(* This file was automatically generated: do not edit *********************)
+include "LambdaDelta-1/subst1/fwd.ma".
-
-include "subst1/fwd.ma".
-
-include "subst0/subst0.ma".
+include "LambdaDelta-1/subst0/subst0.ma".
theorem subst1_subst1:
\forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl
(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1
t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1
-i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (ex2 T
-(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
-t t3))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 y)).(subst1_ind i u u1
-(\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0))
-(\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) (\lambda (H3: (eq T u1
-u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda (t0: T).(subst1 j t t1
-t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3)))) (ex_intro2 T
-(\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
-t t3)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i j)) u t3)) u1
-H3)) (\lambda (t0: T).(\lambda (H3: (subst0 i u u1 t0)).(\lambda (H4: (eq T
-t0 u2)).(let H5 \def (eq_ind T t0 (\lambda (t: T).(subst0 i u u1 t)) H3 u2
-H4) in (ex2_ind T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0
-(S (plus i j)) u t t3)) (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
-(t: T).(subst1 (S (plus i j)) u t t3))) (\lambda (x: T).(\lambda (H6: (subst0
-j u1 t1 x)).(\lambda (H7: (subst0 (S (plus i j)) u x t3)).(ex_intro2 T
+i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S
+(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1
+y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0:
+T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3)))))
+(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda
+(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
+t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t:
+T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0)
+(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3:
+(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0
+(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t:
+T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T
(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
-t t3)) x (subst1_single j u1 t1 x H6) (subst1_single (S (plus i j)) u x t3
-H7))))) (subst0_subst0 t1 t3 u2 j H0 u1 u i H5)))))) y H2))) H1))))))) t2
-H))))).
+t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7:
+(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1
+t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1
+x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j
+H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))).
theorem subst1_subst1_back:
\forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
\def
\lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1)
-(\lambda (t: T).(subst1 i u t0 t)) (\forall (t2: T).((subst1 i u t0 (lift (S
-O) i t2)) \to (eq T t1 t2))) (\lambda (y: T).(\lambda (H0: (subst1 i u t0
-y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i t1)) \to
-(\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))))
-(\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H2:
-(subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda (t:
-T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 \def
-(sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u (lift
-(S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n:
-nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) H3))
-in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: (subst0 i
-u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3:
+(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1
+i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0:
+(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i
+t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1
+t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda
+(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda
+(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4
+\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u
+(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
+(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O)))
+H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1:
+(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3:
T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2
(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T
-(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (eq T t1 t3) (\lambda
-(y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 (\lambda (t:
-T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: (eq T t0
-(lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: T).(subst0 i u t
-(lift (S O) i t1))) H4 (lift (S O) i t3) H6) in (subst0_gen_lift_false t3 u
-(lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
-(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O)))
-H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: (subst0 i u t0
-t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def (eq_ind T t4
-(\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in (sym_eq T t3
-t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) H3))))))) y
-H0))) H))))).
+(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1
+t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0
+(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6:
+(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t:
+T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in
+(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r
+nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i
+(S O)) (plus_comm i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6:
+(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def
+(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in
+(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5)))
+H3))))))) y H0))) H))))).
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