include "drop1/defs.ma".
-include "getl/defs.ma".
+include "getl/drop.ma".
-axiom drop1_getl_trans:
+theorem drop1_getl_trans:
\forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2
(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
-.
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
+C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
+C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
+(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i
+v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to
+(ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with
+[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2:
+(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C
+c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))
+(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2:
+C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i
+c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2
+H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds0 H2) \Rightarrow (\lambda (H3:
+(eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5:
+(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
+(False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1
+hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b)
+v)))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C
+c2) (refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda
+(hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2
+c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i:
+nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl
+(trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
+(_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq
+C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
+h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
+\Rightarrow (ctrans hds0 i v)])))))))))))) with [(drop1_nil c) \Rightarrow
+(\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c
+c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
+\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2)
+in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3
+H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds1 H3) \Rightarrow (\lambda (H4: (eq
+PList (PCons h0 d0 hds1) (PCons h d hds0))).(\lambda (H5: (eq C c0
+c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda
+(e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
+\Rightarrow hds1 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds1) (PCons h
+d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
+in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _
+n _) \Rightarrow n])) (PCons h0 d0 hds1) (PCons h d hds0) H4) in ((let H9
+\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
+(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
+(PCons h0 d0 hds1) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n:
+nat).((eq nat d0 d) \to ((eq PList hds1 hds0) \to ((eq C c0 c2) \to ((eq C c4
+c1) \to ((drop n d0 c0 c3) \to ((drop1 hds1 c3 c4) \to (ex C (\lambda (e2:
+C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i)
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match
+(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))))
+(\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds1
+hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1
+hds1 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d)
+with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0
+i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
+\Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds1
+hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1)
+\to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda
+(H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d
+c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt
+(trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow
+(plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d)
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4
+c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c)
+\to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift
+h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans
+hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1
+hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex
+C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) |
+false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x:
+bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to
+(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i)
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H16: (eq bool
+(blt (trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let
+H17 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2
+(Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2
+(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)))))) (\lambda (x: C).(\lambda (H18: (getl (trans hds0 i) c3 (CHead x (Bind
+b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d
+(le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d)
+(S (trans hds0 i)) H16))) c2 c3 h H14 b x (ctrans hds0 i v) H18) in (let H19
+\def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2
+(Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda
+(e2: C).(drop h (minus d (S (trans hds0 i))) e2 x)) (ex C (\lambda (e2:
+C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H20: (getl (trans hds0
+i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d
+(S (trans hds0 i))) (ctrans hds0 i v))))) x0 H20)))) H19))))) H17))))
+(\lambda (H16: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3
+H15 b e1 v i H1) in (let H17 \def H_x in (ex_ind C (\lambda (e2: C).(getl
+(trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2:
+C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))))
+(\lambda (x: C).(\lambda (H18: (getl (trans hds0 i) c3 (CHead x (Bind b)
+(ctrans hds0 i v)))).(let H19 \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d
+h H14 (CHead x (Bind b) (ctrans hds0 i v)) H18) in (ex_intro C (\lambda (e2:
+C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x
+(H19 (bge_le d (trans hds0 i) H16)))))) H17)))) x_x))))) c4 (sym_eq C c4 c1
+H13))) c0 (sym_eq C c0 c2 H12))) hds1 (sym_eq PList hds1 hds0 H11))) d0
+(sym_eq nat d0 d H10))) h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))])
+in (H2 (refl_equal PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C
+c1))))))))))))))) hds).
include "iso/defs.ma".
-axiom iso_trans:
+theorem iso_trans:
\forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
t3) \to (iso t1 t3)))))
-.
