include "lift/props.ma".
-theorem subst0_gen_sort:
+axiom subst0_gen_sort:
\forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
i v (TSort n) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(let H0 \def (match H in
-subst0 return (\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda
-(t1: T).(\lambda (_: (subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to
-((eq T t0 (TSort n)) \to ((eq T t1 x) \to P))))))))) with [(subst0_lref v0
-i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0
-v)).(\lambda (H2: (eq T (TLRef i0) (TSort n))).(\lambda (H3: (eq T (lift (S
-i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef
-n0) (TSort n)) \to ((eq T (lift (S n0) O v0) x) \to P)))) (\lambda (H4: (eq T
-v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TSort n)) \to ((eq T
-(lift (S i) O t) x) \to P))) (\lambda (H5: (eq T (TLRef i) (TSort n))).(let
-H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (TSort n) H5) in (False_ind ((eq T (lift
-(S i) O v) x) \to P) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1
-H2 H3))))) | (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq
-nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t)
-(TSort n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda
-(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead
-k u2 t) x) \to ((subst0 n0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0
-v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k u1 t) (TSort n)) \to ((eq T
-(THead k u2 t) x) \to ((subst0 i t0 u1 u2) \to P)))) (\lambda (H6: (eq T
-(THead k u1 t) (TSort n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TSort n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2)
-\to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4
-H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat
-i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TSort
-n))).(\lambda (H4: (eq T (THead k u t2) x)).(eq_ind nat i (\lambda (n0:
-nat).((eq T v0 v) \to ((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u
-t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to P))))) (\lambda (H5: (eq T v0
-v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TSort n)) \to ((eq T
-(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to P)))) (\lambda (H6: (eq T
-(THead k u t1) (TSort n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TSort n) H6) in (False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v
-t1 t2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4
-H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2:
-(eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1)
-(TSort n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda
-(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TSort n)) \to ((eq T
-(THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2)
-\to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T
-(THead k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1
-u2) \to ((subst0 (s k i) t t1 t2) \to P))))) (\lambda (H7: (eq T (THead k u1
-t1) (TSort n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
-H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to
-((subst0 (s k i) v t1 t2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq
-nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v)
-(refl_equal T (TSort n)) (refl_equal T x)))))))).
+.
-theorem subst0_gen_lref:
+axiom subst0_gen_lref:
\forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
i v (TLRef n) x) \to (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: (subst0 i v (TLRef n) x)).(let H0 \def (match H in subst0 return (\lambda
-(n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_:
-(subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TLRef n))
-\to ((eq T t1 x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))))))))
-with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda
-(H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TLRef n))).(\lambda (H3:
-(eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v)
-\to ((eq T (TLRef n0) (TLRef n)) \to ((eq T (lift (S n0) O v0) x) \to (land
-(eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H4: (eq T v0
-v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift
-(S i) O t) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) (\lambda
-(H5: (eq T (TLRef i) (TLRef n))).(let H6 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n)
-H5) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) x) \to (land
-(eq nat n n0) (eq T x (lift (S n) O v))))) (\lambda (H7: (eq T (lift (S n) O
-v) x)).(eq_ind T (lift (S n) O v) (\lambda (t: T).(land (eq nat n n) (eq T t
-(lift (S n) O v)))) (conj (eq nat n n) (eq T (lift (S n) O v) (lift (S n) O
-v)) (refl_equal nat n) (refl_equal T (lift (S n) O v))) x H7)) i (sym_eq nat
-i n H6)))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) |
-(subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0
-i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TLRef
-n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0:
-nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2
-t) x) \to ((subst0 n0 v0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O
-v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T
-(THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1
-u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T
-(THead k u1 t) (TLRef n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TLRef n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2)
-\to (land (eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v
-H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0
-u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
-v)).(\lambda (H3: (eq T (THead k u t1) (TLRef n))).(\lambda (H4: (eq T (THead
-k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k
-u t1) (TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2)
-\to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0
-v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TLRef n)) \to ((eq T
-(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T
-x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u t1) (TLRef n))).(let
-H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in
-(False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to (land
-(eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0
-(sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2
-H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0
-v)).(\lambda (H4: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H5: (eq T
-(THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T
-(THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 n0 v0
-u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to (land (eq nat n i) (eq T x (lift
-(S n) O v))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t:
-T).((eq T (THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to
-((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq
-T x (lift (S n) O v)))))))) (\lambda (H7: (eq T (THead k u1 t1) (TLRef
-n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in
-(False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s
-k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) H8))) v0
-(sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0
-(refl_equal nat i) (refl_equal T v) (refl_equal T (TLRef n)) (refl_equal T
-x))))))).
