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 with [(TSort n) \Rightarrow (f n) | (TLRef n) \Rightarrow
-(f0 n) | (THead k t0 t1) \Rightarrow (let TMP_1 \def ((T_rect P f f0 f1) t0)
-in (let TMP_2 \def ((T_rect P f f0 f1) t1) in (f1 k t0 TMP_1 t1 TMP_2)))].
+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_1 \def (\lambda (t0:
-T).((eq T (THead k v t0) t0) \to (\forall (P: Prop).P))) in (let TMP_6 \def
-(\lambda (n: nat).(\lambda (H: (eq T (THead k v (TSort n)) (TSort
-n))).(\lambda (P: Prop).(let TMP_2 \def (TSort n) in (let TMP_3 \def (THead k
-v TMP_2) in (let TMP_4 \def (\lambda (ee: T).(match ee with [(TSort _)
+ \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])) in (let TMP_5 \def (TSort n) in (let H0 \def (eq_ind T TMP_3 TMP_4 I
-TMP_5 H) in (False_ind P H0))))))))) in (let TMP_11 \def (\lambda (n:
-nat).(\lambda (H: (eq T (THead k v (TLRef n)) (TLRef n))).(\lambda (P:
-Prop).(let TMP_7 \def (TLRef n) in (let TMP_8 \def (THead k v TMP_7) in (let
-TMP_9 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in (let
-TMP_10 \def (TLRef n) in (let H0 \def (eq_ind T TMP_8 TMP_9 I TMP_10 H) in
-(False_ind P H0))))))))) in (let TMP_28 \def (\lambda (k0: K).(\lambda (t0:
+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 TMP_12 \def (\lambda (e: T).(match
-e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _)
-\Rightarrow k1])) in (let TMP_13 \def (THead k0 t0 t1) in (let TMP_14 \def
-(THead k v TMP_13) in (let TMP_15 \def (THead k0 t0 t1) in (let H2 \def
-(f_equal T K TMP_12 TMP_14 TMP_15 H1) in (let TMP_16 \def (\lambda (e:
-T).(match e with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead
-_ t2 _) \Rightarrow t2])) in (let TMP_17 \def (THead k0 t0 t1) in (let TMP_18
-\def (THead k v TMP_17) in (let TMP_19 \def (THead k0 t0 t1) in (let H3 \def
-(f_equal T T TMP_16 TMP_18 TMP_19 H1) in (let TMP_20 \def (\lambda (e:
-T).(match e with [(TSort _) \Rightarrow (THead k0 t0 t1) | (TLRef _)
-\Rightarrow (THead k0 t0 t1) | (THead _ _ t2) \Rightarrow t2])) in (let
-TMP_21 \def (THead k0 t0 t1) in (let TMP_22 \def (THead k v TMP_21) in (let
-TMP_23 \def (THead k0 t0 t1) in (let H4 \def (f_equal T T TMP_20 TMP_22
-TMP_23 H1) in (let TMP_26 \def (\lambda (H5: (eq T v t0)).(\lambda (H6: (eq K
-k k0)).(let TMP_24 \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_24 H0 t0 H5) in
-(let TMP_25 \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_25 H7 k0 H6) in (H8 H4
-P))))))) in (let TMP_27 \def (TMP_26 H3) in (TMP_27
-H2))))))))))))))))))))))))) in (T_ind TMP_1 TMP_6 TMP_11 TMP_28 t))))))).
+(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))).
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