include "basic_1/tlist/defs.ma".
-theorem theads_tapp:
+lemma theads_tapp:
\forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T
(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
\def
- \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: TList).(let
-TMP_5 \def (\lambda (t0: TList).(let TMP_1 \def (TApp t0 v) in (let TMP_2
-\def (THeads k TMP_1 t) in (let TMP_3 \def (THead k v t) in (let TMP_4 \def
-(THeads k t0 TMP_3) in (eq T TMP_2 TMP_4)))))) in (let TMP_6 \def (THead k v
-t) in (let TMP_7 \def (refl_equal T TMP_6) in (let TMP_21 \def (\lambda (t0:
-T).(\lambda (t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k
-t1 (THead k v t)))).(let TMP_8 \def (TApp t1 v) in (let TMP_9 \def (THeads k
-TMP_8 t) in (let TMP_14 \def (\lambda (t2: T).(let TMP_10 \def (TApp t1 v) in
-(let TMP_11 \def (THeads k TMP_10 t) in (let TMP_12 \def (THead k t0 TMP_11)
-in (let TMP_13 \def (THead k t0 t2) in (eq T TMP_12 TMP_13)))))) in (let
-TMP_15 \def (TApp t1 v) in (let TMP_16 \def (THeads k TMP_15 t) in (let
-TMP_17 \def (THead k t0 TMP_16) in (let TMP_18 \def (refl_equal T TMP_17) in
-(let TMP_19 \def (THead k v t) in (let TMP_20 \def (THeads k t1 TMP_19) in
-(eq_ind T TMP_9 TMP_14 TMP_18 TMP_20 H))))))))))))) in (TList_ind TMP_5 TMP_7
-TMP_21 vs)))))))).
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads
+k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda
+(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k
+v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0
+(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads
+k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
-theorem tcons_tapp_ex:
+lemma tcons_tapp_ex:
\forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
\def
- \lambda (ts1: TList).(let TMP_7 \def (\lambda (t: TList).(\forall (t1:
-T).(let TMP_3 \def (\lambda (ts2: TList).(\lambda (t2: T).(let TMP_1 \def
-(TCons t1 t) in (let TMP_2 \def (TApp ts2 t2) in (eq TList TMP_1 TMP_2)))))
-in (let TMP_6 \def (\lambda (ts2: TList).(\lambda (_: T).(let TMP_4 \def
-(tslen t) in (let TMP_5 \def (tslen ts2) in (eq nat TMP_4 TMP_5))))) in
-(ex2_2 TList T TMP_3 TMP_6))))) in (let TMP_17 \def (\lambda (t1: T).(let
-TMP_10 \def (\lambda (ts2: TList).(\lambda (t2: T).(let TMP_8 \def (TCons t1
-TNil) in (let TMP_9 \def (TApp ts2 t2) in (eq TList TMP_8 TMP_9))))) in (let
-TMP_12 \def (\lambda (ts2: TList).(\lambda (_: T).(let TMP_11 \def (tslen
-ts2) in (eq nat O TMP_11)))) in (let TMP_13 \def (TApp TNil t1) in (let
-TMP_14 \def (refl_equal TList TMP_13) in (let TMP_15 \def (tslen TNil) in
-(let TMP_16 \def (refl_equal nat TMP_15) in (ex2_2_intro TList T TMP_10
-TMP_12 TNil t1 TMP_14 TMP_16)))))))) in (let TMP_71 \def (\lambda (t:
+ \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
+TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
+ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
+ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
+(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
+TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
-ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in (let
-TMP_20 \def (\lambda (ts2: TList).(\lambda (t2: T).(let TMP_18 \def (TCons t
-t0) in (let TMP_19 \def (TApp ts2 t2) in (eq TList TMP_18 TMP_19))))) in (let
-TMP_23 \def (\lambda (ts2: TList).(\lambda (_: T).(let TMP_21 \def (tslen t0)
-in (let TMP_22 \def (tslen ts2) in (eq nat TMP_21 TMP_22))))) in (let TMP_27
