T).(subst1 (S (plus i j)) u t t2)))))))))))
\def
\lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
-(H: (subst1 j u2 t1 t2)).(let TMP_5 \def (\lambda (t: T).(\forall (u1:
-T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (let TMP_1 \def
-(\lambda (t0: T).(subst1 j u1 t1 t0)) in (let TMP_4 \def (\lambda (t0:
-T).(let TMP_2 \def (plus i j) in (let TMP_3 \def (S TMP_2) in (subst1 TMP_3 u
-t0 t)))) in (ex2 T TMP_1 TMP_4)))))))) in (let TMP_14 \def (\lambda (u1:
-T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (subst1 i u u1 u2)).(let
-TMP_6 \def (\lambda (t: T).(subst1 j u1 t1 t)) in (let TMP_9 \def (\lambda
-(t: T).(let TMP_7 \def (plus i j) in (let TMP_8 \def (S TMP_7) in (subst1
-TMP_8 u t t1)))) in (let TMP_10 \def (subst1_refl j u1 t1) in (let TMP_11
-\def (plus i j) in (let TMP_12 \def (S TMP_11) in (let TMP_13 \def
-(subst1_refl TMP_12 u t1) in (ex_intro2 T TMP_6 TMP_9 t1 TMP_10
-TMP_13))))))))))) in (let TMP_63 \def (\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)).(let TMP_15 \def (\lambda (t: T).(subst1 i u u1 t)) in
-(let TMP_20 \def (\lambda (_: T).(let TMP_16 \def (\lambda (t0: T).(subst1 j
-u1 t1 t0)) in (let TMP_19 \def (\lambda (t0: T).(let TMP_17 \def (plus i j)
-in (let TMP_18 \def (S TMP_17) in (subst1 TMP_18 u t0 t3)))) in (ex2 T TMP_16
-TMP_19)))) in (let TMP_62 \def (\lambda (y: T).(\lambda (H2: (subst1 i u u1
-y)).(let TMP_25 \def (\lambda (t: T).((eq T t u2) \to (let TMP_21 \def
-(\lambda (t0: T).(subst1 j u1 t1 t0)) in (let TMP_24 \def (\lambda (t0:
-T).(let TMP_22 \def (plus i j) in (let TMP_23 \def (S TMP_22) in (subst1
-TMP_23 u t0 t3)))) in (ex2 T TMP_21 TMP_24))))) in (let TMP_40 \def (\lambda
-(H3: (eq T u1 u2)).(let TMP_30 \def (\lambda (t: T).(let TMP_26 \def (\lambda
-(t0: T).(subst1 j t t1 t0)) in (let TMP_29 \def (\lambda (t0: T).(let TMP_27
-\def (plus i j) in (let TMP_28 \def (S TMP_27) in (subst1 TMP_28 u t0 t3))))
-in (ex2 T TMP_26 TMP_29)))) in (let TMP_31 \def (\lambda (t: T).(subst1 j u2
-t1 t)) in (let TMP_34 \def (\lambda (t: T).(let TMP_32 \def (plus i j) in
-(let TMP_33 \def (S TMP_32) in (subst1 TMP_33 u t t3)))) in (let TMP_35 \def
-(subst1_single j u2 t1 t3 H0) in (let TMP_36 \def (plus i j) in (let TMP_37
-\def (S TMP_36) in (let TMP_38 \def (subst1_refl TMP_37 u t3) in (let TMP_39
-\def (ex_intro2 T TMP_31 TMP_34 t3 TMP_35 TMP_38) in (eq_ind_r T u2 TMP_30
-TMP_39 u1 H3)))))))))) in (let TMP_61 \def (\lambda (t0: T).(\lambda (H3:
-(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let TMP_41 \def (\lambda (t:
-T).(subst0 i u u1 t)) in (let H5 \def (eq_ind T t0 TMP_41 H3 u2 H4) in (let
-TMP_42 \def (\lambda (t: T).(subst0 j u1 t1 t)) in (let TMP_45 \def (\lambda
-(t: T).(let TMP_43 \def (plus i j) in (let TMP_44 \def (S TMP_43) in (subst0
-TMP_44 u t t3)))) in (let TMP_46 \def (\lambda (t: T).(subst1 j u1 t1 t)) in
-(let TMP_49 \def (\lambda (t: T).(let TMP_47 \def (plus i j) in (let TMP_48
-\def (S TMP_47) in (subst1 TMP_48 u t t3)))) in (let TMP_50 \def (ex2 T