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
+(TSort n2) t3)).(let H1 \def (match H0 in iso return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to
+(iso (TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H1: (eq
+T (TSort n0) (TSort n2))).(\lambda (H2: (eq T (TSort n3) t3)).((let H3 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n2) H1) in (eq_ind nat n2 (\lambda (_:
+nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H4: (eq T
+(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t))
+(iso_sort n1 n3) t3 H4)) n0 (sym_eq nat n0 n2 H3))) H2))) | (iso_lref i1 i2)
+\Rightarrow (\lambda (H1: (eq T (TLRef i1) (TSort n2))).(\lambda (H2: (eq T
+(TLRef i2) t3)).((let H3 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H1) in
+(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H3)) H2))) |
+(iso_head k v1 v2 t0 t4) \Rightarrow (\lambda (H1: (eq T (THead k v1 t0)
+(TSort n2))).(\lambda (H2: (eq T (THead k v2 t4) t3)).((let H3 \def (eq_ind T
+(THead k v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort n2) H1) in (False_ind ((eq T (THead k v2 t4)
+t3) \to (iso (TSort n1) t3)) H3)) H2)))]) in (H1 (refl_equal T (TSort n2))
+(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3:
+T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 in iso return
+(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef
+i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2)
+\Rightarrow (\lambda (H1: (eq T (TSort n1) (TLRef i2))).(\lambda (H2: (eq T
+(TSort n2) t3)).((let H3 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef
+_) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H1) in
+(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H3)) H2))) |
+(iso_lref i0 i3) \Rightarrow (\lambda (H1: (eq T (TLRef i0) (TLRef
+i2))).(\lambda (H2: (eq T (TLRef i3) t3)).((let H3 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i2) H1) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef
+i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H4: (eq T (TLRef i3) t3)).(eq_ind
+T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H4)) i0
+(sym_eq nat i0 i2 H3))) H2))) | (iso_head k v1 v2 t0 t4) \Rightarrow (\lambda
+(H1: (eq T (THead k v1 t0) (TLRef i2))).(\lambda (H2: (eq T (THead k v2 t4)
+t3)).((let H3 \def (eq_ind T (THead k v1 t0) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H1) in
+(False_ind ((eq T (THead k v2 t4) t3) \to (iso (TLRef i1) t3)) H3)) H2)))])
+in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k:
+K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def
+(match H0 in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t
+t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3)
+t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort n1)
+(THead k v2 t4))).(\lambda (H2: (eq T (TSort n2) t5)).((let H3 \def (eq_ind T
+(TSort n1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow False])) I (THead k v2 t4) H1) in (False_ind ((eq T (TSort n2)
+t5) \to (iso (THead k v1 t3) t5)) H3)) H2))) | (iso_lref i1 i2) \Rightarrow
+(\lambda (H1: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H2: (eq T (TLRef
+i2) t5)).((let H3 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H1)
+in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H3)) H2)))
+| (iso_head k0 v0 v3 t0 t6) \Rightarrow (\lambda (H1: (eq T (THead k0 v0 t0)
+(THead k v2 t4))).(\lambda (H2: (eq T (THead k0 v3 t6) t5)).((let H3 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H1) in ((let H4 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H1) in ((let H5 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 v0 t0) (THead k v2 t4) H1) in (eq_ind K k
+(\lambda (k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t6)
+t5) \to (iso (THead k v1 t3) t5))))) (\lambda (H6: (eq T v0 v2)).(eq_ind T v2
+(\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t6) t5) \to (iso (THead
+k v1 t3) t5)))) (\lambda (H7: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq
+T (THead k v3 t6) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H8: (eq T
+(THead k v3 t6) t5)).(eq_ind T (THead k v3 t6) (\lambda (t: T).(iso (THead k
+v1 t3) t)) (iso_head k v1 v3 t3 t6) t5 H8)) t0 (sym_eq T t0 t4 H7))) v0
+(sym_eq T v0 v2 H6))) k0 (sym_eq K k0 k H5))) H4)) H3)) H2)))]) in (H1
+(refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/asucc".
-include "leq/defs.ma".
+include "leq/props.ma".
include "aplus/props.ma".
a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g a3 a4 H0
(asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
-axiom asucc_inj:
+theorem asucc_inj:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
g a2)) \to (leq g a1 a2))))
-.
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
+(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
+(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
+(asucc g (ASort n1 n2)))).((match n in nat return (\lambda (n3: nat).((leq g
+(asucc g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0)
+(ASort n1 n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O
+n0)) (asucc g (ASort n1 n2)))).((match n1 in nat return (\lambda (n3:
+nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g
+(ASort O n0) (ASort n3 n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc
+g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq return
+(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O
+(next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort O n0)
+(ASort O n2))))))) with [(leq_sort h1 h2 n3 n4 k H2) \Rightarrow (\lambda
+(H3: (eq A (ASort h1 n3) (ASort O (next g n0)))).(\lambda (H4: (eq A (ASort
+h2 n4) (ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n5) \Rightarrow
