--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/pr3/pr1.ma".
+
+include "LambdaDelta-1/pr2/props.ma".
+
+include "LambdaDelta-1/pr1/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_flat:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead
+(Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
+(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f
+u2))))))))).
+
+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)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
+t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_:
+(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k
+u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4:
+(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead
+c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d
+(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t))))
+(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8:
+(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d
+(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda
+(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \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) H9))) in ((let H11
+\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) H9))) in ((let H12 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (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) H9))) in (\lambda (H13: (eq B Abbr
+b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0:
+T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3
+(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4
+t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t)
+(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x
+t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr)
+u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind
+Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17))))))
+(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f:
+F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr)
+u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (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) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda
+(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4
+t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k
+u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(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 t4 t)
+\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda
+(b: B).(\lambda (H9: (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 t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2
+(CHead c (Bind b) u1) t3 t (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 H9) u1) t3 t4 H3 t H8))))) (\lambda (f:
+F).(\lambda (H9: (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 t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2
+(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0)
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f
+u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
+
+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)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3
+(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
+K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
+(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1:
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y:
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5
+(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
+c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n:
+nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
+t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3
+(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
+(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \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 c0 (Bind
+b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14
+\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 c0 (Bind b) t)
+(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \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 c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr)
+u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let
+H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in
+(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind
+T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i
+O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda
+(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let
+H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O
+i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n
+u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7))
+(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda
+(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0
+x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind
+Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead
+c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i)
+(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1)
+(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x
+O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14))
+H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0
+(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6
+(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0
+(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5
+H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0)
+H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0
+(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4
+t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr)
+u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl
+(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1)
+t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t)
+(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta
+(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind
+Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4
+t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat
+f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6
+(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f)
+c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10)))))
+i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))).
+
+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)))))))).
+
+theorem pr3_eta:
+ \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind
+Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v
+(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
+\def
+ \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind
+Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead
+(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c
+(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w
+u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead
+(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat
+Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u
+(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat
+Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)))
+(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S
+O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing
+(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O)
+(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O))
+(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))
+(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind
+Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u
+(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead
+(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
+
projects / helm.git / blobdiff