-TMP_46 TMP_49) in (let TMP_59 \def (\lambda (x: T).(\lambda (H6: (subst0 j u1
-t1 x)).(\lambda (H7: (subst0 (S (plus i j)) u x t3)).(let TMP_51 \def
-(\lambda (t: T).(subst1 j u1 t1 t)) in (let TMP_54 \def (\lambda (t: T).(let
-TMP_52 \def (plus i j) in (let TMP_53 \def (S TMP_52) in (subst1 TMP_53 u t
-t3)))) in (let TMP_55 \def (subst1_single j u1 t1 x H6) in (let TMP_56 \def
-(plus i j) in (let TMP_57 \def (S TMP_56) in (let TMP_58 \def (subst1_single
-TMP_57 u x t3 H7) in (ex_intro2 T TMP_51 TMP_54 x TMP_55 TMP_58)))))))))) in
-(let TMP_60 \def (subst0_subst0 t1 t3 u2 j H0 u1 u i H5) in (ex2_ind T TMP_42
-TMP_45 TMP_50 TMP_59 TMP_60))))))))))))) in (subst1_ind i u u1 TMP_25 TMP_40
-TMP_61 y H2)))))) in (insert_eq T u2 TMP_15 TMP_20 TMP_62 H1)))))))))) in
-(subst1_ind j u2 t1 TMP_5 TMP_14 TMP_63 t2 H)))))))).
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
+t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
+(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
+(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)) (\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))) (\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
T).(subst1 (S (plus i j)) u t2 t)))))))))))
\def
\lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
-(H: (subst1 j u2 t1 t2)).(let TMP_5 \def (\lambda (t: T).(\forall (u1:
-T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (let TMP_1 \def
-(\lambda (t0: T).(subst1 j u1 t1 t0)) in (let TMP_4 \def (\lambda (t0:
-T).(let TMP_2 \def (plus i j) in (let TMP_3 \def (S TMP_2) in (subst1 TMP_3 u
-t t0)))) in (ex2 T TMP_1 TMP_4)))))))) in (let TMP_14 \def (\lambda (u1:
-T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (subst1 i u u2 u1)).(let
-TMP_6 \def (\lambda (t: T).(subst1 j u1 t1 t)) in (let TMP_9 \def (\lambda
-(t: T).(let TMP_7 \def (plus i j) in (let TMP_8 \def (S TMP_7) in (subst1
-TMP_8 u t1 t)))) in (let TMP_10 \def (subst1_refl j u1 t1) in (let TMP_11
-\def (plus i j) in (let TMP_12 \def (S TMP_11) in (let TMP_13 \def
-(subst1_refl TMP_12 u t1) in (ex_intro2 T TMP_6 TMP_9 t1 TMP_10
-TMP_13))))))))))) in (let TMP_49 \def (\lambda (t3: T).(\lambda (H0: (subst0
-j u2 t1 t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1:
-(subst1 i u u2 u1)).(let TMP_19 \def (\lambda (t: T).(let TMP_15 \def
-(\lambda (t0: T).(subst1 j t t1 t0)) in (let TMP_18 \def (\lambda (t0:
-T).(let TMP_16 \def (plus i j) in (let TMP_17 \def (S TMP_16) in (subst1
-TMP_17 u t3 t0)))) in (ex2 T TMP_15 TMP_18)))) in (let TMP_20 \def (\lambda
-(t: T).(subst1 j u2 t1 t)) in (let TMP_23 \def (\lambda (t: T).(let TMP_21
-\def (plus i j) in (let TMP_22 \def (S TMP_21) in (subst1 TMP_22 u t3 t))))
-in (let TMP_24 \def (subst1_single j u2 t1 t3 H0) in (let TMP_25 \def (plus i
-j) in (let TMP_26 \def (S TMP_25) in (let TMP_27 \def (subst1_refl TMP_26 u
-t3) in (let TMP_28 \def (ex_intro2 T TMP_20 TMP_23 t3 TMP_24 TMP_27) in (let
-TMP_48 \def (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(let TMP_29