+n5 | (AHead _ _) \Rightarrow n3])) (ASort h1 n3) (ASort O (next g n0)) H3) in
+((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda
+(_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow h1]))
+(ASort h1 n3) (ASort O (next g n0)) H3) in (eq_ind nat O (\lambda (n5:
+nat).((eq nat n3 (next g n0)) \to ((eq A (ASort h2 n4) (ASort O (next g n2)))
+\to ((eq A (aplus g (ASort n5 n3) k) (aplus g (ASort h2 n4) k)) \to (leq g
+(ASort O n0) (ASort O n2)))))) (\lambda (H7: (eq nat n3 (next g n0))).(eq_ind
+nat (next g n0) (\lambda (n5: nat).((eq A (ASort h2 n4) (ASort O (next g
+n2))) \to ((eq A (aplus g (ASort O n5) k) (aplus g (ASort h2 n4) k)) \to (leq
+g (ASort O n0) (ASort O n2))))) (\lambda (H8: (eq A (ASort h2 n4) (ASort O
+(next g n2)))).(let H9 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n5) \Rightarrow n5 | (AHead _ _)
+\Rightarrow n4])) (ASort h2 n4) (ASort O (next g n2)) H8) in ((let H10 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow h2])) (ASort h2 n4)
+(ASort O (next g n2)) H8) in (eq_ind nat O (\lambda (n5: nat).((eq nat n4
+(next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n5
+n4) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H11: (eq nat n4
+(next g n2))).(eq_ind nat (next g n2) (\lambda (n5: nat).((eq A (aplus g
+(ASort O (next g n0)) k) (aplus g (ASort O n5) k)) \to (leq g (ASort O n0)
+(ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort O (next g n0)) k) (aplus
+g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A (aplus g (ASort O
+(next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k)))
+H12 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H14
+\def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A
+(aplus g (ASort O n0) (S k)) a)) H13 (aplus g (ASort O n2) (S k))
+(aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H14)))) n4
+(sym_eq nat n4 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9))) n3 (sym_eq
+nat n3 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head
+a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort O
+(next g n0)))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g
+n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A
+(AHead a3 a5) (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5)
+\to (leq g (ASort O n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2
+(refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O (next g n2))))))
+| (S n3) \Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g
+(ASort (S n3) n2)))).(let H2 \def (match H1 in leq return (\lambda (a:
+A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O (next g
+n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq g (ASort O n0) (ASort (S n3)
+n2))))))) with [(leq_sort h1 h2 n4 n5 k H2) \Rightarrow (\lambda (H3: (eq A
+(ASort h1 n4) (ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n5)
+(ASort n3 n2))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _ _)
+\Rightarrow n4])) (ASort h1 n4) (ASort O (next g n0)) H3) in ((let H6 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4)
+(ASort O (next g n0)) H3) in (eq_ind nat O (\lambda (n6: nat).((eq nat n4
+(next g n0)) \to ((eq A (ASort h2 n5) (ASort n3 n2)) \to ((eq A (aplus g
+(ASort n6 n4) k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort O n0) (ASort (S
+n3) n2)))))) (\lambda (H7: (eq nat n4 (next g n0))).(eq_ind nat (next g n0)
+(\lambda (n6: nat).((eq A (ASort h2 n5) (ASort n3 n2)) \to ((eq A (aplus g
+(ASort O n6) k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort O n0) (ASort (S
+n3) n2))))) (\lambda (H8: (eq A (ASort h2 n5) (ASort n3 n2))).(let H9 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n6) \Rightarrow n6 | (AHead _ _) \Rightarrow n5])) (ASort h2 n5)
+(ASort n3 n2) H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e
+in A return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow n6 | (AHead _
+_) \Rightarrow h2])) (ASort h2 n5) (ASort n3 n2) H8) in (eq_ind nat n3
+(\lambda (n6: nat).((eq nat n5 n2) \to ((eq A (aplus g (ASort O (next g n0))
+k) (aplus g (ASort n6 n5) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))
+(\lambda (H11: (eq nat n5 n2)).(eq_ind nat n2 (\lambda (n6: nat).((eq A
+(aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n6) k)) \to (leq g
+(ASort O n0) (ASort (S n3) n2)))) (\lambda (H12: (eq A (aplus g (ASort O
+(next g n0)) k) (aplus g (ASort n3 n2) k))).(let H13 \def (eq_ind_r A (aplus
+g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort n3 n2)
+k))) H12 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let
+H14 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a: A).(eq A (aplus g
+(ASort O n0) (S k)) a)) H13 (aplus g (ASort (S n3) n2) (S k))
+(aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H14))))
+n5 (sym_eq nat n5 n2 H11))) h2 (sym_eq nat h2 n3 H10))) H9))) n4 (sym_eq nat
+n4 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head a0
+a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort O (next
+g n0)))).(\lambda (H5: (eq A (AHead a3 a5) (ASort n3 n2))).((let H6 \def
+(eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A (AHead a3 a5) (ASort
+n3 n2)) \to ((leq g a0 a3) \to ((leq g a4 a5) \to (leq g (ASort O n0) (ASort
+(S n3) n2))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (ASort O (next g n0)))
+(refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0:
+(leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 in
+nat return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g
+(ASort n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+O n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O
+(next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2))))))) with [(leq_sort
