(* This file was automatically generated: do not edit *********************)
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
-include "pr0/props.ma".
+include "LambdaDelta-1/pr0/props.ma".
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
theorem pr2_thin_dx:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
t2)))))))
\def
\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead
-k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind
-b) u) t1 t2)).(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 (CHead c (Bind
-b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1)
-(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
-(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_:
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind
-b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u
-t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead
-(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1)
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda
-(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda
-(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
-t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind
-b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4:
-T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3)
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u
-t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind
-(\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u
-t2))))) (\lambda (H11: (getl O (CHead c (Bind b) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H13 \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) u0) (CHead c (Bind b)
-u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind
-b) u) (CHead d (Bind Abbr) u0) H11))) 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 d
+(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u)
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead
+k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c
+(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k
+u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3
+t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d:
+C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0))
+\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t:
+T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u
+t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4)
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u))
+\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1:
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4
+t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b)
+u) H4) in (let H6 \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) u0) (CHead c (Bind b) u) (clear_gen_bind b c
+(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind
+Abbr) u0) H5))) in ((let H7 \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) u0)
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d
(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr)
-u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in
-((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t4) \Rightarrow t4]))
-(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d
-(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)
-H11))) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H18
-\def (eq_ind T u0 (\lambda (t4: T).(subst0 O t4 t3 t2)) H12 u H15) in (eq_ind
-B Abbr (\lambda (b0: B).(pr2 c (THead (Bind b0) u t1) (THead (Bind b0) u
-t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2)
-(pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H18)) b H16))))) H14)) H13))))
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) u) (CHead d
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Bind b) u t1)
-(THead (Bind b) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Bind b) u)
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3
-t2)).(pr2_delta c d u0 (r (Bind b) i0) (getl_gen_S (Bind b) c (CHead d (Bind
-Abbr) u0) u i0 H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u
-(pr0_refl u) t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0
-t2 t3 (r (Bind b) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1
-H2))))]) in (H0 (refl_equal C (CHead c (Bind b) u)) (refl_equal T t1)
-(refl_equal T t2))))) (\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u)
-t1 t2)).(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 (CHead c (Flat f)
-u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Flat f) u t1)
-(THead (Flat f) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
-(\lambda (H1: (eq C c0 (CHead c (Flat f) u))).(\lambda (H2: (eq T t0
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (_:
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Flat
-f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Flat f) u
-t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f)
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Flat f) u t1) (THead
-(Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Flat f)))) t3 (sym_eq T
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) u) H1)
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda
-(H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: (eq T t0 t1)).(\lambda
-(H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (c1: C).((eq T t0
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f)
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
-t t2) \to ((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat
-f) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4:
-T).((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3)
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u
-t2)))))) (\lambda (H8: (getl i (CHead c (Flat f) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind
-(\lambda (n: nat).((getl n (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u
-t2))))) (\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c
-(CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind
-Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11)))
-(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3
-H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u))))
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Flat f) u) (CHead d
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Flat f) u t1)
-(THead (Flat f) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Flat f) u)
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3
-t2)).(pr2_delta c d u0 (r (Flat f) i0) (getl_gen_S (Flat f) c (CHead d (Bind
-Abbr) u0) u i0 H11) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u
-(pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0
-t2 t3 (r (Flat f) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1
-H2))))]) in (H0 (refl_equal C (CHead c (Flat f) u)) (refl_equal T t1)
-(refl_equal T t2))))) k))))).
+u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in
+(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T
+u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda
+(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u)
+t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1:
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
+c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b)
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
+Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
+(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u
+t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n)
+(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3)
+(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead
+(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u)))))))))))))
+i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
+T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind
+Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall
+(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c
+(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0
+(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4:
+(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1:
+C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in
+(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c)
+(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f)
+u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4)
+(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t)
+(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1:
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
+c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f)
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
+Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
+(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u
+t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n)
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3)
+(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead
+(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u)))))))))))))
+i)))))) k) y t1 t2 H0))) H)))))).
theorem clear_pr2_trans:
\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
\def
\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H in
-pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2
-c t t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1
-t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c
-c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1
-t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
-t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind
-T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1
-t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1
-H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t
-H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1)
-\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3)
-\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1
-t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2
-(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1
-t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10:
-(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i
-(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11))))
-t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1
-H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T
-t2)))))))).
+t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1:
+C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_:
+(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1:
+C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c
+(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
theorem pr2_cflat:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f:
-F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f:
-F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda
-(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0
-(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))).
+t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free
+(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u
+i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c
+t1 t2 H)))))).
theorem pr2_ctail:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
\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: (pr2 e 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 e) \to ((eq T t t1)
-\to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with
-[(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3:
-(eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T
-t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d
-t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
-t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6:
-(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d
-t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1)
-(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T
-t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
-t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1)
-\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3)
-\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda
-(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e
-(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c
-(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2
-(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to
-((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9:
-(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
-(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2))
-(\lambda (H12: (lt i d)).(let H13 \def (drop_getl_trans_le i d (le_S_n i d
-(le_S (S i) d H12)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C
-(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda
-(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear
-e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda
-(x0: C).(\lambda (x1: C).(\lambda (H14: (drop i O c x0)).(\lambda (H15: (drop
-h (minus d i) x0 x1)).(\lambda (H16: (clear x1 (CHead d0 (Bind Abbr)
-u))).(let H17 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0
-x1)) H15 (S (minus d (S i))) (minus_x_Sy d i H12)) in (let H18 \def
-(drop_clear_S x1 x0 h (minus d (S i)) H17 Abbr d0 u H16) in (ex2_ind C
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i))
-u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1)
-(lift h d t2)) (\lambda (x: C).(\lambda (H19: (clear x0 (CHead x (Bind Abbr)
-(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x
-d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x
-(Bind Abbr) (lift h (minus d (S i)) u)) x0 H14 H19) (lift h d t1) (lift h d
-t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d
-H12 h))))) H18)))))))) H13))) (\lambda (H12: (le d i)).(pr2_delta c d0 u
-(plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H12)
-(lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2)
-(subst0_lift_ge t3 t2 u i h H11 d H12))))))) t (sym_eq T t t2 H8))) t0
-(sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1
-(refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))).
+t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c
+(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e)
+\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0
+e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d)))))))
+(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H2: (getl i c0 (CHead d0 (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 e)).(let H6 \def (eq_ind C c0 (\lambda (c1:
+C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def
+(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0
+(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9:
+(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11:
+(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i)
+(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i
+H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u
+H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h
+(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x
+(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S
+i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead
+x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d
+t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7
+h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h)
+(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3)
+(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h
+H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
projects / helm.git / blobdiff