--- /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/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+include "LambdaDelta-1/lift/tlt.ma".
+
+theorem subst0_weight_le:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift
+(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O
+v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd
+g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g
+H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
+u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1)
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2))
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i)))
+\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S
+(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f))))))))))))))))
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0)
+(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g
+O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map
+f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b))
+(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g
+m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g
+u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0)
+(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2
+H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall
+(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f
+m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le
+(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda
+(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s
+k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map
+f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b:
+B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s
+(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
+(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f
+t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2))
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f
+g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1)))
+(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1))
+(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5
+(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v))
+(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O
+(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5:
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b))
+(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
+(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g
+t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5:
+(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2)
+(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus
+(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1
+f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
+
+theorem subst0_weight_lt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v:
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
+(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd
+f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n
+(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O
+O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
+u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1)
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2))
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i:
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i)))
+\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f))))))))))))))))
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f
+u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd
+g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
+(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
+(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
+f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1))
+(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
+(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
+(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt
+(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map
+g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1:
+T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
+k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v
+t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2)
+(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
+(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2))
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
+t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1
+f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2)))
+(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S
+(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2)
+(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O
+v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f
+u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i)
+f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v
+t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m)
+(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m)
+(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O)
+m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g
+i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v
+(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m))))
+\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2)
+(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
+(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
+f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
+(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
+(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t
+z H))))).
+
+theorem subst0_tlt_head:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt
+(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
+(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map
+(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n
+(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m)))
+(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n:
+nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda
+(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_:
+nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u)
+(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda
+(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda
+(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
+
+theorem subst0_tlt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z
+(THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx
+(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
+
projects / helm.git / blobdiff