+h1 h2 n4 n5 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n4) (ASort n3
+n0))).(\lambda (H4: (eq A (ASort h2 n5) (ASort O (next g n2)))).((let H5 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n6) \Rightarrow n6 | (AHead _ _) \Rightarrow n4])) (ASort h1 n4)
+(ASort n3 n0) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in
+A return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow n6 | (AHead _ _)
+\Rightarrow h1])) (ASort h1 n4) (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda
+(n6: nat).((eq nat n4 n0) \to ((eq A (ASort h2 n5) (ASort O (next g n2))) \to
+((eq A (aplus g (ASort n6 n4) k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort
+(S n3) n0) (ASort O n2)))))) (\lambda (H7: (eq nat n4 n0)).(eq_ind nat n0
+(\lambda (n6: nat).((eq A (ASort h2 n5) (ASort O (next g n2))) \to ((eq A
+(aplus g (ASort n3 n6) k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort (S n3)
+n0) (ASort O n2))))) (\lambda (H8: (eq A (ASort h2 n5) (ASort O (next g
+n2)))).(let H9 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _ _)
+\Rightarrow n5])) (ASort h2 n5) (ASort O (next g n2)) H8) in ((let H10 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h2])) (ASort h2 n5)
+(ASort O (next g n2)) H8) in (eq_ind nat O (\lambda (n6: nat).((eq nat n5
+(next g n2)) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n6 n5) k))
+\to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H11: (eq nat n5 (next
+g n2))).(eq_ind nat (next g n2) (\lambda (n6: nat).((eq A (aplus g (ASort n3
+n0) k) (aplus g (ASort O n6) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))
+(\lambda (H12: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2))
+k))).(let H13 \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq
+A a (aplus g (ASort O (next g n2)) k))) H12 (aplus g (ASort (S n3) n0) (S k))
+(aplus_sort_S_S_simpl g n0 n3 k)) in (let H14 \def (eq_ind_r A (aplus g
+(ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
+k)) a)) H13 (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in
+(leq_sort g (S n3) O n0 n2 (S k) H14)))) n5 (sym_eq nat n5 (next g n2) H11)))
+h2 (sym_eq nat h2 O H10))) H9))) n4 (sym_eq nat n4 n0 H7))) h1 (sym_eq nat h1
+n3 H6))) H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda
+(H4: (eq A (AHead a0 a4) (ASort n3 n0))).(\lambda (H5: (eq A (AHead a3 a5)
+(ASort O (next g n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H4) in (False_ind
+((eq A (AHead a3 a5) (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4
+a5) \to (leq g (ASort (S n3) n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2
+(refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4)
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+(S n4) n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda
+(a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0
+(ASort n4 n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))))) with
+[(leq_sort h1 h2 n5 n6 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n5)
+(ASort n3 n0))).(\lambda (H4: (eq A (ASort h2 n6) (ASort n4 n2))).((let H5
+\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
+with [(ASort _ n7) \Rightarrow n7 | (AHead _ _) \Rightarrow n5])) (ASort h1
+n5) (ASort n3 n0) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match
+e in A return (\lambda (_: A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead
+_ _) \Rightarrow h1])) (ASort h1 n5) (ASort n3 n0) H3) in (eq_ind nat n3
+(\lambda (n7: nat).((eq nat n5 n0) \to ((eq A (ASort h2 n6) (ASort n4 n2))
+\to ((eq A (aplus g (ASort n7 n5) k) (aplus g (ASort h2 n6) k)) \to (leq g
+(ASort (S n3) n0) (ASort (S n4) n2)))))) (\lambda (H7: (eq nat n5
+n0)).(eq_ind nat n0 (\lambda (n7: nat).((eq A (ASort h2 n6) (ASort n4 n2))
+\to ((eq A (aplus g (ASort n3 n7) k) (aplus g (ASort h2 n6) k)) \to (leq g
+(ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda (H8: (eq A (ASort h2 n6)
+(ASort n4 n2))).(let H9 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n7) \Rightarrow n7 | (AHead _ _)
+\Rightarrow n6])) (ASort h2 n6) (ASort n4 n2) H8) in ((let H10 \def (f_equal
+A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
+n7 _) \Rightarrow n7 | (AHead _ _) \Rightarrow h2])) (ASort h2 n6) (ASort n4
+n2) H8) in (eq_ind nat n4 (\lambda (n7: nat).((eq nat n6 n2) \to ((eq A
+(aplus g (ASort n3 n0) k) (aplus g (ASort n7 n6) k)) \to (leq g (ASort (S n3)
+n0) (ASort (S n4) n2))))) (\lambda (H11: (eq nat n6 n2)).(eq_ind nat n2
+(\lambda (n7: nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n4 n7)
+k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))) (\lambda (H12: (eq A
+(aplus g (ASort n3 n0) k) (aplus g (ASort n4 n2) k))).(let H13 \def (eq_ind_r
+A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort n4 n2)
+k))) H12 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k))
+in (let H14 \def (eq_ind_r A (aplus g (ASort n4 n2) k) (\lambda (a: A).(eq A
+(aplus g (ASort (S n3) n0) (S k)) a)) H13 (aplus g (ASort (S n4) n2) (S k))
+(aplus_sort_S_S_simpl g n2 n4 k)) in (leq_sort g (S n3) (S n4) n0 n2 (S k)
+H14)))) n6 (sym_eq nat n6 n2 H11))) h2 (sym_eq nat h2 n4 H10))) H9))) n5
+(sym_eq nat n5 n0 H7))) h1 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) | (leq_head
+a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort n3
+n0))).(\lambda (H5: (eq A (AHead a3 a5) (ASort n4 n2))).((let H6 \def (eq_ind
+A (AHead a0 a4) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
+(ASort n3 n0) H4) in (False_ind ((eq A (AHead a3 a5) (ASort n4 n2)) \to ((leq
+g a0 a3) \to ((leq g a4 a5) \to (leq g (ASort (S n3) n0) (ASort (S n4)
+n2))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A
