include "basic_1/tlist/props.ma".
-theorem tslt_wf__q_ind:
+fact tslt_wf__q_ind:
\forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
-ts) n) \to (P ts)))))).(\lambda (ts: TList).(let TMP_1 \def (tslen ts) in
-(let TMP_2 \def (tslen ts) in (let TMP_3 \def (refl_equal nat TMP_2) in (H
-TMP_1 ts TMP_3))))))).
+ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
+(tslen ts)))))).
-theorem tslt_wf_ind:
+lemma tslt_wf_ind:
\forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
TList).(P ts)))
TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
-TList).(let TMP_1 \def (\lambda (t: TList).(P t)) in (let TMP_11 \def
-(\lambda (n: nat).(let TMP_2 \def (\lambda (t: TList).(P t)) in (let TMP_3
-\def (Q TMP_2) in (let TMP_10 \def (\lambda (n0: nat).(\lambda (H0: ((\forall
-(m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) m))))).(\lambda (ts0:
-TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let TMP_4 \def (\lambda (n1:
-nat).(\forall (m: nat).((lt m n1) \to (\forall (ts1: TList).((eq nat (tslen
-ts1) m) \to (P ts1)))))) in (let TMP_5 \def (tslen ts0) in (let H2 \def
-(eq_ind_r nat n0 TMP_4 H0 TMP_5 H1) in (let TMP_9 \def (\lambda (ts1:
-TList).(\lambda (H3: (lt (tslen ts1) (tslen ts0))).(let TMP_6 \def (tslen
-ts1) in (let TMP_7 \def (tslen ts1) in (let TMP_8 \def (refl_equal nat TMP_7)
-in (H2 TMP_6 H3 ts1 TMP_8)))))) in (H ts0 TMP_9))))))))) in (lt_wf_ind n
-TMP_3 TMP_10))))) in (tslt_wf__q_ind TMP_1 TMP_11 ts)))))).
+TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
+nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
+(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
+m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
+\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
+(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
+H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
+ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
-theorem tlist_ind_rev:
+lemma tlist_ind_rev:
\forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
TList).(P ts))))
\def
\lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
-t))))))).(\lambda (ts: TList).(let TMP_1 \def (\lambda (t: TList).(P t)) in
-(let TMP_28 \def (\lambda (ts2: TList).(let TMP_2 \def (\lambda (t:
-TList).(((\forall (ts1: TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) in
-(let TMP_3 \def (\lambda (_: ((\forall (ts1: TList).((tslt ts1 TNil) \to (P
-ts1))))).H) in (let TMP_27 \def (\lambda (t: T).(\lambda (t0: TList).(\lambda
-(_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) \to (P
-t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) \to (P
-ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in (let TMP_6
-\def (\lambda (ts3: TList).(\lambda (t2: T).(let TMP_4 \def (TCons t t0) in
-(let TMP_5 \def (TApp ts3 t2) in (eq TList TMP_4 TMP_5))))) in (let TMP_9
-\def (\lambda (ts3: TList).(\lambda (_: T).(let TMP_7 \def (tslen t0) in (let
-TMP_8 \def (tslen ts3) in (eq nat TMP_7 TMP_8))))) in (let TMP_10 \def (TCons
-t t0) in (let TMP_11 \def (P TMP_10) in (let TMP_26 \def (\lambda (x0:
-TList).(\lambda (x1: T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0
-x1))).(\lambda (H5: (eq nat (tslen t0) (tslen x0))).(let TMP_12 \def (TApp x0
-x1) in (let TMP_13 \def (\lambda (t1: TList).(P t1)) in (let TMP_14 \def
-(tslen t0) in (let TMP_17 \def (\lambda (n: nat).(let TMP_15 \def (TCons t
-t0) in (let TMP_16 \def (tslen TMP_15) in (lt n TMP_16)))) in (let TMP_18
-\def (TCons t t0) in (let TMP_19 \def (tslen TMP_18) in (let TMP_20 \def
-(le_n TMP_19) in (let TMP_21 \def (tslen x0) in (let TMP_22 \def (eq_ind nat
-TMP_14 TMP_17 TMP_20 TMP_21 H5) in (let TMP_23 \def (H2 x0 TMP_22) in (let
-TMP_24 \def (H0 x0 x1 TMP_23) in (let TMP_25 \def (TCons t t0) in (eq_ind_r
-TList TMP_12 TMP_13 TMP_24 TMP_25 H4))))))))))))))))) in (ex2_2_ind TList T
-TMP_6 TMP_9 TMP_11 TMP_26 H3)))))))))))) in (TList_ind TMP_2 TMP_3 TMP_27
-ts2))))) in (tslt_wf_ind TMP_1 TMP_28 ts)))))).
+t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
+(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
+TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
+TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
+\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
+\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
+(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
+T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
+(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
+t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
+(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
+H4))))) H3))))))) ts2)) ts)))).
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