-\def (\lambda (t: T).(subst0 j t0 t1 t)) in (let TMP_32 \def (\lambda (t:
-T).(let TMP_30 \def (plus i j) in (let TMP_31 \def (S TMP_30) in (subst0
-TMP_31 u t3 t)))) in (let TMP_33 \def (\lambda (t: T).(subst1 j t0 t1 t)) in
-(let TMP_36 \def (\lambda (t: T).(let TMP_34 \def (plus i j) in (let TMP_35
-\def (S TMP_34) in (subst1 TMP_35 u t3 t)))) in (let TMP_37 \def (ex2 T
-TMP_33 TMP_36) in (let TMP_46 \def (\lambda (x: T).(\lambda (H3: (subst0 j t0
-t1 x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(let TMP_38 \def
-(\lambda (t: T).(subst1 j t0 t1 t)) in (let TMP_41 \def (\lambda (t: T).(let
-TMP_39 \def (plus i j) in (let TMP_40 \def (S TMP_39) in (subst1 TMP_40 u t3
-t)))) in (let TMP_42 \def (subst1_single j t0 t1 x H3) in (let TMP_43 \def
-(plus i j) in (let TMP_44 \def (S TMP_43) in (let TMP_45 \def (subst1_single
-TMP_44 u t3 x H4) in (ex_intro2 T TMP_38 TMP_41 x TMP_42 TMP_45)))))))))) in
-(let TMP_47 \def (subst0_subst0_back t1 t3 u2 j H0 t0 u i H2) in (ex2_ind T
-TMP_29 TMP_32 TMP_37 TMP_46 TMP_47)))))))))) in (subst1_ind i u u2 TMP_19
-TMP_28 TMP_48 u1 H1)))))))))))))))) in (subst1_ind j u2 t1 TMP_5 TMP_14
-TMP_49 t2 H)))))))).
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t
+t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
+(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
+(t: T).(subst1 (S (plus i j)) u t1 t)) 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 u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0:
+T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0))))
+(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S
+(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i
+j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T
+(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u
+t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S
+(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1
+x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t:
+T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x
+(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4)))))
+(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))).
theorem subst1_trans:
\forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1
t3)))))))
\def
\lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i v t1 t2)).(let TMP_1 \def (\lambda (t: T).(\forall (t3:
-T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) in (let TMP_2 \def (\lambda
-(t3: T).(\lambda (H0: (subst1 i v t1 t3)).H0)) in (let TMP_7 \def (\lambda
-(t3: T).(\lambda (H0: (subst0 i v t1 t3)).(\lambda (t4: T).(\lambda (H1:
-(subst1 i v t3 t4)).(let TMP_3 \def (\lambda (t: T).(subst1 i v t1 t)) in
-(let TMP_4 \def (subst1_single i v t1 t3 H0) in (let TMP_6 \def (\lambda (t0:
-T).(\lambda (H2: (subst0 i v t3 t0)).(let TMP_5 \def (subst0_trans t3 t1 v i
-H0 t0 H2) in (subst1_single i v t1 t0 TMP_5)))) in (subst1_ind i v t3 TMP_3
-TMP_4 TMP_6 t4 H1)))))))) in (subst1_ind i v t1 TMP_1 TMP_2 TMP_7 t2
-H)))))))).
