X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Flift%2Ftlt.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Flift%2Ftlt.ma;h=4e088a12270bd030cc74ed1d1e7b8b7bca045471;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma new file mode 100644 index 000000000..4e088a122 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma @@ -0,0 +1,284 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "LambdaDelta-1/lift/fwd.ma". + +include "LambdaDelta-1/tlt/props.ma". + +theorem lift_weight_map: + \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to +nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat +(weight_map f (lift h d t)) (weight_map f t)))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0))))))) +(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m) +O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n: +nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m) +O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f +(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: +T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat +(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) +(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq +nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda +(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) +(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda +(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f +t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f +t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) +O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 +t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead +(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat +(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda +(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift +h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) +t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f +(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with +[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S +(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) +(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) +(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: +nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S +(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) +(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: +nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) +(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus +(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) +(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) +(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda +(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda +(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) +(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r +nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map +f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) +t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 +nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map +(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat +(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h +(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat +(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd +f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le +d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x +H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) +(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus +(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f +t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f +(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd +f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) +(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: +nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S +n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: +nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d +m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h +d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s +(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f +(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d +t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) +(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) +(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) +(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) +k)))))))))) t). + +theorem lift_weight: + \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d +t)) (weight t)))) +\def + \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d +(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat +O)))))). + +theorem lift_weight_add: + \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall +(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to +(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat +(weight_map f (lift h d t)) (weight_map g (lift (S h) d t))))))))))) +\def + \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: +nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat +(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) +\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d +t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: +nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) +w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f +m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n)))))))))))) +(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m +d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: +((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d +(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d +(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: +T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) +(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) +(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef +n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d +H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: +T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) +(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f +(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda +(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f +(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) +(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift +h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda +(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat +\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to +(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d +m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0)) +(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0: +((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall +(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f +m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g +(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift +(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat +\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: +nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) +w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f +m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 +t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b: +B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) +(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead +(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h) +(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b) +(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind +(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d +t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h +d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S +(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) +t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h) +d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h) +(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0)) +(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus +(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) +t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus +(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift +(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map +f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S +(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S +(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2 +H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S +(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S +d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) +(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d +t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m +O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift +(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat +nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq +nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g +H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S +m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat +m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g +(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda +(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r +nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d +t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) +H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: +nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) +n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus +(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) +t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g +(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map +(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) +(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) +(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) +(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat +O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m +H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda +(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) +(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: +nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) +H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S +x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g +n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat +S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) +t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S +h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) +(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) +(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S +d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S +d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) +(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda +(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) +(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: +nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda +(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O +m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda +(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) +(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: +nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S +n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: +nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) +(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head +(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind +b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) +(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) +(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead +(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: +T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) +d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d +t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) +(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f +(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) +(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) +(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) +(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) +k))))))))))))) t)). + +theorem lift_weight_add_O: + \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to +nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) +O t)))))) +\def + \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to +nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m: +nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m))))) +(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal +nat (f m)))))))). + +theorem lift_tlt_dx: + \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall +(d: nat).(tlt t (THead k u (lift h d t))))))) +\def + \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda +(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight +(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t) +(lift_weight t h d)))))). +