+.
-theorem subst0_gen_head:
+axiom subst0_gen_head:
\forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T
(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (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: (subst0 i v (THead k u1 t1) x)).(let H0
-\def (match H in subst0 return (\lambda (n: nat).(\lambda (t: T).(\lambda
-(t0: T).(\lambda (t2: T).(\lambda (_: (subst0 n t t0 t2)).((eq nat n i) \to
-((eq T t v) \to ((eq T t0 (THead k u1 t1)) \to ((eq T t2 x) \to (or3 (ex2 T
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
-u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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))))))))))))))
-with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda
-(H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (THead k u1 t1))).(\lambda
-(H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0
-v) \to ((eq T (TLRef n) (THead k u1 t1)) \to ((eq T (lift (S n) O v0) x) \to
-(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
-T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
-t2))))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T
-(TLRef i) (THead k u1 t1)) \to ((eq T (lift (S i) O t) x) \to (or3 (ex2 T
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
-u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
-T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))
-(\lambda (H5: (eq T (TLRef i) (THead k u1 t1))).(let H6 \def (eq_ind T (TLRef
-i) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead k u1 t1) H5) in (False_ind ((eq T (lift (S i) O v) x) \to
-(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
-T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
-t2)))))) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3)))))
-| (subst0_fst v0 u2 u0 i0 H0 t k0) \Rightarrow (\lambda (H1: (eq nat i0
-i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k
-u1 t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(eq_ind nat i (\lambda (n:
-nat).((eq T v0 v) \to ((eq T (THead k0 u0 t) (THead k u1 t1)) \to ((eq T
-(THead k0 u2 t) x) \to ((subst0 n v0 u0 u2) \to (or3 (ex2 T (\lambda (u3:
-T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T
-(\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v
-t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3
-t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H5: (eq T v0
-v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k0 u0 t) (THead k u1 t1)) \to
-((eq T (THead k0 u2 t) x) \to ((subst0 i t0 u0 u2) \to (or3 (ex2 T (\lambda
-(u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2
-T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i)
-v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3
-t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H6: (eq T
-(THead k0 u0 t) (THead k u1 t1))).(let H7 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t |
-(TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t)
-(THead k u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t) (THead k u1
-t1) H6) in ((let H9 \def (f_equal T K (\lambda (e: T).(match e in T return
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0
-| (THead k _ _) \Rightarrow k])) (THead k0 u0 t) (THead k u1 t1) H6) in
-(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t t1) \to ((eq T (THead
-k1 u2 t) x) \to ((subst0 i v u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x
-(THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2:
-T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)))
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H10: (eq T u0
-u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t t1) \to ((eq T (THead k u2 t) x)
-\to ((subst0 i v t0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3
-t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H11: (eq T t t1)).(eq_ind T t1
-(\lambda (t0: T).((eq T (THead k u2 t0) x) \to ((subst0 i v u1 u2) \to (or3
-(ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i
-v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
-T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1
-u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))
-(\lambda (H12: (eq T (THead k u2 t1) x)).(eq_ind T (THead k u2 t1) (\lambda
-(t0: T).((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead
-k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq
-T t0 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2))))))) (\lambda (H13: (subst0 i v u1
-u2)).(or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3
-t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T
-(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)))
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) (THead k
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (u3:
-T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1
-u3)) u2 (refl_equal T (THead k u2 t1)) H13))) x H12)) t (sym_eq T t t1 H11)))
-u0 (sym_eq T u0 u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0
-v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k0 v0 t2 t0 i0
-H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
-v)).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 t1))).(\lambda (H4: (eq T
-(THead k0 u t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T
-(THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0
-(s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq
-T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0
-(s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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))))))))) (\lambda (H6: (eq T (THead k0 u t0) (THead k u1 t1))).(let
-H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H8 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H9 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _)
-\Rightarrow k])) (THead k0 u t0) (THead k u1 t1) H6) in (eq_ind K k (\lambda
-(k1: K).((eq T u u1) \to ((eq T t0 t1) \to ((eq T (THead k1 u t2) x) \to
-((subst0 (s k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k
-u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T
-x (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 x (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)))))))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1
-(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead k t t2) x) \to ((subst0 (s k
-i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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))))))))) (\lambda (H11: (eq T t0 t1)).(eq_ind T t1 (\lambda (t:
-T).((eq T (THead k u1 t2) x) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
-u2))) (ex2 T (\lambda (t3: T).(eq T x (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 x (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))))))))
-(\lambda (H12: (eq T (THead k u1 t2) x)).(eq_ind T (THead k u1 t2) (\lambda
-(t: T).((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T t
-(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3:
-T).(eq T t (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 t (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))))))) (\lambda (H13: (subst0 (s
-k i) v t1 t2)).(or3_intro1 (ex2 T (\lambda (u2: T).(eq T (THead k u1 t2)
-(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3:
-T).(eq T (THead k u1 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 (THead k u1 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)))) (ex_intro2 T
-(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3:
-T).(subst0 (s k i) v t1 t3)) t2 (refl_equal T (THead k u1 t2)) H13))) x H12))
-t0 (sym_eq T t0 t1 H11))) u (sym_eq T u u1 H10))) k0 (sym_eq K k0 k H9)))
-H8)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) |
-(subst0_both v0 u0 u2 i0 H0 k0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0
-i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k
-u1 t1))).(\lambda (H5: (eq T (THead k0 u2 t2) x)).(eq_ind nat i (\lambda (n:
-nat).((eq T v0 v) \to ((eq T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T
-(THead k0 u2 t2) x) \to ((subst0 n v0 u0 u2) \to ((subst0 (s k0 n) v0 t0 t2)
-\to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3:
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3)))
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
-t3))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T
-(THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to ((subst0
-i t u0 u2) \to ((subst0 (s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u3:
-T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T
-(\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v
-t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H7: (eq T
-(THead k0 u0 t0) (THead k u1 t1))).(let H8 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t0)
-(THead k u1 t1) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t0) (THead k u1
-t1) H7) in ((let H10 \def (f_equal T K (\lambda (e: T).(match e in T return
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0
-| (THead k _ _) \Rightarrow k])) (THead k0 u0 t0) (THead k u1 t1) H7) in
-(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T
-(THead k1 u2 t2) x) \to ((subst0 i v u0 u2) \to ((subst0 (s k1 i) v t0 t2)
-\to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3:
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3)))
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
-t3))))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq
-T t0 t1) \to ((eq T (THead k u2 t2) x) \to ((subst0 i v t u2) \to ((subst0 (s
-k i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1)))
-(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead
-k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
-(u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda
-(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
-v t1 t3)))))))))) (\lambda (H12: (eq T t0 t1)).(eq_ind T t1 (\lambda (t:
-T).((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t
-t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3:
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3)))
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
-t3))))))))) (\lambda (H13: (eq T (THead k u2 t2) x)).(eq_ind T (THead k u2
-t2) (\lambda (t: T).((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to
-(or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1))) (\lambda (u3:
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T t (THead k u1 t3)))
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T t (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
-t3)))))))) (\lambda (H14: (subst0 i v u1 u2)).(\lambda (H15: (subst0 (s k i)
-v t1 t2)).(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k
-u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T
-(THead k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)))
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2_intro T T (\lambda
-(u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda
-(u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s k i) v t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H14
-H15)))) x H13)) t0 (sym_eq T t0 t1 H12))) u0 (sym_eq T u0 u1 H11))) k0
-(sym_eq K k0 k H10))) H9)) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i
-H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v)
-(refl_equal T (THead k u1 t1)) (refl_equal T x))))))))).
+.
-theorem subst0_gen_lift_lt:
+axiom subst0_gen_lift_lt:
\forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
(h: nat).(\forall (d: nat).((subst0 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
(t2: T).(subst0 i u t1 t2)))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d
-u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n:
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n))
-x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t:
-T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d))))
-in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n)
-t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S
-(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2:
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
-n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
-(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
-(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (and_ind (eq nat n i) (eq
-T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
-(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
-(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))))
-(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0)
-O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda
-(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
-(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T
-(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O
-(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3)))
-(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
-n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
-T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
-(plus i d)) H0)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
-h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
-(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
-u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n))
-H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2:
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
-n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4))))
-(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t)
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
-(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift
-h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0
-t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t
-t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0))
-(\lambda (t: T).(subst0 i (lift h d u) t x)) H1 (THead k (lift h (S (plus i
-d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d))))
-in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus
-i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d))
-t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t)
-t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
-d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S
-(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h
-d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
-t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k
-(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S
-(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
-(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h
-(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
-T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d)))
-t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t)
-x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T
-x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T
-(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
-(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda
-(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d)))
-t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
-t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T
-(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h
-(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k
-(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1
-t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3:
-(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2)))
-(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d)))
-t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i
-d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
-(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
-T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda
-(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0)
-x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2
-T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3:
-T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i
-d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S
-(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus
-i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0))
-H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
-(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
-(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d))
-x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s
-k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h
-(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i
-u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T
-(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1))
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
-t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda
-(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind
-T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
-(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i
-d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d))
-(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i
-d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d))))
-(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0
-H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2)))
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
-(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq
-T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d
-u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
-(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda
-(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x
-(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d))
-t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
-d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
-(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda
-(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i
-d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d))
-(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7
-(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
-(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda
-(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k
-i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d))
-t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T
-(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
-(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S
-(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S
-(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
-x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
-t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T
-(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S
-(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind
-nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k
-(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus
-i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S
-(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2:
-T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead
-k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus
-i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T
-(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k
-x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u
-t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
-(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d)))
-(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0
-i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
-(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i
-H2))))))))))))) t1)).