+(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3:
+T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda
+(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1
+t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3
+(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0:
+T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans
+t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))).
theorem subst1_confluence_neq:
\forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t))))))))))))
\def
\lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
-nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(let TMP_3 \def (\lambda (t:
+nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t:
T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2)
-\to ((not (eq nat i1 i2)) \to (let TMP_1 \def (\lambda (t3: T).(subst1 i2 u2
-t t3)) in (let TMP_2 \def (\lambda (t3: T).(subst1 i1 u1 t2 t3)) in (ex2 T
-TMP_1 TMP_2))))))))) in (let TMP_7 \def (\lambda (t2: T).(\lambda (u2:
+\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3))
+(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2:
T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not
-(eq nat i1 i2))).(let TMP_4 \def (\lambda (t: T).(subst1 i2 u2 t0 t)) in (let
-TMP_5 \def (\lambda (t: T).(subst1 i1 u1 t2 t)) in (let TMP_6 \def
-(subst1_refl i1 u1 t2) in (ex_intro2 T TMP_4 TMP_5 t2 H0 TMP_6))))))))) in
-(let TMP_29 \def (\lambda (t2: T).(\lambda (H0: (subst0 i1 u1 t0
-t2)).(\lambda (t3: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H1:
-(subst1 i2 u2 t0 t3)).(\lambda (H2: (not (eq nat i1 i2))).(let TMP_10 \def
-(\lambda (t: T).(let TMP_8 \def (\lambda (t4: T).(subst1 i2 u2 t2 t4)) in
-(let TMP_9 \def (\lambda (t4: T).(subst1 i1 u1 t t4)) in (ex2 T TMP_8
-TMP_9)))) in (let TMP_11 \def (\lambda (t: T).(subst1 i2 u2 t2 t)) in (let
-TMP_12 \def (\lambda (t: T).(subst1 i1 u1 t0 t)) in (let TMP_13 \def
-(subst1_refl i2 u2 t2) in (let TMP_14 \def (subst1_single i1 u1 t0 t2 H0) in
-(let TMP_15 \def (ex_intro2 T TMP_11 TMP_12 t2 TMP_13 TMP_14) in (let TMP_28
-\def (\lambda (t4: T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(let TMP_16 \def
-(\lambda (t: T).(subst0 i1 u1 t4 t)) in (let TMP_17 \def (\lambda (t:
-T).(subst0 i2 u2 t2 t)) in (let TMP_18 \def (\lambda (t: T).(subst1 i2 u2 t2
-t)) in (let TMP_19 \def (\lambda (t: T).(subst1 i1 u1 t4 t)) in (let TMP_20
-\def (ex2 T TMP_18 TMP_19) in (let TMP_25 \def (\lambda (x: T).(\lambda (H4:
-(subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(let TMP_21 \def
-(\lambda (t: T).(subst1 i2 u2 t2 t)) in (let TMP_22 \def (\lambda (t:
-T).(subst1 i1 u1 t4 t)) in (let TMP_23 \def (subst1_single i2 u2 t2 x H5) in
-(let TMP_24 \def (subst1_single i1 u1 t4 x H4) in (ex_intro2 T TMP_21 TMP_22
-x TMP_23 TMP_24)))))))) in (let TMP_26 \def (sym_not_eq nat i1 i2 H2) in (let
-TMP_27 \def (subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 TMP_26) in
-(ex2_ind T TMP_16 TMP_17 TMP_20 TMP_25 TMP_27))))))))))) in (subst1_ind i2 u2
-t0 TMP_10 TMP_15 TMP_28 t3 H1))))))))))))))) in (subst1_ind i1 u1 t0 TMP_3
-TMP_7 TMP_29 t1 H)))))))).
