--- /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 "LambdaDelta-1/clear/fwd.ma".
+
+theorem clear_clear:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
+(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear
+(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c:
+C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2
+c2))))).(\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
+(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t)
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0))
+(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f:
+F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c
+c2 t H1)))) k H0))))))) c1).
+
+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).
+
+theorem clear_trans:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c
+c2) \to (clear c1 c2)))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2:
+C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n)
+c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t)
+c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0:
+K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b:
+B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0
+(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c
+c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead
+c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3)))))
+(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c
+c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
+
+theorem clear_ctail:
+ \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1
+(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k
+u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
+K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b)
+u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H:
+(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1:
+T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1
+c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n
+H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind
+b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear
+(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2))))
+k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2:
+T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1:
+T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear
+(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1:
+T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to
+(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2))))
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind
+b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t)
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_:
+K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2
+(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b)
+u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0)
+\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t)
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b
+b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead
+(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r
+C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail
+k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0
+u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail
+k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1:
+(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1
+c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead
+c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)).
+
+theorem 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).
+
projects / helm.git / blobdiff