+(ASort n4 n2)))))]) H0))]) H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc
+g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0:
+A).(\lambda (H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g
+(ASort n n0) a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g
+(AHead a a0)))).((match n in nat return (\lambda (n1: nat).((((leq g (asucc g
+(ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g
+(asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to
+((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1
+n0) (AHead a a0)))))) with [O \Rightarrow (\lambda (_: (((leq g (asucc g
+(ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda (_: (((leq
+g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0)
+a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a
+a0)))).(let H5 \def (match H4 in leq return (\lambda (a3: A).(\lambda (a4:
+A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort O (next g n0))) \to ((eq A a4
+(AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a a0))))))) with
+[(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n1)
+(ASort O (next g n0)))).(\lambda (H7: (eq A (ASort h2 n2) (AHead a (asucc g
+a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead _ _)
+\Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H6) in ((let H9 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
+(ASort O (next g n0)) H6) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
+(next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
+(aplus g (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
+(AHead a a0)))))) (\lambda (H10: (eq nat n1 (next g n0))).(eq_ind nat (next g
+n0) (\lambda (n3: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq
+A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
+(AHead a a0))))) (\lambda (H11: (eq A (ASort h2 n2) (AHead a (asucc g
+a0)))).(let H12 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead a (asucc g a0)) H11) in (False_ind ((eq A
+(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g
+(ASort O n0) (AHead a a0))) H12))) n1 (sym_eq nat n1 (next g n0) H10))) h1
+(sym_eq nat h1 O H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6 H6)
+\Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort O (next g
+n0)))).(\lambda (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9
+\def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O (next g n0)) H7) in (False_ind ((eq A (AHead a4 a6) (AHead
+a (asucc g a0))) \to ((leq g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort O
+n0) (AHead a a0))))) H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort O (next g
+n0))) (refl_equal A (AHead a (asucc g a0)))))))) | (S n1) \Rightarrow
+(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to (leq g
+(ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0))
+(asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g (asucc
+g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H5 \def (match H4 in leq
+return (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A
+a3 (ASort n1 n0)) \to ((eq A a4 (AHead a (asucc g a0))) \to (leq g (ASort (S
+n1) n0) (AHead a a0))))))) with [(leq_sort h1 h2 n2 n3 k H5) \Rightarrow
+(\lambda (H6: (eq A (ASort h1 n2) (ASort n1 n0))).(\lambda (H7: (eq A (ASort
+h2 n3) (AHead a (asucc g a0)))).((let H8 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow
+n4 | (AHead _ _) \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0) H6) in ((let
+H9 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_:
+A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h1]))
+(ASort h1 n2) (ASort n1 n0) H6) in (eq_ind nat n1 (\lambda (n4: nat).((eq nat
+n2 n0) \to ((eq A (ASort h2 n3) (AHead a (asucc g a0))) \to ((eq A (aplus g
+(ASort n4 n2) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n1) n0)
+(AHead a a0)))))) (\lambda (H10: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4:
+nat).((eq A (ASort h2 n3) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort
+n1 n4) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a
+a0))))) (\lambda (H11: (eq A (ASort h2 n3) (AHead a (asucc g a0)))).(let H12
+\def (eq_ind A (ASort h2 n3) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a (asucc g a0)) H11) in (False_ind ((eq A (aplus g (ASort
+n1 n0) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a
+a0))) H12))) n2 (sym_eq nat n2 n0 H10))) h1 (sym_eq nat h1 n1 H9))) H8)) H7
+H5))) | (leq_head a3 a4 H5 a5 a6 H6) \Rightarrow (\lambda (H7: (eq A (AHead
+a3 a5) (ASort n1 n0))).(\lambda (H8: (eq A (AHead a4 a6) (AHead a (asucc g
+a0)))).((let H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n1 n0) H7) in (False_ind ((eq A (AHead a4 a6)
+(AHead a (asucc g a0))) \to ((leq g a3 a4) \to ((leq g a5 a6) \to (leq g
+(ASort (S n1) n0) (AHead a a0))))) H9)) H8 H5 H6)))]) in (H5 (refl_equal A
+(ASort n1 n0)) (refl_equal A (AHead a (asucc g a0))))))))]) H H0 H1))))))
+a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a)
+(asucc g a2)) \to (leq g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall
+(a2: A).((leq g (asucc g a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda
+(a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g (AHead a a0)) (asucc g a3))
+\to (leq g (AHead a a0) a3))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
+(H1: (leq g (asucc g (AHead a a0)) (asucc g (ASort n n0)))).((match n in nat
+return (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) (asucc g (ASort n1
+n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) with [O \Rightarrow (\lambda
+(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H3 \def
+(match H2 in leq return (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq ?
