include "basic_1/subst0/fwd.ma".
-theorem subst1_ind:
+implied lemma subst1_ind:
\forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (P: ((T \to
Prop))).((P t1) \to (((\forall (t2: T).((subst0 i v t1 t2) \to (P t2)))) \to
(\forall (t: T).((subst1 i v t1 t) \to (P t))))))))
with [subst1_refl \Rightarrow f | (subst1_single x x0) \Rightarrow (f0 x
x0)])))))))).
-theorem subst1_gen_sort:
+lemma subst1_gen_sort:
\forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
i v (TSort n) x) \to (eq T x (TSort n))))))
\def
\lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst1 i v (TSort n) x)).(let TMP_1 \def (TSort n) in (let TMP_3 \def
-(\lambda (t: T).(let TMP_2 \def (TSort n) in (eq T t TMP_2))) in (let TMP_4
-\def (TSort n) in (let TMP_5 \def (refl_equal T TMP_4) in (let TMP_8 \def
-(\lambda (t2: T).(\lambda (H0: (subst0 i v (TSort n) t2)).(let TMP_6 \def
-(TSort n) in (let TMP_7 \def (eq T t2 TMP_6) in (subst0_gen_sort v t2 i n H0
-TMP_7))))) in (subst1_ind i v TMP_1 TMP_3 TMP_5 TMP_8 x H)))))))))).
+(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T
+t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0
+i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x
+H))))).
-theorem subst1_gen_lref:
+lemma subst1_gen_lref:
\forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift
(S n) O v))))))))
\def
\lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst1 i v (TLRef n) x)).(let TMP_1 \def (TLRef n) in (let TMP_9 \def
-(\lambda (t: T).(let TMP_2 \def (TLRef n) in (let TMP_3 \def (eq T t TMP_2)
-in (let TMP_4 \def (eq nat n i) in (let TMP_5 \def (S n) in (let TMP_6 \def
-(lift TMP_5 O v) in (let TMP_7 \def (eq T t TMP_6) in (let TMP_8 \def (land
-TMP_4 TMP_7) in (or TMP_3 TMP_8))))))))) in (let TMP_10 \def (TLRef n) in
-(let TMP_11 \def (TLRef n) in (let TMP_12 \def (eq T TMP_10 TMP_11) in (let
-TMP_13 \def (eq nat n i) in (let TMP_14 \def (TLRef n) in (let TMP_15 \def (S
-n) in (let TMP_16 \def (lift TMP_15 O v) in (let TMP_17 \def (eq T TMP_14
-TMP_16) in (let TMP_18 \def (land TMP_13 TMP_17) in (let TMP_19 \def (TLRef
-n) in (let TMP_20 \def (refl_equal T TMP_19) in (let TMP_21 \def (or_introl
-TMP_12 TMP_18 TMP_20) in (let TMP_48 \def (\lambda (t2: T).(\lambda (H0:
-(subst0 i v (TLRef n) t2)).(let TMP_22 \def (eq nat n i) in (let TMP_23 \def
-(S n) in (let TMP_24 \def (lift TMP_23 O v) in (let TMP_25 \def (eq T t2
-TMP_24) in (let TMP_26 \def (TLRef n) in (let TMP_27 \def (eq T t2 TMP_26) in
-(let TMP_28 \def (eq nat n i) in (let TMP_29 \def (S n) in (let TMP_30 \def
-(lift TMP_29 O v) in (let TMP_31 \def (eq T t2 TMP_30) in (let TMP_32 \def
-(land TMP_28 TMP_31) in (let TMP_33 \def (or TMP_27 TMP_32) in (let TMP_46
-\def (\lambda (H1: (eq nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O
-v))).(let TMP_34 \def (TLRef n) in (let TMP_35 \def (eq T t2 TMP_34) in (let
-TMP_36 \def (eq nat n i) in (let TMP_37 \def (S n) in (let TMP_38 \def (lift
-TMP_37 O v) in (let TMP_39 \def (eq T t2 TMP_38) in (let TMP_40 \def (land
-TMP_36 TMP_39) in (let TMP_41 \def (eq nat n i) in (let TMP_42 \def (S n) in
-(let TMP_43 \def (lift TMP_42 O v) in (let TMP_44 \def (eq T t2 TMP_43) in
-(let TMP_45 \def (conj TMP_41 TMP_44 H1 H2) in (or_intror TMP_35 TMP_40
-TMP_45))))))))))))))) in (let TMP_47 \def (subst0_gen_lref v t2 i n H0) in
-(land_ind TMP_22 TMP_25 TMP_33 TMP_46 TMP_47))))))))))))))))) in (subst1_ind
-i v TMP_1 TMP_9 TMP_21 TMP_48 x H)))))))))))))))))))).
