(* This file was automatically generated: do not edit *********************)
-include "Basic-1/clear/defs.ma".
+include "basic_1/clear/defs.ma".
-theorem clear_gen_sort:
+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
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 in C return
-(\lambda (_: C).Prop) 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 in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
-(False_ind P H4))))))))) y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 215
-END *)
+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)))).
-theorem clear_gen_bind:
+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
(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 in C return
-(\lambda (_: C).C) 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 in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) 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 in C return (\lambda (_: C).T) 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 in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) 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))))).
-(* COMMENTS
-Initial nodes: 525
-END *)
+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))))).
-theorem clear_gen_flat:
+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
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
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
-_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 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
+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 in C return (\lambda (_: C).C)
-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 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow
-f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) 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 in C return (\lambda (_: C).T) 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))))).
-(* COMMENTS
-Initial nodes: 453
-END *)
+\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))))).
-theorem clear_gen_flat_r:
+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
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 in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 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)
+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)))))).
-(* COMMENTS
-Initial nodes: 303
-END *)
-theorem clear_gen_all:
+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
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))).
-(* COMMENTS
-Initial nodes: 381
-END *)
+
+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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