+(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda
+(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2:
+T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not
+(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2
+(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4:
+T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1
+u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1
+i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda
+(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x
+(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4)))))
+(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2
+H2))))) t3 H1)))))))) t1 H))))).
theorem subst1_confluence_eq:
\forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t)))))))))
\def
\lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u t0 t1)).(let TMP_3 \def (\lambda (t: T).(\forall (t2:
-T).((subst1 i u t0 t2) \to (let TMP_1 \def (\lambda (t3: T).(subst1 i u t
-t3)) in (let TMP_2 \def (\lambda (t3: T).(subst1 i u t2 t3)) in (ex2 T TMP_1
-TMP_2)))))) in (let TMP_7 \def (\lambda (t2: T).(\lambda (H0: (subst1 i u t0
-t2)).(let TMP_4 \def (\lambda (t: T).(subst1 i u t0 t)) in (let TMP_5 \def
-(\lambda (t: T).(subst1 i u t2 t)) in (let TMP_6 \def (subst1_refl i u t2) in
-(ex_intro2 T TMP_4 TMP_5 t2 H0 TMP_6)))))) in (let TMP_57 \def (\lambda (t2:
+(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2:
+T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3))
+(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0:
+(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda
+(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2:
T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i
-u t0 t3)).(let TMP_10 \def (\lambda (t: T).(let TMP_8 \def (\lambda (t4:
-T).(subst1 i u t2 t4)) in (let TMP_9 \def (\lambda (t4: T).(subst1 i u t t4))
-in (ex2 T TMP_8 TMP_9)))) in (let TMP_11 \def (\lambda (t: T).(subst1 i u t2
-t)) in (let TMP_12 \def (\lambda (t: T).(subst1 i u t0 t)) in (let TMP_13
-\def (subst1_refl i u t2) in (let TMP_14 \def (subst1_single i u t0 t2 H0) in
-(let TMP_15 \def (ex_intro2 T TMP_11 TMP_12 t2 TMP_13 TMP_14) in (let TMP_56
-\def (\lambda (t4: T).(\lambda (H2: (subst0 i u t0 t4)).(let TMP_16 \def (eq
-T t4 t2) in (let TMP_17 \def (\lambda (t: T).(subst0 i u t4 t)) in (let
-TMP_18 \def (\lambda (t: T).(subst0 i u t2 t)) in (let TMP_19 \def (ex2 T
-TMP_17 TMP_18) in (let TMP_20 \def (subst0 i u t4 t2) in (let TMP_21 \def
-(subst0 i u t2 t4) in (let TMP_22 \def (\lambda (t: T).(subst1 i u t2 t)) in
-(let TMP_23 \def (\lambda (t: T).(subst1 i u t4 t)) in (let TMP_24 \def (ex2
-T TMP_22 TMP_23) in (let TMP_33 \def (\lambda (H3: (eq T t4 t2)).(let TMP_27
-\def (\lambda (t: T).(let TMP_25 \def (\lambda (t5: T).(subst1 i u t2 t5)) in
-(let TMP_26 \def (\lambda (t5: T).(subst1 i u t t5)) in (ex2 T TMP_25
-TMP_26)))) in (let TMP_28 \def (\lambda (t: T).(subst1 i u t2 t)) in (let
-TMP_29 \def (\lambda (t: T).(subst1 i u t2 t)) in (let TMP_30 \def
-(subst1_refl i u t2) in (let TMP_31 \def (subst1_refl i u t2) in (let TMP_32
-\def (ex_intro2 T TMP_28 TMP_29 t2 TMP_30 TMP_31) in (eq_ind_r T t2 TMP_27
-TMP_32 t4 H3)))))))) in (let TMP_44 \def (\lambda (H3: (ex2 T (\lambda (t:
-T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 t)))).(let TMP_34 \def
-(\lambda (t: T).(subst0 i u t4 t)) in (let TMP_35 \def (\lambda (t:
-T).(subst0 i u t2 t)) in (let TMP_36 \def (\lambda (t: T).(subst1 i u t2 t))