+(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or
+(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl
+(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O
+v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v
+(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq
+nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i)
+(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x
+H))))).
-theorem subst1_gen_head:
+lemma subst1_gen_head:
\forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T
(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
T).(subst1 (s k i) v t1 t2))))))))))
\def
\lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) x)).(let
-TMP_1 \def (THead k u1 t1) in (let TMP_7 \def (\lambda (t: T).(let TMP_3 \def
-(\lambda (u2: T).(\lambda (t2: T).(let TMP_2 \def (THead k u2 t2) in (eq T t
-TMP_2)))) in (let TMP_4 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1
-u2))) in (let TMP_6 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_5 \def (s
-k i) in (subst1 TMP_5 v t1 t2)))) in (ex3_2 T T TMP_3 TMP_4 TMP_6))))) in
-(let TMP_10 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_8 \def (THead k
-u1 t1) in (let TMP_9 \def (THead k u2 t2) in (eq T TMP_8 TMP_9))))) in (let
-TMP_11 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in (let
-TMP_13 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_12 \def (s k i) in
-(subst1 TMP_12 v t1 t2)))) in (let TMP_14 \def (THead k u1 t1) in (let TMP_15
-\def (refl_equal T TMP_14) in (let TMP_16 \def (subst1_refl i v u1) in (let
-TMP_17 \def (s k i) in (let TMP_18 \def (subst1_refl TMP_17 v t1) in (let
-TMP_19 \def (ex3_2_intro T T TMP_10 TMP_11 TMP_13 u1 t1 TMP_15 TMP_16 TMP_18)
-in (let TMP_102 \def (\lambda (t2: T).(\lambda (H0: (subst0 i v (THead k u1
-t1) t2)).(let TMP_21 \def (\lambda (u2: T).(let TMP_20 \def (THead k u2 t1)
-in (eq T t2 TMP_20))) in (let TMP_22 \def (\lambda (u2: T).(subst0 i v u1
-u2)) in (let TMP_23 \def (ex2 T TMP_21 TMP_22) in (let TMP_25 \def (\lambda
-(t3: T).(let TMP_24 \def (THead k u1 t3) in (eq T t2 TMP_24))) in (let TMP_27
-\def (\lambda (t3: T).(let TMP_26 \def (s k i) in (subst0 TMP_26 v t1 t3)))
-in (let TMP_28 \def (ex2 T TMP_25 TMP_27) in (let TMP_30 \def (\lambda (u2:
-T).(\lambda (t3: T).(let TMP_29 \def (THead k u2 t3) in (eq T t2 TMP_29))))
-in (let TMP_31 \def (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) in
-(let TMP_33 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_32 \def (s k i) in
-(subst0 TMP_32 v t1 t3)))) in (let TMP_34 \def (ex3_2 T T TMP_30 TMP_31
-TMP_33) in (let TMP_36 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_35
-\def (THead k u2 t3) in (eq T t2 TMP_35)))) in (let TMP_37 \def (\lambda (u2:
-T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_39 \def (\lambda (_:
-T).(\lambda (t3: T).(let TMP_38 \def (s k i) in (subst1 TMP_38 v t1 t3)))) in
-(let TMP_40 \def (ex3_2 T T TMP_36 TMP_37 TMP_39) in (let TMP_59 \def
-(\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda
-(u2: T).(subst0 i v u1 u2)))).(let TMP_42 \def (\lambda (u2: T).(let TMP_41
-\def (THead k u2 t1) in (eq T t2 TMP_41))) in (let TMP_43 \def (\lambda (u2:
-T).(subst0 i v u1 u2)) in (let TMP_45 \def (\lambda (u2: T).(\lambda (t3:
-T).(let TMP_44 \def (THead k u2 t3) in (eq T t2 TMP_44)))) in (let TMP_46
-\def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_48
-\def (\lambda (_: T).(\lambda (t3: T).(let TMP_47 \def (s k i) in (subst1
-TMP_47 v t1 t3)))) in (let TMP_49 \def (ex3_2 T T TMP_45 TMP_46 TMP_48) in
-(let TMP_58 \def (\lambda (x0: T).(\lambda (H2: (eq T t2 (THead k x0
-t1))).(\lambda (H3: (subst0 i v u1 x0)).(let TMP_51 \def (\lambda (u2:
-T).(\lambda (t3: T).(let TMP_50 \def (THead k u2 t3) in (eq T t2 TMP_50))))
-in (let TMP_52 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in
-(let TMP_54 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_53 \def (s k i) in
-(subst1 TMP_53 v t1 t3)))) in (let TMP_55 \def (subst1_single i v u1 x0 H3)
