--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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)))))).
+
projects / helm.git / blobdiff