-in (let TMP_37 \def (\lambda (t: T).(subst1 i u t4 t)) in (let TMP_38 \def
-(ex2 T TMP_36 TMP_37) in (let TMP_43 \def (\lambda (x: T).(\lambda (H4:
-(subst0 i u t4 x)).(\lambda (H5: (subst0 i u t2 x)).(let TMP_39 \def (\lambda
-(t: T).(subst1 i u t2 t)) in (let TMP_40 \def (\lambda (t: T).(subst1 i u t4
-t)) in (let TMP_41 \def (subst1_single i u t2 x H5) in (let TMP_42 \def
-(subst1_single i u t4 x H4) in (ex_intro2 T TMP_39 TMP_40 x TMP_41
-TMP_42)))))))) in (ex2_ind T TMP_34 TMP_35 TMP_38 TMP_43 H3)))))))) in (let
-TMP_49 \def (\lambda (H3: (subst0 i u t4 t2)).(let TMP_45 \def (\lambda (t:
-T).(subst1 i u t2 t)) in (let TMP_46 \def (\lambda (t: T).(subst1 i u t4 t))
-in (let TMP_47 \def (subst1_refl i u t2) in (let TMP_48 \def (subst1_single i
-u t4 t2 H3) in (ex_intro2 T TMP_45 TMP_46 t2 TMP_47 TMP_48)))))) in (let
-TMP_54 \def (\lambda (H3: (subst0 i u t2 t4)).(let TMP_50 \def (\lambda (t:
-T).(subst1 i u t2 t)) in (let TMP_51 \def (\lambda (t: T).(subst1 i u t4 t))
-in (let TMP_52 \def (subst1_single i u t2 t4 H3) in (let TMP_53 \def
-(subst1_refl i u t4) in (ex_intro2 T TMP_50 TMP_51 t4 TMP_52 TMP_53)))))) in
-(let TMP_55 \def (subst0_confluence_eq t0 t4 u i H2 t2 H0) in (or4_ind TMP_16
-TMP_19 TMP_20 TMP_21 TMP_24 TMP_33 TMP_44 TMP_49 TMP_54
-TMP_55))))))))))))))))) in (subst1_ind i u t0 TMP_10 TMP_15 TMP_56 t3
-H1)))))))))))) in (subst1_ind i u t0 TMP_3 TMP_7 TMP_57 t1 H)))))))).
+u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1
+i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t:
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u
+t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u
+t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t))
+(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4)
+(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)))
+(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda
+(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2
+T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2
+(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T
+(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i
+u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i
+u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5:
+(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda
+(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4
+x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t:
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u
+t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2
+t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1
+i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4)))
+(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))).
theorem subst1_confluence_lift:
\forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
t2)) \to (eq T t1 t2)))))))
\def
\lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u t0 (lift (S O) i t1))).(let TMP_1 \def (S O) in (let TMP_2
-\def (lift TMP_1 i t1) in (let TMP_3 \def (\lambda (t: T).(subst1 i u t0 t))
-in (let TMP_4 \def (\lambda (_: T).(\forall (t2: T).((subst1 i u t0 (lift (S
-O) i t2)) \to (eq T t1 t2)))) in (let TMP_70 \def (\lambda (y: T).(\lambda
-(H0: (subst1 i u t0 y)).(let TMP_5 \def (\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))))) in (let TMP_32 \def (\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
-TMP_8 \def (\lambda (t: T).(let TMP_6 \def (S O) in (let TMP_7 \def (lift
-TMP_6 i t2) in (subst1 i u t TMP_7)))) in (let TMP_9 \def (S O) in (let