-\def (\lambda (ts2: TList).(\lambda (t2: T).(let TMP_24 \def (TCons t t0) in
-(let TMP_25 \def (TCons t1 TMP_24) in (let TMP_26 \def (TApp ts2 t2) in (eq
-TList TMP_25 TMP_26)))))) in (let TMP_31 \def (\lambda (ts2: TList).(\lambda
-(_: T).(let TMP_28 \def (tslen t0) in (let TMP_29 \def (S TMP_28) in (let
-TMP_30 \def (tslen ts2) in (eq nat TMP_29 TMP_30)))))) in (let TMP_32 \def
-(ex2_2 TList T TMP_27 TMP_31) in (let TMP_70 \def (\lambda (x0:
-TList).(\lambda (x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0
-x1))).(\lambda (H2: (eq nat (tslen t0) (tslen x0))).(let TMP_33 \def (TApp x0
-x1) in (let TMP_41 \def (\lambda (t2: TList).(let TMP_36 \def (\lambda (ts2:
-TList).(\lambda (t3: T).(let TMP_34 \def (TCons t1 t2) in (let TMP_35 \def
-(TApp ts2 t3) in (eq TList TMP_34 TMP_35))))) in (let TMP_40 \def (\lambda
-(ts2: TList).(\lambda (_: T).(let TMP_37 \def (tslen t0) in (let TMP_38 \def
-(S TMP_37) in (let TMP_39 \def (tslen ts2) in (eq nat TMP_38 TMP_39)))))) in
-(ex2_2 TList T TMP_36 TMP_40)))) in (let TMP_42 \def (tslen x0) in (let
-TMP_50 \def (\lambda (n: nat).(let TMP_46 \def (\lambda (ts2: TList).(\lambda
-(t2: T).(let TMP_43 \def (TApp x0 x1) in (let TMP_44 \def (TCons t1 TMP_43)
-in (let TMP_45 \def (TApp ts2 t2) in (eq TList TMP_44 TMP_45)))))) in (let
-TMP_49 \def (\lambda (ts2: TList).(\lambda (_: T).(let TMP_47 \def (S n) in
-(let TMP_48 \def (tslen ts2) in (eq nat TMP_47 TMP_48))))) in (ex2_2 TList T
-TMP_46 TMP_49)))) in (let TMP_54 \def (\lambda (ts2: TList).(\lambda (t2:
-T).(let TMP_51 \def (TApp x0 x1) in (let TMP_52 \def (TCons t1 TMP_51) in
-(let TMP_53 \def (TApp ts2 t2) in (eq TList TMP_52 TMP_53)))))) in (let
-TMP_58 \def (\lambda (ts2: TList).(\lambda (_: T).(let TMP_55 \def (tslen x0)
-in (let TMP_56 \def (S TMP_55) in (let TMP_57 \def (tslen ts2) in (eq nat
-TMP_56 TMP_57)))))) in (let TMP_59 \def (TCons t1 x0) in (let TMP_60 \def
-(TCons t1 x0) in (let TMP_61 \def (TApp TMP_60 x1) in (let TMP_62 \def
-(refl_equal TList TMP_61) in (let TMP_63 \def (TCons t1 x0) in (let TMP_64
-\def (tslen TMP_63) in (let TMP_65 \def (refl_equal nat TMP_64) in (let
-TMP_66 \def (ex2_2_intro TList T TMP_54 TMP_58 TMP_59 x1 TMP_62 TMP_65) in
-(let TMP_67 \def (tslen t0) in (let TMP_68 \def (eq_ind_r nat TMP_42 TMP_50
-TMP_66 TMP_67 H2) in (let TMP_69 \def (TCons t t0) in (eq_ind_r TList TMP_33
-TMP_41 TMP_68 TMP_69 H1)))))))))))))))))))))) in (ex2_2_ind TList T TMP_20
-TMP_23 TMP_32 TMP_70 H0))))))))))))) in (TList_ind TMP_7 TMP_17 TMP_71
-ts1)))).
+ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
+(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
+TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
+(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
+(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
+nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
+TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
+t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
+(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
+nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
+t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
+nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
+(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
+(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
+(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
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