 u0) H4) in ((let H7 \def 
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u2) 
+(CHead e1 k0 u0) H4) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T 
+u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c0 k 
+u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))))) (\lambda (H8: (eq K k 
+k0)).(eq_ind K k0 (\lambda (k1: K).((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop O O (CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u3 u0)))))) (\lambda (H9: (eq T u2 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C 
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))))) (let H10 \def (eq_ind T u2 (\lambda (t: 
+T).(pr0 u1 t)) H2 u0 H9) in (let H11 \def (eq_ind C c3 (\lambda (c: C).(wcpr0 
+c0 c)) H0 e1 H7) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop 
+O O (CHead c0 k0 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) c0 u1 
+(drop_refl (CHead c0 k0 u1)) H11 H10))) u2 (sym_eq T u2 u0 H9))) k (sym_eq K 
+k k0 H8))) c3 (sym_eq C c3 e1 H7))) H6)) H5)))]) in (H4 (refl_equal C (CHead 
+e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u2) (CHead e1 k1 
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 k0 
+u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u2) (CHead 
+e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O 
+(CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: 
+B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
+(k0: K).((drop n O (CHead c3 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 (Bind b) u1) (CHead e2 
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: 
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u2) (CHead 
+e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c0 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c0 (CHead 
+x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 
+u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
+c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop 
+(Bind b) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 n e1 u0 k0 
+(drop_gen_drop (Bind b) c3 (CHead e1 k0 u0) u2 n H4)))))))))) (\lambda (f: 
+F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
+(k0: K).((drop n O (CHead c3 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 (Flat f) u1) (CHead e2 
+k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: 
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u2) (CHead 
+e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c0 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(drop (S n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c0 
+(CHead x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 
+u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
+c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop 
+(Flat f) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 (S n) e1 u0 k0 
+(drop_gen_drop (Flat f) c3 (CHead e1 k0 u0) u2 n H4)))))))))) k) h)))))))))) 
+c2 c1 H))).
+
+theorem wcpr0_getl:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
+(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 k u1) (CHead e1 
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k 
+u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c0 k u1) 
+(CHead e1 k0 u0))).((match k in K return (\lambda (k1: K).((clear (CHead c0 
+k1 u1) (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) with 
+[(Bind b) \Rightarrow (\lambda (H4: (clear (CHead c0 (Bind b) u1) (CHead e1 
+k0 u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow 
+c])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 
+k0 u0) u1 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C 
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b 
+c0 (CHead e1 k0 u0) u1 H4)) in ((let H7 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) 
+(clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in (\lambda (H8: (eq K k0 (Bind 
+b))).(\lambda (H9: (eq C e1 c0)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead 
+e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r T u1 (\lambda (t: T).(ex3_2 C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind_r C c0 (\lambda (c: C).(ex3_2 C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: 
+C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 c0 e2))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u1 u3))) c3 u2 (getl_refl b c3 u2) H0 H2) e1 H9) u0 H7) k0 
+H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c0 (Flat 
+f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c0 (CHead 
+e1 k0 u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e1 k0 u0) u1 H4))) in 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u2) (CHead 
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f 
+u2) H7 H8)))))) H5)))]) (getl_gen_O (CHead c0 k u1) (CHead e1 k0 u0) H3)))))) 
+(K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: 
+T).(\forall (k1: K).((getl n (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C 
+T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 u2) (CHead e2 k1 
+u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: 
+T).(\forall (k1: K).((getl (S n) (CHead c0 k0 u1) (CHead e1 k1 u3)) \to 
+(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u2) 
+(CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: 
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
+n (CHead c0 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u2) (CHead e2 k0 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
+K).(\lambda (H4: (getl (S n) (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let 
+H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c0 (CHead e1 k0 u0) u1 n H4)) in 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2) 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c3 (CHead 
+x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
+n (CHead c0 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c3 (Flat f) u2) (CHead e2 k0 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
+K).(\lambda (H4: (getl (S n) (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let 
+H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c0 (CHead e1 k0 u0) u1 n H4)) 
+in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead 
+x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 
+x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 
+(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat 
+f) n c3 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 
+H))).
+
+theorem wcpr0_getl_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
+u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
+(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u2) (CHead e1 
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 k 
+u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u2) 
+(CHead e1 k0 u0))).((match k in K return (\lambda (k1: K).((clear (CHead c3 
+k1 u2) (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) with 
+[(Bind b) \Rightarrow (\lambda (H4: (clear (CHead c3 (Bind b) u2) (CHead e1 
+k0 u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow 
+c])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 
+k0 u0) u2 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C 
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b 
+c3 (CHead e1 k0 u0) u2 H4)) in ((let H7 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) 
+(clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in (\lambda (H8: (eq K k0 (Bind 
+b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead 
+e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r T u2 (\lambda (t: T).(ex3_2 C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: 
+C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c3))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u3 u2))) c0 u1 (getl_refl b c0 u1) H0 H2) e1 H9) u0 H7) k0 
+H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c3 (Flat 
+f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c3 (CHead 
+e1 k0 u0) c3 (drop_refl c3) (clear_gen_flat f c3 (CHead e1 k0 u0) u2 H4))) in 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c0 (CHead e2 k0 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c0 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f) u1) (CHead 
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_flat c0 (CHead x0 k0 x1) O H6 f 
+u1) H7 H8)))))) H5)))]) (getl_gen_O (CHead c3 k u2) (CHead e1 k0 u0) H3)))))) 
+(K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: 
+T).(\forall (k1: K).((getl n (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C 
+T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 k0 u1) (CHead e2 k1 
+u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: 
+T).(\forall (k1: K).((getl (S n) (CHead c3 k0 u2) (CHead e1 k1 u3)) \to 
+(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c0 k0 u1) 
+(CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: 
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
+n (CHead c3 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c0 (Bind b) u1) (CHead e2 k0 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
+K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let 
+H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u2 n H4)) in 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c0 (CHead e2 k0 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c0 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1) 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c0 (CHead 
+x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
+n (CHead c3 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c0 (Flat f) u1) (CHead e2 k0 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
+K).(\lambda (H4: (getl (S n) (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let 
+H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u2 n H4)) 
+in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c0 (CHead e2 
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c0 (CHead 
+x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 
+u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 
+(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat 
+f) n c0 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 
+H))).
+