+++ /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_gen_sort:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
-i v (TSort n) x) \to (eq T x (TSort n))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T
-t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0
-i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x
-H))))).
-(* COMMENTS
-Initial nodes: 89
-END *)
-
-theorem subst1_gen_lref:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
-i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift
-(S n) O v))))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or
-(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl
-(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O
-v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v
-(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
-(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq
-nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
-(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i)
-(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x
-H))))).
-(* COMMENTS
-Initial nodes: 305
-END *)
-
-theorem subst1_gen_head:
- \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
-(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst1 (s k i) v t1 t2))))))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
-(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1)
-x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1
-t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1
-t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal
-T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda
-(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T
-(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
-u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
-T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2
-(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda
-(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda
-(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1
-x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1
-x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3:
-T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
-T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0:
-T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v
-t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1)
-(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda
-(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1
-i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0
-x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4)))))))
-H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
-(* COMMENTS
-Initial nodes: 1199
-END *)
-
-theorem subst1_gen_lift_lt:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1)
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
-(t2: T).(subst1 i u t1 t2)))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S
-(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1)
-(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2)))
-(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
-(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2:
-T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1))
-(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u)
-(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h
-(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1
-t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d))
-x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T
-t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1
-(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x
-H))))))).
-(* COMMENTS
-Initial nodes: 395
-END *)
-
-theorem subst1_gen_lift_eq:
- \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
-(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u
-(lift h d t) x) \to (eq T x (lift h d t))))))))))
-\def
- \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d
-h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t)
-(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda
-(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t
-u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
-(* COMMENTS
-Initial nodes: 141
-END *)
-
-theorem subst1_gen_lift_ge:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h)
-i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
-T).(subst1 (minus i h) u t1 t2))))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1)
-x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda
-(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2:
-T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
-h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1
-(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2:
-T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3:
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3))
-(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1
-(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d
-x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3:
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0
-H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h
-d H1 H0)))) x H)))))))).
-(* COMMENTS
-Initial nodes: 355
-END *)
-
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