+++ /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/lift/defs.ma".
-
-theorem lift_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
-n)) (TSort n))))
-\def
- \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort
-n)))).
-(* COMMENTS
-Initial nodes: 13
-END *)
-
-theorem lift_lref_lt:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T
-(lift h d (TLRef n)) (TLRef n)))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
-d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T
-(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
-(* COMMENTS
-Initial nodes: 72
-END *)
-
-theorem lift_lref_ge:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
-(lift h d (TLRef n)) (TLRef (plus n h))))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
-n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h))))
-(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false
-(le_bge d n H)))))).
-(* COMMENTS
-Initial nodes: 80
-END *)
-
-theorem lift_head:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d)
-t)))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
-(* COMMENTS
-Initial nodes: 37
-END *)
-
-theorem lift_bind:
- \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u)
-(lift h (S d) t)))))))
-\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
-(* COMMENTS
-Initial nodes: 37
-END *)
-
-theorem lift_flat:
- \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u)
-(lift h d t)))))))
-\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
-(* COMMENTS
-Initial nodes: 35
-END *)
-
-theorem lift_gen_sort:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T
-(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind
-(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))
-(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort
-n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H:
-(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort
-n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
-(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let
-H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
-(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0)
-H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d
-n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
-(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
-(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef
-(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda
-(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq
-T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d
-t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d
-(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
-(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1))
-(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k
-t0 t1) (TSort n)) H3))))))))) t)))).
-(* COMMENTS
-Initial nodes: 613
-END *)
-
-theorem lift_gen_lref:
- \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
-(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le
-(plus d h) i) (eq T t (TLRef (minus i h)))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h:
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
-(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
-h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda
-(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def
-(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort
-n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i)))
-(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1))))))))
-(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
-nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or
-(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T
-(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
-T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n)
-(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
-n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef
-n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0
-h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus
-d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n)
-(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d
-n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef
-i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h)
-(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le
-(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n
-(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n
-h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0)))))
-(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
-(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h)
-(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h))
-(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i
-H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d:
-nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to
-(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0
-(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d:
-nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to
-(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1
-(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
-nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def
-(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1
-(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
-H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d)
-t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i)))
-(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h)))))
-H3)))))))))))) t).
-(* COMMENTS
-Initial nodes: 1221
-END *)
-
-theorem lift_gen_lref_lt:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall
-(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n
-d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x
-\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d)
-(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq
-T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind
-(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n
-d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2:
-(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d
-h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus
-d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
-(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq
-T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S
-n) d h H))) t H4))) H2)) H1)))))))).
-(* COMMENTS
-Initial nodes: 363
-END *)
-
-theorem lift_gen_lref_false:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
-(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall
-(P: Prop).P)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
-n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T
-(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d
-h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n)))
-(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land
-(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P
-(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H
-H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n
-h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
-(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false
-(plus d h) n P H4 H0))) H3)) H2)))))))))).
-(* COMMENTS
-Initial nodes: 269
-END *)
-
-theorem lift_gen_lref_ge:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
-(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
-n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
-t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in
-(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus
-d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n))
-(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind
-(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda
-(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r
-T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq
-T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d
-(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2))
-(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
-h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
-h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
-(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n
-h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus
-(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
-(* COMMENTS
-Initial nodes: 473
-END *)
-
-theorem lift_gen_head:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z)))))))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
-(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
-(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n:
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
-(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda
-(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1
-\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef
-n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
-(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0:
-(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
-(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def
-(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def
-(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H
-(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead
-k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0:
-T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
-(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1:
-T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
-(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0
-t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2:
-T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
-(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0
-(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2]))
-(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2)
-\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1))
-H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let
-H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k
-H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z:
-T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
-k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let
-H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0:
-nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift
-h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k
-d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
-(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d)
-z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
-t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10
-(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
-t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9
-(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z))))))
-(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d)
-z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0))
-(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4))
-H3))))))))))) x)))).
-(* COMMENTS
-Initial nodes: 2083
-END *)
-
-theorem lift_gen_bind:
- \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T
-T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (S d) z)))))))))))
-\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d
-x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
-h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
-(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
-(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d)
-x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
-(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d)
-z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro
-T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind
-b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
-z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d
-x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
-(* COMMENTS
-Initial nodes: 637
-END *)
-
-theorem lift_gen_flat:
- \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T
-T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h d z)))))))))))
-\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d
-x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
-h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
-(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
-(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T
-T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0:
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1)
-(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T
-(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
-T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d
-x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
-(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
-(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1))
-(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x
-H1)))))) H0))))))))).
-(* COMMENTS
-Initial nodes: 615
-END *)
-
projects / helm.git / blobdiff