+++ /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/subst1/defs.ma".
-
-include "Basic-1/subst0/props.ma".
-
-theorem subst1_head:
- \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1
-i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s
-k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2))))))))))
-\def
- \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k:
-K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i
-v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1:
-T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k
-i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t)))
-(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k
-i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k
-v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
-t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1
-(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead
-k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1)
-(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k
-i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both
-v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
-(* COMMENTS
-Initial nodes: 369
-END *)
-
-theorem subst1_lift_lt:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1
-i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i
-(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d:
-nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u)
-(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i
-d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d
-t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d:
-nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h
-(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d
-H1 h))))))) t2 H))))).
-(* COMMENTS
-Initial nodes: 185
-END *)
-
-theorem subst1_lift_ge:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
-(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1
-(plus i h) u (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t:
-T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h
-d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u
-(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda
-(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1)
-(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
-(* COMMENTS
-Initial nodes: 157
-END *)
-
-theorem subst1_ex:
- \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2:
-T).(subst1 d u t1 (lift (S O) d t2))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex
-T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n:
-nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n)
-(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d
-u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n))
-(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d
-(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda
-(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O)
-d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n)
-t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S
-O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T
-(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T
-(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u)
-(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n)
-t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S
-O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d
-H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n)
-(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t:
-T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef
-(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t:
-T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift
-(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T
-(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d:
-nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2:
-T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u
-(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u
-t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
-(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex
-T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda
-(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d)
-x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d
-t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k
-d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t
-(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k
-x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
-(* COMMENTS
-Initial nodes: 925
-END *)
-
-theorem subst1_lift_S:
- \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i
-(TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
-\def
- \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h:
-nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i
-t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_:
-(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift
-(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef
-h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n))
-(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S
-i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H:
-(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n))
-(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n)
-(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T
-(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i
-(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0))
-(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0))))
-(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0:
-nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef
-h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T
-(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n))))
-(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef
-n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h)
-(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n
-(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0:
-nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O
-(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n
-(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n))
-(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n))
-(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h)
-(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n))
-(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt
-n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T
-(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i
-(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i
-(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n
-(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S
-(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i)
-H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i:
-nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t)
-(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i:
-nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i)
-t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1:
-(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
-t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0))))
-(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1:
-T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i))
-t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i
-h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S
-(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift
-(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k
-(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i))
-(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u).
-(* COMMENTS
-Initial nodes: 1421
-END *)
-
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