+.
-theorem subst0_gen_lift_false:
+axiom subst0_gen_lift_false:
\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 ((subst0 i u
(lift h d t) x) \to (\forall (P: Prop).P)))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: 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 ((subst0 i u (lift h d t0) x) \to (\forall (P:
-Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda
-(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n))
-x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda
-(t: T).(subst0 i u t x)) H1 (TSort n) (lift_sort n h d)) in (subst0_gen_sort
-u x i n H2 P)))))))))))) (\lambda (n: nat).(\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: (subst0 i u (lift h d
-(TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P (\lambda (H2: (lt n d)).(let
-H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H1
-(TLRef n) (lift_lref_lt n h d H2)) in (and_ind (eq nat n i) (eq T x (lift (S
-n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O
-u))).(let H6 \def (eq_ind nat n (\lambda (n: nat).(lt n d)) H2 i H4) in
-(le_false d i P H H6)))) (subst0_gen_lref u x i n H3)))) (\lambda (H2: (le d
-n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t
-x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d H2)) in (and_ind (eq nat (plus
-n h) i) (eq T x (lift (S (plus n h)) O u)) P (\lambda (H4: (eq nat (plus n h)
-i)).(\lambda (_: (eq T x (lift (S (plus n h)) O u))).(let H6 \def (eq_ind_r
-nat i (\lambda (n: nat).(lt n (plus d h))) H0 (plus n h) H4) in (le_false d n
-P H2 (lt_le_S n d (simpl_lt_plus_r h n d H6)))))) (subst0_gen_lref u x i
-(plus n h) H3))))))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H:
-((\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 ((subst0 i u (lift h d t0) x)
-\to (\forall (P: Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\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 ((subst0 i u (lift h d t1) x) \to
-(\forall (P: Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda
-(H2: (lt i (plus d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1))
-x)).(\lambda (P: Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1))
-(\lambda (t: T).(subst0 i u t x)) H3 (THead k (lift h d t0) (lift h (s k d)
-t1)) (lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x
-(THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0)
-u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
-(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
-(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2:
-T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u
-(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
-(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda
-(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7:
-(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda
-(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
-(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2:
-T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
-(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k
-(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1)
-x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h))
-(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i
-(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7:
-(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d)
-t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d
-t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
+.
-theorem subst0_gen_lift_ge:
+axiom subst0_gen_lift_ge:
\forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
(h: nat).(\forall (d: nat).((subst0 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:
T).(subst0 (minus i h) u t1 t2))))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h
-d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
-t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n:
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus
-d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0
-i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2
-T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i
-h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i:
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d
-(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
-n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
-(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
-(and_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq T
-x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
-(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
-\def (eq_ind nat n (\lambda (n: nat).(lt n d)) H1 i H3) in (le_false (plus d
-h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0
-(minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
-(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
-T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
-(lift_lref_ge n h d H1)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S
-(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
-(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
-h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
-h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
-(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S
-(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d
-t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2))))
-(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S
-(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n)
-t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h
-d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u)
-(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n
-h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
-u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
-nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
-(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_comm n
-h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans d (S n)
-(plus O (S n)) (le_S d n H1) (le_n (plus O (S n)))) (le_O_n d))) (subst0_lref
-u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i H3))) (subst0_gen_lref
-u x i (plus n h) H2)))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H:
-((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
-nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t
-t2))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (i:
-nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h d t0) x) \to
-((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