-in (let TMP_56 \def (s k i) in (let TMP_57 \def (subst1_refl TMP_56 v t1) in
-(ex3_2_intro T T TMP_51 TMP_52 TMP_54 x0 t1 H2 TMP_55 TMP_57)))))))))) in
-(ex2_ind T TMP_42 TMP_43 TMP_49 TMP_58 H1))))))))) in (let TMP_79 \def
-(\lambda (H1: (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda
-(t3: T).(subst0 (s k i) v t1 t3)))).(let TMP_61 \def (\lambda (t3: T).(let
-TMP_60 \def (THead k u1 t3) in (eq T t2 TMP_60))) in (let TMP_63 \def
-(\lambda (t3: T).(let TMP_62 \def (s k i) in (subst0 TMP_62 v t1 t3))) in
-(let TMP_65 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_64 \def (THead k
-u2 t3) in (eq T t2 TMP_64)))) in (let TMP_66 \def (\lambda (u2: T).(\lambda
-(_: T).(subst1 i v u1 u2))) in (let TMP_68 \def (\lambda (_: T).(\lambda (t3:
-T).(let TMP_67 \def (s k i) in (subst1 TMP_67 v t1 t3)))) in (let TMP_69 \def
-(ex3_2 T T TMP_65 TMP_66 TMP_68) in (let TMP_78 \def (\lambda (x0:
+(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1)
+x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1
+t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1
+t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal
+T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda
+(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2
+(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda
+(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1
+x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1
+x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3:
+T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0:
T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v
-t1 x0)).(let TMP_71 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_70 \def
-(THead k u2 t3) in (eq T t2 TMP_70)))) in (let TMP_72 \def (\lambda (u2:
-T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_74 \def (\lambda (_:
-T).(\lambda (t3: T).(let TMP_73 \def (s k i) in (subst1 TMP_73 v t1 t3)))) in
-(let TMP_75 \def (subst1_refl i v u1) in (let TMP_76 \def (s k i) in (let
-TMP_77 \def (subst1_single TMP_76 v t1 x0 H3) in (ex3_2_intro T T TMP_71
-TMP_72 TMP_74 u1 x0 H2 TMP_75 TMP_77)))))))))) in (ex2_ind T TMP_61 TMP_63
-TMP_69 TMP_78 H1))))))))) in (let TMP_100 \def (\lambda (H1: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
+t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1)
+(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s k i) v t1 t3))))).(let TMP_81 \def (\lambda (u2: T).(\lambda
-(t3: T).(let TMP_80 \def (THead k u2 t3) in (eq T t2 TMP_80)))) in (let
-TMP_82 \def (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) in (let
-TMP_84 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_83 \def (s k i) in
-(subst0 TMP_83 v t1 t3)))) in (let TMP_86 \def (\lambda (u2: T).(\lambda (t3:
-T).(let TMP_85 \def (THead k u2 t3) in (eq T t2 TMP_85)))) in (let TMP_87
-\def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_89
-\def (\lambda (_: T).(\lambda (t3: T).(let TMP_88 \def (s k i) in (subst1
-TMP_88 v t1 t3)))) in (let TMP_90 \def (ex3_2 T T TMP_86 TMP_87 TMP_89) in
-(let TMP_99 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2
-(THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda (H4: (subst0 (s
-k i) v t1 x1)).(let TMP_92 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_91
-\def (THead k u2 t3) in (eq T t2 TMP_91)))) in (let TMP_93 \def (\lambda (u2:
-T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_95 \def (\lambda (_:
-T).(\lambda (t3: T).(let TMP_94 \def (s k i) in (subst1 TMP_94 v t1 t3)))) in
-(let TMP_96 \def (subst1_single i v u1 x0 H3) in (let TMP_97 \def (s k i) in
-(let TMP_98 \def (subst1_single TMP_97 v t1 x1 H4) in (ex3_2_intro T T TMP_92
-TMP_93 TMP_95 x0 x1 H2 TMP_96 TMP_98)))))))))))) in (ex3_2_ind T T TMP_81
-TMP_82 TMP_84 TMP_90 TMP_99 H1)))))))))) in (let TMP_101 \def
-(subst0_gen_head k v u1 t1 t2 i H0) in (or3_ind TMP_23 TMP_28 TMP_34 TMP_40
-TMP_59 TMP_79 TMP_100 TMP_101))))))))))))))))))))) in (subst1_ind i v TMP_1
-TMP_7 TMP_19 TMP_102 x H))))))))))))))))))).
