include "basic_1/T/defs.ma".
-let rec T_rect (P: (T \to Type[0])) (f: (\forall (n: nat).(P (TSort n))))
-(f0: (\forall (n: nat).(P (TLRef n)))) (f1: (\forall (k: K).(\forall (t:
-T).((P t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) (t: T) on
-t: P t \def match t in T with [(TSort n) \Rightarrow (f n) | (TLRef n)
-\Rightarrow (f0 n) | (THead k t0 t1) \Rightarrow (let TMP_2 \def ((T_rect P f
-f0 f1) t0) in (let TMP_1 \def ((T_rect P f f0 f1) t1) in (f1 k t0 TMP_2 t1
-TMP_1)))].
+implied rec lemma T_rect (P: (T \to Type[0])) (f: (\forall (n: nat).(P (TSort
+n)))) (f0: (\forall (n: nat).(P (TLRef n)))) (f1: (\forall (k: K).(\forall
+(t: T).((P t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) (t:
+T) on t: P t \def match t with [(TSort n) \Rightarrow (f n) | (TLRef n)
+\Rightarrow (f0 n) | (THead k t0 t1) \Rightarrow (f1 k t0 ((T_rect P f f0 f1)
+t0) t1 ((T_rect P f f0 f1) t1))].
-theorem T_ind:
+implied lemma T_ind:
\forall (P: ((T \to Prop))).(((\forall (n: nat).(P (TSort n)))) \to
(((\forall (n: nat).(P (TLRef n)))) \to (((\forall (k: K).(\forall (t: T).((P
t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) \to (\forall (t:
\def
\lambda (P: ((T \to Prop))).(T_rect P).
-theorem thead_x_y_y:
+lemma thead_x_y_y:
\forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to
(\forall (P: Prop).P))))
\def
- \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(let TMP_676 \def (\lambda
-(t0: T).((eq T (THead k v t0) t0) \to (\forall (P: Prop).P))) in (let TMP_675
-\def (\lambda (n: nat).(\lambda (H: (eq T (THead k v (TSort n)) (TSort
-n))).(\lambda (P: Prop).(let TMP_673 \def (TSort n) in (let TMP_674 \def
-(THead k v TMP_673) in (let TMP_672 \def (\lambda (ee: T).(match ee in T with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) in (let TMP_671 \def (TSort n) in (let H0 \def (eq_ind T
-TMP_674 TMP_672 I TMP_671 H) in (False_ind P H0))))))))) in (let TMP_670 \def
-(\lambda (n: nat).(\lambda (H: (eq T (THead k v (TLRef n)) (TLRef
-n))).(\lambda (P: Prop).(let TMP_668 \def (TLRef n) in (let TMP_669 \def
-(THead k v TMP_668) in (let TMP_667 \def (\lambda (ee: T).(match ee in T with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) in (let TMP_666 \def (TLRef n) in (let H0 \def (eq_ind T
-TMP_669 TMP_667 I TMP_666 H) in (False_ind P H0))))))))) in (let TMP_665 \def
-(\lambda (k0: K).(\lambda (t0: T).(\lambda (_: (((eq T (THead k v t0) t0) \to
-(\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v
-t1) t1) \to (\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0
-t0 t1)) (THead k0 t0 t1))).(\lambda (P: Prop).(let TMP_652 \def (\lambda (e:
-T).(match e in T with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
-(THead k1 _ _) \Rightarrow k1])) in (let TMP_650 \def (THead k0 t0 t1) in
-(let TMP_651 \def (THead k v TMP_650) in (let TMP_649 \def (THead k0 t0 t1)
-in (let H2 \def (f_equal T K TMP_652 TMP_651 TMP_649 H1) in (let TMP_656 \def
-(\lambda (e: T).(match e in T with [(TSort _) \Rightarrow v | (TLRef _)
-\Rightarrow v | (THead _ t2 _) \Rightarrow t2])) in (let TMP_654 \def (THead
-k0 t0 t1) in (let TMP_655 \def (THead k v TMP_654) in (let TMP_653 \def
-(THead k0 t0 t1) in (let H3 \def (f_equal T T TMP_656 TMP_655 TMP_653 H1) in
-(let TMP_660 \def (\lambda (e: T).(match e in T with [(TSort _) \Rightarrow
-(THead k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2)
-\Rightarrow t2])) in (let TMP_658 \def (THead k0 t0 t1) in (let TMP_659 \def
-(THead k v TMP_658) in (let TMP_657 \def (THead k0 t0 t1) in (let H4 \def
-(f_equal T T TMP_660 TMP_659 TMP_657 H1) in (let TMP_663 \def (\lambda (H5:
-(eq T v t0)).(\lambda (H6: (eq K k k0)).(let TMP_661 \def (\lambda (t2:
-T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) in (let H7 \def
-(eq_ind T v TMP_661 H0 t0 H5) in (let TMP_662 \def (\lambda (k1: K).((eq T
-(THead k1 t0 t1) t1) \to (\forall (P0: Prop).P0))) in (let H8 \def (eq_ind K
-k TMP_662 H7 k0 H6) in (H8 H4 P))))))) in (let TMP_664 \def (TMP_663 H3) in
-(TMP_664 H2))))))))))))))))))))))))) in (T_ind TMP_676 TMP_675 TMP_670
-TMP_665 t))))))).
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq
+T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda
+(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H) in (False_ind P H0))))) (\lambda (n: nat).(\lambda (H:
+(eq T (THead k v (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (THead k v (TLRef n)) (\lambda (ee: T).(match ee with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0: K).(\lambda (t0:
+T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P:
+Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to
+(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1))
+(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead
+k1 _ _) \Rightarrow k1])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1)
+in ((let H3 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
+\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2]))
+(THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T
+T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead k0 t0 t1) |
+(TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) \Rightarrow t2]))
+(THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in (\lambda (H5: (eq T v
+t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T v (\lambda (t2:
+T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) H0 t0 H5) in (let
+H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0 t1) t1) \to (\forall
+(P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3)) H2))))))))) t))).