+a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A a4 (ASort O (next g
+n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with [(leq_sort h1 h2 n1 n2
+k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1) (AHead a (asucc g
+a0)))).(\lambda (H5: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H6
+\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort h2 n2)
+(ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H6)) H5 H3))) | (leq_head
+a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5) (AHead a
+(asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort O (next g
+n0)))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
+\Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
+(AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
+a0)) \to ((eq A (AHead a4 a6) (ASort O (next g n0))) \to ((leq g a7 a4) \to
+((leq g a5 a6) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H9: (eq A
+a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4
+a6) (ASort O (next g n0))) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g
+(AHead a a0) (ASort O n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort O
+(next g n0)))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e
+in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H10) in (False_ind
+((leq g a a4) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort O
+n0)))) H11))) a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7))
+H6 H3 H4)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A
+(ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H2: (leq g (asucc g
+(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H3 \def (match H2 in leq
+return (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A
+a3 (AHead a (asucc g a0))) \to ((eq A a4 (ASort n1 n0)) \to (leq g (AHead a
+a0) (ASort (S n1) n0))))))) with [(leq_sort h1 h2 n2 n3 k H3) \Rightarrow
+(\lambda (H4: (eq A (ASort h1 n2) (AHead a (asucc g a0)))).(\lambda (H5: (eq
+A (ASort h2 n3) (ASort n1 n0))).((let H6 \def (eq_ind A (ASort h1 n2)
+(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
+H4) in (False_ind ((eq A (ASort h2 n3) (ASort n1 n0)) \to ((eq A (aplus g
+(ASort h1 n2) k) (aplus g (ASort h2 n3) k)) \to (leq g (AHead a a0) (ASort (S
+n1) n0)))) H6)) H5 H3))) | (leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda
+(H5: (eq A (AHead a3 a5) (AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead
+a4 a6) (ASort n1 n0))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
+\Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
+(AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
+a0)) \to ((eq A (AHead a4 a6) (ASort n1 n0)) \to ((leq g a7 a4) \to ((leq g
+a5 a6) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H9: (eq A a5
+(asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4 a6)
+(ASort n1 n0)) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g (AHead a a0)
+(ASort (S n1) n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort n1
+n0))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n1 n0) H10) in (False_ind ((leq g a a4) \to
+((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H11)))
+a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7)) H6 H3 H4)))])
+in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1
+n0)))))]) H1)))) (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a
+a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda
+(_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0)
+a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3
+a4)))).(let H4 \def (match H3 in leq return (\lambda (a5: A).(\lambda (a6:
+A).(\lambda (_: (leq ? a5 a6)).((eq A a5 (AHead a (asucc g a0))) \to ((eq A
+a6 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4))))))) with
+[(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1)
+(AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g
+a4)))).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort
+h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) |
+(leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7)
+(AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a6 a8) (AHead a3 (asucc g
+a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a9)
+\Rightarrow a9])) (AHead a5 a7) (AHead a (asucc g a0)) H6) in ((let H9 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a5 | (AHead a9 _) \Rightarrow a9])) (AHead a5 a7)
+(AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a9: A).((eq A a7 (asucc g
+a0)) \to ((eq A (AHead a6 a8) (AHead a3 (asucc g a4))) \to ((leq g a9 a6) \to
+((leq g a7 a8) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq
+A a7 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a9: A).((eq A (AHead a6
+a8) (AHead a3 (asucc g a4))) \to ((leq g a a6) \to ((leq g a9 a8) \to (leq g
+(AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a6 a8) (AHead a3
+(asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a8 | (AHead _ a9)
+\Rightarrow a9])) (AHead a6 a8) (AHead a3 (asucc g a4)) H11) in ((let H13
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a6 | (AHead a9 _) \Rightarrow a9])) (AHead a6
+a8) (AHead a3 (asucc g a4)) H11) in (eq_ind A a3 (\lambda (a9: A).((eq A a8
+(asucc g a4)) \to ((leq g a a9) \to ((leq g (asucc g a0) a8) \to (leq g
+(AHead a a0) (AHead a3 a4)))))) (\lambda (H14: (eq A a8 (asucc g
+a4))).(eq_ind A (asucc g a4) (\lambda (a9: A).((leq g a a3) \to ((leq g
+(asucc g a0) a9) \to (leq g (AHead a a0) (AHead a3 a4))))) (\lambda (H15:
+(leq g a a3)).(\lambda (H16: (leq g (asucc g a0) (asucc g a4))).(leq_head g a
+a3 H15 a0 a4 (H0 a4 H16)))) a8 (sym_eq A a8 (asucc g a4) H14))) a6 (sym_eq A
+a6 a3 H13))) H12))) a7 (sym_eq A a7 (asucc g a0) H10))) a5 (sym_eq A a5 a
+H9))) H8)) H7 H4 H5)))]) in (H4 (refl_equal A (AHead a (asucc g a0)))
+(refl_equal A (AHead a3 (asucc g a4)))))))))) a2)))))) a1)).
