+++ /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 "Basic-1/subst0/defs.ma".
-
-include "Basic-1/lift/props.ma".
-
-include "Basic-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))))).
-(* COMMENTS
-Initial nodes: 4101
-END *)
-
-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))))).
-(* COMMENTS
-Initial nodes: 4207
-END *)
-
-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))))))))).
-(* COMMENTS
-Initial nodes: 347
-END *)
-
-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))))).
-(* COMMENTS
-Initial nodes: 59
-END *)
-
projects / helm.git / blobdiff