-(\lambda (t2: T).(subst0 (minus i h) u t0 t2))))))))))).(\lambda (x:
-T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
-(subst0 i u (lift h d (THead k t t0)) x)).(\lambda (H2: (le (plus d h)
-i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) (\lambda (t: T).(subst0
-i u t x)) H1 (THead k (lift h d t) (lift h (s k d) t0)) (lift_head k t t0 h
-d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d)
-t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2))) (ex2 T (\lambda (t2:
-T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u
-(lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
-x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d
-t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d)
-t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
-T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (H4: (ex2 T (\lambda
-(u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i
-u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
-(s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2)) (ex2 T (\lambda
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
-k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x (THead k x0 (lift h (s k
-d) t0)))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(eq_ind_r T (THead k x0
-(lift h (s k d) t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift
-h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
-(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0
-(minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 (lift h (s k
-d) t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0)
-t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h d x1))).(\lambda (H8:
-(subst0 (minus i h) u t x1)).(eq_ind_r T (lift h d x1) (\lambda (t2: T).(ex2
-T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k d) t0)) (lift h d t3)))
-(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift
-h d (THead k x1 t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift
-h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
-(ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k x1 t0)) (lift h d
-t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x1
-t0) (refl_equal T (lift h d (THead k x1 t0))) (subst0_fst u x1 t (minus i h)
-H8 t0 k)) (THead k (lift h d x1) (lift h (s k d) t0)) (lift_head k x1 t0 h
-d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) (\lambda (H4: (ex2 T (\lambda
-(t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i)
-u (lift h (s k d) t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k
-(lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0)
-t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0
-(minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x
-(THead k (lift h d t) x0))).(\lambda (H6: (subst0 (s k i) u (lift h (s k d)
-t0) x0)).(eq_ind_r T (THead k (lift h d t) x0) (\lambda (t2: T).(ex2 T
-(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i
-h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (s k
-d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda
-(t2: T).(eq T (THead k (lift h d t) x0) (lift h d t2))) (\lambda (t2:
-T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7:
-(eq T x0 (lift h (s k d) x1))).(\lambda (H8: (subst0 (minus (s k i) h) u t0
-x1)).(eq_ind_r T (lift h (s k d) x1) (\lambda (t2: T).(ex2 T (\lambda (t3:
-T).(eq T (THead k (lift h d t) t2) (lift h d t3))) (\lambda (t3: T).(subst0
-(minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d (THead k t x1))
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda
-(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H9 \def (eq_ind_r
-nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x1)) H8 (s k (minus i
-h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2:
-T).(eq T (lift h d (THead k t x1)) (lift h d t2))) (\lambda (t2: T).(subst0
-(minus i h) u (THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h d
-(THead k t x1))) (subst0_snd k u x1 t0 (minus i h) H9 t))) (THead k (lift h d
-t) (lift h (s k d) x1)) (lift_head k t x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s
-k d) H6 (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le
-k (plus d h) i H2) (plus (s k d) h) (s_plus k d h)))) x H5)))) H4)) (\lambda
-(H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex2 T
-(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h)
-u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T
-x (THead k x0 x1))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(\lambda (H7:
-(subst0 (s k i) u (lift h (s k d) t0) x1)).(eq_ind_r T (THead k x0 x1)
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda
-(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2:
-T).(eq T x1 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h)
-u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2)))
-(\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x2:
-T).(\lambda (H8: (eq T x1 (lift h (s k d) x2))).(\lambda (H9: (subst0 (minus
-(s k i) h) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2)))
-(\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T
-(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
-k t t0) t2))) (\lambda (x3: T).(\lambda (H10: (eq T x0 (lift h d
-x3))).(\lambda (H11: (subst0 (minus i h) u t x3)).(eq_ind_r T (lift h d x3)
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 x1) (lift h d
-t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind_r
-T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k
-(lift h d x3) t2) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u
-(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x3 x2)) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
-(minus i h) u (THead k t t0) t3)))) (let H12 \def (eq_ind_r nat (minus (s k
-i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 (s k (minus i h)) (s_minus k i
-h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h
-d (THead k x3 x2)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u
-(THead k t t0) t2)) (THead k x3 x2) (refl_equal T (lift h d (THead k x3 x2)))
-(subst0_both u t x3 (minus i h) H11 k t0 x2 H12))) (THead k (lift h d x3)
-(lift h (s k d) x2)) (lift_head k x3 x2 h d)) x1 H8) x0 H10)))) (H x0 i h d
-H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind nat (s k (plus d h)) (\lambda
-(n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) (s_plus k
-d h)))) x H5)))))) H4)) (subst0_gen_head k u (lift h d t) (lift h (s k d) t0)
-x i H3)))))))))))))) t1)).
+.
projects / helm.git / blobdiff