X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Flift%2Fprops.ma;h=3e230188360cf9d32db9845ab624dccf06a8ad65;hb=92e6d4bc77a154dde1df3a25b0004d8fd46cc8b3;hp=7c205f57ad34ffdebb9fc1cb47662774fe123391;hpb=592dca18b8f2bd19c3020bc1bb1f0270188e347a;p=helm.git diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma index 7c205f57a..3e2301883 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma @@ -45,40 +45,41 @@ h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) -\Rightarrow t])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) -in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow -(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda -(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat) -(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef -i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) -(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0 -d) t1)) | (TLRef _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat \to -nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow -(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda -(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat) -(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef -i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) -(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0 -d) t1)) | (THead _ _ t) \Rightarrow t])) (THead k v (lift h d (THead k0 t0 -t1))) (THead k0 t0 t1) H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k -k0)).(let H7 \def (eq_ind K k (\lambda (k: K).(\forall (v: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead k v (lift h d t1)) t1) \to (\forall (P: -Prop).P)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1)) -(\lambda (t: T).(eq T t t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -(lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3)) -H2)))))))))))) t)). +[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) +\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) +H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map +(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort +n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) +with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat +\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow +(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T +\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f +i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f +(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec +lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with +[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i +d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) +\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) +H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def +(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall +(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: +Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 +t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 +d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) +H3)) H2)))))))))))) t)). theorem lift_r: \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t)) @@ -94,11 +95,11 @@ t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) -t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_equal T (THead k t0 t1) -(THead k (lift O d t0) (lift O (s k d) t1)) (sym_equal T (THead k (lift O d -t0) (lift O (s k d) t1)) (THead k t0 t1) (sym_equal T (THead k t0 t1) (THead -k (lift O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d -t0) t1 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) +t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1) +(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0) +(lift O (s k d) t1)) (THead k t0 t1) (sym_eq T (THead k t0 t1) (THead k (lift +O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d t0) t1 +(lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) (sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t). @@ -135,17 +136,17 @@ h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) -(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to -(eq T t0 t)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: +(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: +T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) +\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to -(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: T).(\forall -(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to (eq T t0 -t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: +(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall +(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 +t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T -(lift h d (THead (Bind b) t t0)) (\lambda (t: T).(eq T t (lift h d t1))) H1 +(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) @@ -154,32 +155,31 @@ z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) (H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) -(sym_equal T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_equal T (THead -(Bind b) t t0) (THead (Bind b) x0 x1) (sym_equal T (THead (Bind b) x0 x1) -(THead (Bind b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 -(refl_equal K (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 -h (S d) H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) -t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall -(t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d -t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to -(eq T t0 t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) (lift h d -t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) (\lambda (t: -T).(eq T t (lift h d t1))) H1 (THead (Flat f) (lift h d t) (lift h d t0)) -(lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq -T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d -t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift -h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift -h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d -x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat -f) t t0) t2)) (sym_equal T (THead (Flat f) x0 x1) (THead (Flat f) t t0) -(sym_equal T (THead (Flat f) t t0) (THead (Flat f) x0 x1) (sym_equal T (THead -(Flat f) x0 x1) (THead (Flat f) t t0) (f_equal3 K T T T THead (Flat f) (Flat -f) x0 t x1 t0 (refl_equal K (Flat f)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T -t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) (lift_gen_flat f (lift h d t) (lift h d -t0) t1 h d H2)))))))))))) k)) x). +(sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind +b) t t0) (THead (Bind b) x0 x1) (sym_eq T (THead (Bind b) x0 x1) (THead (Bind +b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 (refl_equal K +(Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h (S d) +H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d +H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0: +T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to +(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall +(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 +t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: +(eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T +(lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 +(THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in +(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) +(\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T +(THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq +T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d +x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead +(Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T +(THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0) +(THead (Flat f) x0 x1) (sym_eq T (THead (Flat f) x0 x1) (THead (Flat f) t t0) +(f_equal3 K T T T THead (Flat f) (Flat f) x0 t x1 t0 (refl_equal K (Flat f)) +(sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) +(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x). theorem lift_gen_lift: \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: @@ -242,8 +242,8 @@ H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt (plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4: -(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n: -nat).(eq T (TLRef n) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1) +(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n0: +nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1 (le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda @@ -283,9 +283,9 @@ t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1 x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2 t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead -(Bind b) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H3 (THead -(Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y +(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 +(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) +in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) @@ -333,7 +333,7 @@ H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind -T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus +T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift @@ -413,12 +413,12 @@ nat T TLRef (plus (plus n h) k) (plus n (plus k h)) (lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1 (le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k: +(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0)))))))))).(\lambda (t1: -T).(\lambda (H0: ((\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e -(lift h d t1)) (lift (plus k h) d t1)))))))))).(\lambda (h: nat).(\lambda +(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda +(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: +nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e +(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 @@ -489,13 +489,13 @@ n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1 (le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n)) (lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda -(H: ((\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e: -nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t0)) (lift k e (lift h -d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k -d) (lift k e t1)) (lift k e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda -(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e -d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: +(H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: +nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift +h d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall +(k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h +(plus k0 d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: +nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le +e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) (eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d