-TMP_10 \def (lift TMP_9 i t1) in (let H3 \def (eq_ind T t0 TMP_8 H2 TMP_10
-H1) in (let TMP_11 \def (S O) in (let TMP_12 \def (lift TMP_11 i t2) in (let
-TMP_13 \def (S O) in (let TMP_14 \def (lift TMP_13 i t1) in (let TMP_15 \def
-(S O) in (let TMP_16 \def (lift TMP_15 i t2) in (let TMP_17 \def (S O) in
-(let TMP_18 \def (le_n i) in (let TMP_19 \def (S O) in (let TMP_20 \def (plus
-TMP_19 i) in (let TMP_21 \def (\lambda (n: nat).(lt i n)) in (let TMP_22 \def
-(S O) in (let TMP_23 \def (plus TMP_22 i) in (let TMP_24 \def (le_n TMP_23)
-in (let TMP_25 \def (S O) in (let TMP_26 \def (plus i TMP_25) in (let TMP_27
-\def (S O) in (let TMP_28 \def (plus_sym i TMP_27) in (let TMP_29 \def
-(eq_ind_r nat TMP_20 TMP_21 TMP_24 TMP_26 TMP_28) in (let TMP_30 \def
-(subst1_gen_lift_eq t1 u TMP_16 TMP_17 i i TMP_18 TMP_29 H3) in (let H4 \def
-(sym_eq T TMP_12 TMP_14 TMP_30) in (let TMP_31 \def (S O) in (lift_inj t1 t2
-TMP_31 i H4)))))))))))))))))))))))))))))) in (let TMP_69 \def (\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
-TMP_33 \def (\lambda (t: T).(subst0 i u t0 t)) in (let TMP_34 \def (S O) in
-(let TMP_35 \def (lift TMP_34 i t1) in (let H4 \def (eq_ind T t2 TMP_33 H1
-TMP_35 H2) in (let TMP_36 \def (S O) in (let TMP_37 \def (lift TMP_36 i t3)
-in (let TMP_38 \def (\lambda (t: T).(subst1 i u t0 t)) in (let TMP_39 \def
-(\lambda (_: T).(eq T t1 t3)) in (let TMP_68 \def (\lambda (y0: T).(\lambda
-(H5: (subst1 i u t0 y0)).(let TMP_40 \def (\lambda (t: T).((eq T t (lift (S
-O) i t3)) \to (eq T t1 t3))) in (let TMP_62 \def (\lambda (H6: (eq T t0 (lift
-(S O) i t3))).(let TMP_43 \def (\lambda (t: T).(let TMP_41 \def (S O) in (let
-TMP_42 \def (lift TMP_41 i t1) in (subst0 i u t TMP_42)))) in (let TMP_44
-\def (S O) in (let TMP_45 \def (lift TMP_44 i t3) in (let H7 \def (eq_ind T
-t0 TMP_43 H4 TMP_45 H6) in (let TMP_46 \def (S O) in (let TMP_47 \def (lift
-TMP_46 i t1) in (let TMP_48 \def (S O) in (let TMP_49 \def (le_n i) in (let
-TMP_50 \def (S O) in (let TMP_51 \def (plus TMP_50 i) in (let TMP_52 \def
-(\lambda (n: nat).(lt i n)) in (let TMP_53 \def (S O) in (let TMP_54 \def
-(plus TMP_53 i) in (let TMP_55 \def (le_n TMP_54) in (let TMP_56 \def (S O)
-in (let TMP_57 \def (plus i TMP_56) in (let TMP_58 \def (S O) in (let TMP_59
-\def (plus_sym i TMP_58) in (let TMP_60 \def (eq_ind_r nat TMP_51 TMP_52
-TMP_55 TMP_57 TMP_59) in (let TMP_61 \def (eq T t1 t3) in
-(subst0_gen_lift_false t3 u TMP_47 TMP_48 i i TMP_49 TMP_60 H7
-TMP_61)))))))))))))))))))))) in (let TMP_67 \def (\lambda (t4: T).(\lambda
-(H6: (subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let
-TMP_63 \def (\lambda (t: T).(subst0 i u t0 t)) in (let TMP_64 \def (S O) in
-(let TMP_65 \def (lift TMP_64 i t3) in (let H8 \def (eq_ind T t4 TMP_63 H6
-TMP_65 H7) in (let TMP_66 \def (subst0_confluence_lift t0 t3 u i H8 t1 H4) in
-(sym_eq T t3 t1 TMP_66))))))))) in (subst1_ind i u t0 TMP_40 TMP_62 TMP_67 y0
-H5)))))) in (insert_eq T TMP_37 TMP_38 TMP_39 TMP_68 H3))))))))))))))) in
-(subst1_ind i u t0 TMP_5 TMP_32 TMP_69 y H0)))))) in (insert_eq T TMP_2 TMP_3
-TMP_4 TMP_70 H)))))))))).
+(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)) (\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_sym 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)) (\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_sym 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))))).
projects / helm.git / blobdiff