+
+theorem leq_asucc:
+ \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
+a0)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
+A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
+A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
+(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
+(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
+(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
+(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
+(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
+g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
+(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
+a)).
+
+theorem leq_ahead_asucc_false:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
+(asucc g a1)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
+(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).((match n in nat return
+(\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O
+\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P))
+with [O \Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O
+(next g n0)))).(let H1 \def (match H0 in leq return (\lambda (a: A).(\lambda
+(a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort O n0) a2)) \to ((eq
+A a0 (ASort O (next g n0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H1)
+\Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead (ASort O n0)
+a2))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H4 \def
+(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead (ASort O n0) a2) H2) in (False_ind ((eq A (ASort h2 n2)
+(ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow
+(\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort O n0) a2))).(\lambda (H4: (eq
+A (AHead a3 a5) (ASort O (next g n0)))).((let H5 \def (f_equal A A (\lambda
+(e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
+a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3)
+in ((let H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a]))
+(AHead a0 a4) (AHead (ASort O n0) a2) H3) in (eq_ind A (ASort O n0) (\lambda
+(a: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) (ASort O (next g n0))) \to
+((leq g a a3) \to ((leq g a4 a5) \to P))))) (\lambda (H7: (eq A a4
+a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5) (ASort O (next g n0)))
+\to ((leq g (ASort O n0) a3) \to ((leq g a a5) \to P)))) (\lambda (H8: (eq A
+(AHead a3 a5) (ASort O (next g n0)))).(let H9 \def (eq_ind A (AHead a3 a5)
+(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0))
+H8) in (False_ind ((leq g (ASort O n0) a3) \to ((leq g a2 a5) \to P)) H9)))
+a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort O n0) H6))) H5)) H4 H1
+H2)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O
+(next g n0)))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S
+n1) n0) a2) (ASort n1 n0))).(let H1 \def (match H0 in leq return (\lambda (a:
+A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort (S n1)
+n0) a2)) \to ((eq A a0 (ASort n1 n0)) \to P))))) with [(leq_sort h1 h2 n2 n3
+k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n2) (AHead (ASort (S n1) n0)
+a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort n1 n0))).((let H4 \def (eq_ind
+A (ASort h1 n2) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
+(AHead (ASort (S n1) n0) a2) H2) in (False_ind ((eq A (ASort h2 n3) (ASort n1
+n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g (ASort h2 n3) k)) \to P))
+H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
+(AHead a0 a4) (AHead (ASort (S n1) n0) a2))).(\lambda (H4: (eq A (AHead a3
+a5) (ASort n1 n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
+\Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3) in ((let H6
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
+(AHead (ASort (S n1) n0) a2) H3) in (eq_ind A (ASort (S n1) n0) (\lambda (a:
+A).((eq A a4 a2) \to ((eq A (AHead a3 a5) (ASort n1 n0)) \to ((leq g a a3)
+\to ((leq g a4 a5) \to P))))) (\lambda (H7: (eq A a4 a2)).(eq_ind A a2
+(\lambda (a: A).((eq A (AHead a3 a5) (ASort n1 n0)) \to ((leq g (ASort (S n1)
+n0) a3) \to ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort
+n1 n0))).(let H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n1 n0) H8) in (False_ind ((leq g (ASort (S
+n1) n0) a3) \to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0
+(sym_eq A a0 (ASort (S n1) n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A
+(AHead (ASort (S n1) n0) a2)) (refl_equal A (ASort n1 n0)))))]) H))))))
+(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g
+a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall
+(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P:
+Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2)
+(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H2 \def (match H1 in leq
+return (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A
+a3 (AHead (AHead a a0) a2)) \to ((eq A a4 (AHead a (asucc g a0))) \to P)))))
+with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1
+n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a
+(asucc g a0)))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match
+e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
+(AHead _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind
+((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1
+n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2
+a5 a6 H3) \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0)
+a2))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H6
