+++ /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/clear/defs.ma".
-
-include "basic_1/C/fwd.ma".
-
-implied rec lemma clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b:
-B).(\forall (e: C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b)
-u)))))) (f0: (\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to
-(\forall (f0: F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C)
-(c0: C) (c1: clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u)
-\Rightarrow (f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3
-((clear_ind P f f0) e c2 c3) f1 u)].
-
-lemma clear_gen_sort:
- \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P:
-Prop).P)))
-\def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda
-(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_:
-C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c:
-C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let
-H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee with
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n)
-H1) in (False_ind P H2)))))) (\lambda (e: C).(\lambda (c: C).(\lambda (_:
-(clear e c)).(\lambda (_: (((eq C e (CSort n)) \to P))).(\lambda (f:
-F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e (Flat f) u) (CSort n))).(let
-H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda (ee: C).(match ee with
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n)
-H3) in (False_ind P H4))))))))) y x H0))) H)))).
-
-lemma clear_gen_bind:
- \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
-(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
-\def
- \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
-(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda
-(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0:
-(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e
-(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda
-(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b)
-u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _)
-\Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind b0) u0)
-(CHead e (Bind b) u) H1) in ((let H3 \def (f_equal C B (\lambda (e1:
-C).(match e1 with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
-(match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead
-e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def (f_equal C T
-(\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
-\Rightarrow t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in (\lambda
-(H5: (eq B b0 b)).(\lambda (H6: (eq C e0 e)).(eq_ind_r T u (\lambda (t:
-T).(eq C (CHead e0 (Bind b0) t) (CHead e0 (Bind b0) t))) (eq_ind_r C e
-(\lambda (c: C).(eq C (CHead c (Bind b0) u) (CHead c (Bind b0) u))) (eq_ind_r
-B b (\lambda (b1: B).(eq C (CHead e (Bind b1) u) (CHead e (Bind b1) u)))
-(refl_equal C (CHead e (Bind b) u)) b0 H5) e0 H6) u0 H4)))) H3)) H2))))))
-(\lambda (e0: C).(\lambda (c: C).(\lambda (_: (clear e0 c)).(\lambda (_:
-(((eq C e0 (CHead e (Bind b) u)) \to (eq C c e0)))).(\lambda (f: F).(\lambda
-(u0: T).(\lambda (H3: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b)
-u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) (\lambda (ee: C).(match ee
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead e (Bind
-b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) u0)) H4))))))))) y x H0)))
-H))))).
-
-lemma clear_gen_flat:
- \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
-(CHead e (Flat f) u) x) \to (clear e x)))))
-\def
- \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
-(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda
-(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda
-(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead
-e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda
-(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f)
-u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat
-f) u) H1) in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda
-(e0: C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0
-(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0:
-T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4
-\def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow e0 |
-(CHead c0 _ _) \Rightarrow c0])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u)
-H3) in ((let H5 \def (f_equal C F (\lambda (e1: C).(match e1 with [(CSort _)
-\Rightarrow f0 | (CHead _ k _) \Rightarrow (match k with [(Bind _)
-\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) u0) (CHead
-e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: C).(match e1
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0
-(Flat f0) u0) (CHead e (Flat f) u) H3) in (\lambda (_: (eq F f0 f)).(\lambda
-(H8: (eq C e0 e)).(let H9 \def (eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead
-e (Flat f) u)) \to (clear e c))) H2 e H8) in (let H10 \def (eq_ind C e0
-(\lambda (c0: C).(clear c0 c)) H1 e H8) in H10))))) H5)) H4))))))))) y x
-H0))) H))))).
-
-lemma clear_gen_flat_r:
- \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x
-(CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
-\def
- \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H:
-(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e
-(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y:
-C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0:
-C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0:
-C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat
-f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match
-ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k
-with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e
-(Flat f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c:
-C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u))
-\to P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat
-f) u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f)
-u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda
-(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C
-(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
-
-lemma clear_gen_all:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind
-(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0:
-B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0
-(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e:
-C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b)
-u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b)
-u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c
-(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
-T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c
-(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C
-(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda
-(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B
-C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind
-x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0)
-x2))) c H3)))))) H2)))))))) c1 c2 H))).
-
-theorem clear_mono:
- \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c
-c2) \to (eq C c1 c2)))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to
-(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n:
-nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2:
-C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1
-c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to
-(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t)
-c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind
-(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2)
-\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b)
-t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0
-(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t)
-(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0
-(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t
-H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t)
-c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f
-c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
-
-lemma clear_cle:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
-(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda
-(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O)))))
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight
-c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
-C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear
-(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t)))))
-(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C
-(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c)
-(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c
-c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t)
-c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2
-(clear_gen_flat f c c2 t H1))))) k H0))))))) c1).
-
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