+T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda
+(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1
+i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0
+x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4)))))))
+H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
-theorem subst1_gen_lift_lt:
+lemma subst1_gen_lift_lt:
\forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1)
x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
\def
\lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S
-(plus i d)) t1) x)).(let TMP_1 \def (lift h d u) in (let TMP_2 \def (plus i
-d) in (let TMP_3 \def (S TMP_2) in (let TMP_4 \def (lift h TMP_3 t1) in (let
-TMP_10 \def (\lambda (t: T).(let TMP_8 \def (\lambda (t2: T).(let TMP_5 \def
-(plus i d) in (let TMP_6 \def (S TMP_5) in (let TMP_7 \def (lift h TMP_6 t2)
-in (eq T t TMP_7))))) in (let TMP_9 \def (\lambda (t2: T).(subst1 i u t1 t2))
-in (ex2 T TMP_8 TMP_9)))) in (let TMP_17 \def (\lambda (t2: T).(let TMP_11
-\def (plus i d) in (let TMP_12 \def (S TMP_11) in (let TMP_13 \def (lift h
-TMP_12 t1) in (let TMP_14 \def (plus i d) in (let TMP_15 \def (S TMP_14) in
-(let TMP_16 \def (lift h TMP_15 t2) in (eq T TMP_13 TMP_16)))))))) in (let
-TMP_18 \def (\lambda (t2: T).(subst1 i u t1 t2)) in (let TMP_19 \def (plus i
-d) in (let TMP_20 \def (S TMP_19) in (let TMP_21 \def (lift h TMP_20 t1) in
-(let TMP_22 \def (refl_equal T TMP_21) in (let TMP_23 \def (subst1_refl i u
-t1) in (let TMP_24 \def (ex_intro2 T TMP_17 TMP_18 t1 TMP_22 TMP_23) in (let
-TMP_44 \def (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) (lift h (S
-(plus i d)) t1) t2)).(let TMP_28 \def (\lambda (t3: T).(let TMP_25 \def (plus
-i d) in (let TMP_26 \def (S TMP_25) in (let TMP_27 \def (lift h TMP_26 t3) in
-(eq T t2 TMP_27))))) in (let TMP_29 \def (\lambda (t3: T).(subst0 i u t1 t3))
-in (let TMP_33 \def (\lambda (t3: T).(let TMP_30 \def (plus i d) in (let
-TMP_31 \def (S TMP_30) in (let TMP_32 \def (lift h TMP_31 t3) in (eq T t2
-TMP_32))))) in (let TMP_34 \def (\lambda (t3: T).(subst1 i u t1 t3)) in (let
-TMP_35 \def (ex2 T TMP_33 TMP_34) in (let TMP_42 \def (\lambda (x0:
-T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) x0))).(\lambda (H2: (subst0
-i u t1 x0)).(let TMP_39 \def (\lambda (t3: T).(let TMP_36 \def (plus i d) in
-(let TMP_37 \def (S TMP_36) in (let TMP_38 \def (lift h TMP_37 t3) in (eq T
-t2 TMP_38))))) in (let TMP_40 \def (\lambda (t3: T).(subst1 i u t1 t3)) in
-(let TMP_41 \def (subst1_single i u t1 x0 H2) in (ex_intro2 T TMP_39 TMP_40
-x0 H1 TMP_41))))))) in (let TMP_43 \def (subst0_gen_lift_lt u t1 t2 i h d H0)
-in (ex2_ind T TMP_28 TMP_29 TMP_35 TMP_42 TMP_43)))))))))) in (subst1_ind i
-TMP_1 TMP_4 TMP_10 TMP_24 TMP_44 x H))))))))))))))))))))).
