+++ /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/leq/defs.ma".
-
-theorem leq_gen_sort1:
- \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
-g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
-(ASort h2 n2))))))))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda
-(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort
-h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A
-a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g
-(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat
-nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a
-k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0:
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
-k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal
-A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
-n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1
-n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0
-h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0
-n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
-nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3
-n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort
-n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0)))))
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2)
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3
-n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
-(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0
-H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
-(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k)
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
-nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2
-n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def
-(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k)
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0)))
-H))))).
-(* COMMENTS
-Initial nodes: 913
-END *)
-
-theorem leq_gen_head1:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
-(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1
-a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
-g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
-a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
-(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq
-A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1)
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
-(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
-(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2)
-(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
-(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A
-(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda
-(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4
-a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
-a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
-A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2
-a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6
-a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4)
-(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6)
-\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0
-a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6:
-A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6:
-A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
-A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda
-(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12
-\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A
-A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda
-(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5)
-(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5)))))))))
-H6))))))))))) y a H0))) H))))).
-(* COMMENTS
-Initial nodes: 797
-END *)
-
-theorem leq_gen_sort2:
- \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
-g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1)
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
-(ASort h2 n2))))))))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda
-(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
-(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq
-A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind
-g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat
-nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus
-g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0:
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
-k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal
-A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
-n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1
-n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2
-h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0
-n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
-nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3
-n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort
-h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0)))))
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0)
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1
-n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
-(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2
-H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
-(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
-nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2
-n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def
-(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
-(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0)))
-H))))).
-(* COMMENTS
-Initial nodes: 913
-END *)
-
-theorem leq_gen_head2:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a
-(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3
-a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
-g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
-a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
-(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq
-A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2)
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
-(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
-(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda
-(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1)
-(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
-(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A
-(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda
-(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4
-a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
-a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
-A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7
-a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6
-a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5)
-(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6)
-\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3
-a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A
-a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6:
-A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6:
-A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
-A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda
-(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12
-\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A
-A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda
-(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4)
-(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4)))))))))
-H6))))))))))) a y H0))) H))))).
-(* COMMENTS
-Initial nodes: 797
-END *)
-
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