--- /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 "basic_1A/pc3/defs.ma".
+
+include "basic_1A/pr3/pr3.ma".
+
+lemma clear_pc3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind
+T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2
+t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1
+H0))))) H1))))))).
+
+lemma pc3_pr2_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
+
+lemma pc3_pr2_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
+
+lemma pc3_pr3_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 H (pr3_refl c t2))))).
+
+lemma pc3_pr3_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) H)))).
+
+lemma pc3_pr3_t:
+ \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall
+(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1
+t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t:
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
+
+lemma pc3_refl:
+ \forall (c: C).(\forall (t: T).(pc3 c t t))
+\def
+ \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0))
+(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
+
+lemma pc3_s:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c
+t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t))
+(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
+
+lemma pc3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u
+t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead
+(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead
+(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f)))))
+H0))))))).
+
+lemma pc3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda
+(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t)
+(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2:
+(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0))
+(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x
+H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl
+(CHead c k x) t)))))) H0))))))).
+
+lemma pc3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 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: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T
+(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u)
+t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1:
+(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2
+T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u
+t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1)
+(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
+
+lemma pc3_pr2_u:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t))
+x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
+
+theorem pc3_t:
+ \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1
+x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t))
+(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7:
+(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c
+H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2)))))
+H4))))) H1))))))).
+
+lemma pc3_pr2_u2:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
+(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
+t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x
+c t1 t0 H) t2 H0)))))).
+
+lemma pc3_pr3_conf:
+ \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall
+(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t
+t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c
+t2 t H0) t1 H)))))).
+
+theorem pc3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
+(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c
+u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pc3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
+(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c
+u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
+
+lemma pc3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\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).(pc3 (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 (pc3 (CHead c k u1) t t0))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3:
+(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0
+(CHead c k u2) H3) in (pc3_pr2_r (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 (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in
+(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr)
+u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k
+u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1)
+t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr)
+u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead
+c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda
+(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr)
+u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e 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 H10)) in
+((let H12 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 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 H10)) in
+((let H13 \def (f_equal C T (\lambda (e: C).(match e 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 H10)) in
+(\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H16 \def (eq_ind
+T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) in (eq_ind B Abbr
+(\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0:
+T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) (pc3 (CHead c (Bind
+Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O u1 t4 x)).(\lambda
+(H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x (pr3_pr2 (CHead c
+(Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl
+Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) u1) t x (pr2_free
+(CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 t4 t O H16 u1 H)) b
+H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: (clear (CHead c (Flat
+f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t3
+t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t (pr2_delta (CHead d (Bind Abbr) u)
+d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) (CHead c (Flat f) u1) (clear_flat
+c (CHead d (Bind Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H10)
+f u1)))) k (getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H8)))) (\lambda
+(i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u))
+\to ((subst0 i0 u t4 t) \to (pc3 (CHead c k u1) t3 t))))).(\lambda (H8: (getl
+(S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H9: (subst0 (S i0)
+u t4 t)).(K_ind (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind
+Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c k0 u1) t3 t)))) \to
+((getl (r k0 i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))))
+(\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind
+Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Bind b) u1) t3
+t))))).(\lambda (H10: (getl (r (Bind b) i0) c (CHead d (Bind Abbr)
+u))).(pc3_pr2_r (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) H10 u1) t3 t4 H3 t
+H9))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead
+d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Flat f) u1) t3
+t))))).(\lambda (H10: (getl (r (Flat f) i0) c (CHead d (Bind Abbr)
+u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u
+(r (Flat f) i0) H10 t3 t4 H3 t H9) f u1))))) k IHi (getl_gen_S k c (CHead d
+(Bind Abbr) u) u2 i0 H8)))))) i H7 H4)))))))))))))) y t1 t2 H1))) H0)))))))).
+
+lemma pc3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2
+u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3
+(CHead c0 k t0) 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 t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 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 t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1:
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) 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 t1)) \to (pc3 (CHead c0 k t) t4
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4
+t5 (pr2_free (CHead c0 k t) 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 t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n:
+nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
+t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3
+(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
+(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
+(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow
+c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0
+(CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 \def (f_equal C B (\lambda
+(e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])]))
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead
+d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def (f_equal C T (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7]))
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead
+d0 (Bind Abbr) u0) t1 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 t1 H15) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4
+t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0
+t6 t7)) (pc3 (CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19:
+(subst0 O t2 t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7:
+T).(subst0 O t t5 t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3
+(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t
+t5 x0)).(\lambda (H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal
+nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24
+\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S
+i) H23) in (pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0
+(Bind Abbr) t) c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x
+(CHead c0 (Bind Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i)
+(getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0
+H24)))))))) (subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd
+t1 t5 t6 O H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda
+(H12: (clear (CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr)
+u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0
+(Bind Abbr) u0) t4 t6 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl
+Abbr d0 u0) t4 t5 H6 t6 H11)) (CHead c0 (Flat f) t) (clear_flat c0 (CHead d0
+(Bind Abbr) u0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f
+t)))) k (getl_gen_O (CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10))))
+(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind
+Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pc3 (CHead c0 k t) t4
+t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t1) (CHead d0 (Bind Abbr)
+u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl
+(r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c0 k0 t) t4 t6)))
+(\lambda (b: B).(\lambda (H12: (getl (r (Bind b) i1) c0 (CHead d0 (Bind Abbr)
+u0))).(pc3_pr2_r (CHead c0 (Bind b) t) t4 t6 (pr2_delta (CHead c0 (Bind b) t)
+d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind Abbr) u0) H12 t) t4 t5
+H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (r (Flat f) i1) c0 (CHead
+d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4
+t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) H12 t4 t5 H6 t6 H11) f t)))) k
+(getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 i1 H10)))))) i0 H9
+H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 H)))).
+
+lemma pc3_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 u2 u1) \to
+(pc3 (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 u2 u1) \to (pc3
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
+u2 u1)).(pc3_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 u2 u1)
+\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2
+u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
+u1 H3)))))))))) t1 t2 H)))))).
+
+lemma pc3_pr3_pc3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2
+u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H0: (pc3 (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).((pc3
+(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0:
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0
+t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c
+k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2)
+t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6:
+(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0
+x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2
+H0)))))) H4))))))))))))) u2 u1 H)))).
+
+lemma pc3_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).((pc3 e t1 t2) \to (pc3 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: (pc3 e t1
+t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t:
+T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda
+(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1)
+(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H
+t2 x H3))))) H1))))))))).
+
+lemma pc3_eta:
+ \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t
+(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead
+(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H:
+(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v
+w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
+(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef
+O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl)
+(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead
+(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t)
+(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S
+O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u)
+H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w
+(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u))))
+(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O
+(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl
+c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))).
+
projects / helm.git / blobdiff