+(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1)
+(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2:
+T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1))
+(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u)
+(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h
+(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1
+t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d))
+x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T
+t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1
+(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x
+H))))))).
-theorem subst1_gen_lift_eq:
+lemma subst1_gen_lift_eq:
\forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u
(lift h d t) x) \to (eq T x (lift h d t))))))))))
\def
\lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d
-h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(let TMP_1 \def (lift h d t)
-in (let TMP_3 \def (\lambda (t0: T).(let TMP_2 \def (lift h d t) in (eq T t0
-TMP_2))) in (let TMP_4 \def (lift h d t) in (let TMP_5 \def (refl_equal T
-TMP_4) in (let TMP_8 \def (\lambda (t2: T).(\lambda (H2: (subst0 i u (lift h
-d t) t2)).(let TMP_6 \def (lift h d t) in (let TMP_7 \def (eq T t2 TMP_6) in
-(subst0_gen_lift_false t u t2 h d i H H0 H2 TMP_7))))) in (subst1_ind i u
-TMP_1 TMP_3 TMP_5 TMP_8 x H1)))))))))))))).
+h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t)
+(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda
+(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t
+u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
-theorem subst1_gen_lift_ge:
+lemma subst1_gen_lift_ge:
\forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h)
i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
\def
\lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1)
-x)).(\lambda (H0: (le (plus d h) i)).(let TMP_1 \def (lift h d t1) in (let
-TMP_6 \def (\lambda (t: T).(let TMP_3 \def (\lambda (t2: T).(let TMP_2 \def
-(lift h d t2) in (eq T t TMP_2))) in (let TMP_5 \def (\lambda (t2: T).(let
-TMP_4 \def (minus i h) in (subst1 TMP_4 u t1 t2))) in (ex2 T TMP_3 TMP_5))))
-in (let TMP_9 \def (\lambda (t2: T).(let TMP_7 \def (lift h d t1) in (let
-TMP_8 \def (lift h d t2) in (eq T TMP_7 TMP_8)))) in (let TMP_11 \def
-(\lambda (t2: T).(let TMP_10 \def (minus i h) in (subst1 TMP_10 u t1 t2))) in
-(let TMP_12 \def (lift h d t1) in (let TMP_13 \def (refl_equal T TMP_12) in
-(let TMP_14 \def (minus i h) in (let TMP_15 \def (subst1_refl TMP_14 u t1) in
-(let TMP_16 \def (ex_intro2 T TMP_9 TMP_11 t1 TMP_13 TMP_15) in (let TMP_34
-\def (\lambda (t2: T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(let
-TMP_18 \def (\lambda (t3: T).(let TMP_17 \def (lift h d t3) in (eq T t2
-TMP_17))) in (let TMP_20 \def (\lambda (t3: T).(let TMP_19 \def (minus i h)
-in (subst0 TMP_19 u t1 t3))) in (let TMP_22 \def (\lambda (t3: T).(let TMP_21
-\def (lift h d t3) in (eq T t2 TMP_21))) in (let TMP_24 \def (\lambda (t3:
-T).(let TMP_23 \def (minus i h) in (subst1 TMP_23 u t1 t3))) in (let TMP_25
-\def (ex2 T TMP_22 TMP_24) in (let TMP_32 \def (\lambda (x0: T).(\lambda (H2:
-(eq T t2 (lift h d x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(let
-TMP_27 \def (\lambda (t3: T).(let TMP_26 \def (lift h d t3) in (eq T t2
-TMP_26))) in (let TMP_29 \def (\lambda (t3: T).(let TMP_28 \def (minus i h)
-in (subst1 TMP_28 u t1 t3))) in (let TMP_30 \def (minus i h) in (let TMP_31
-\def (subst1_single TMP_30 u t1 x0 H3) in (ex_intro2 T TMP_27 TMP_29 x0 H2
-TMP_31)))))))) in (let TMP_33 \def (subst0_gen_lift_ge u t1 t2 i h d H1 H0)
-in (ex2_ind T TMP_18 TMP_20 TMP_25 TMP_32 TMP_33)))))))))) in (subst1_ind i u
-TMP_1 TMP_6 TMP_16 TMP_34 x H)))))))))))))))))).
+x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2:
+T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
+h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1
+(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2:
+T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3:
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3))
+(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1
+(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d
+x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3:
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0
+H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h
+d H1 H0)))) x H)))))))).
projects / helm.git / blobdiff