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3
+a5) (AHead (AHead a a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 |
+(AHead a7 _) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4) in
+(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A a5 a2) \to ((eq A (AHead a4
+a6) (AHead a (asucc g a0))) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P)))))
+(\lambda (H8: (eq A a5 a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4
+a6) (AHead a (asucc g a0))) \to ((leq g (AHead a a0) a4) \to ((leq g a7 a6)
+\to P)))) (\lambda (H9: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).(let H10
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4
+a6) (AHead a (asucc g a0)) H9) in ((let H11 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 |
+(AHead a7 _) \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc g a0)) H9) in
+(eq_ind A a (\lambda (a7: A).((eq A a6 (asucc g a0)) \to ((leq g (AHead a a0)
+a7) \to ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 (asucc g
+a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((leq g (AHead a a0) a) \to
+((leq g a2 a7) \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_:
+(leq g a2 (asucc g a0))).(leq_ahead_false g a a0 H13 P))) a6 (sym_eq A a6
+(asucc g a0) H12))) a4 (sym_eq A a4 a H11))) H10))) a5 (sym_eq A a5 a2 H8)))
+a3 (sym_eq A a3 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A
+(AHead (AHead a a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
+
+theorem leq_asucc_false:
+ \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
+a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
+(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).((match n in nat
+return (\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O
+(next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) with
+[O \Rightarrow (\lambda (H0: (leq g (ASort O (next g n0)) (ASort O n0))).(let
+H1 \def (match H0 in leq return (\lambda (a0: A).(\lambda (a1: A).(\lambda
+(_: (leq ? a0 a1)).((eq A a0 (ASort O (next g n0))) \to ((eq A a1 (ASort O
+n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2:
+(eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H3: (eq A (ASort h2 n2)
+(ASort O n0))).((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A
+return (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead _ _)
+\Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H2) in ((let H5 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
+(ASort O (next g n0)) H2) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
+(next g n0)) \to ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g
+(ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H6: (eq nat
+n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n3: nat).((eq A (ASort h2
+n2) (ASort O n0)) \to ((eq A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2)
+k)) \to P))) (\lambda (H7: (eq A (ASort h2 n2) (ASort O n0))).(let H8 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow n2])) (ASort h2 n2)
+(ASort O n0) H7) in ((let H9 \def (f_equal A nat (\lambda (e: A).(match e in
+A return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
+\Rightarrow h2])) (ASort h2 n2) (ASort O n0) H7) in (eq_ind nat O (\lambda
+(n3: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus
+g (ASort n3 n2) k)) \to P))) (\lambda (H10: (eq nat n2 n0)).(eq_ind nat n0
+(\lambda (n3: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O
+n3) k)) \to P)) (\lambda (H11: (eq A (aplus g (ASort O (next g n0)) k) (aplus
+g (ASort O n0) k))).(let H12 \def (eq_ind_r A (aplus g (ASort O (next g n0))
+k) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) k))) H11 (aplus g (ASort O
+n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k)
+k (ASort O n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n3: nat).(le n3
+k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H10))) h2 (sym_eq nat h2 O
+H9))) H8))) n1 (sym_eq nat n1 (next g n0) H6))) h1 (sym_eq nat h1 O H5)))
+H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
+(AHead a1 a3) (ASort O (next g n0)))).(\lambda (H4: (eq A (AHead a2 a4)
+(ASort O n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e
+in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
+(AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H3) in (False_ind
+((eq A (AHead a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to
+P))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (ASort O (next g n0)))
+(refl_equal A (ASort O n0))))) | (S n1) \Rightarrow (\lambda (H0: (leq g
+(ASort n1 n0) (ASort (S n1) n0))).(let H1 \def (match H0 in leq return
+(\lambda (a0: A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A a0
+(ASort n1 n0)) \to ((eq A a1 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1
+h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n2) (ASort n1
+n0))).(\lambda (H3: (eq A (ASort h2 n3) (ASort (S n1) n0))).((let H4 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort h1 n2)
+(ASort n1 n0) H2) in ((let H5 \def (f_equal A nat (\lambda (e: A).(match e in
+A return (\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
+\Rightarrow h1])) (ASort h1 n2) (ASort n1 n0) H2) in (eq_ind nat n1 (\lambda
+(n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to
+((eq A (aplus g (ASort n4 n2) k) (aplus g (ASort h2 n3) k)) \to P))))
+(\lambda (H6: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort
+h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g
+(ASort h2 n3) k)) \to P))) (\lambda (H7: (eq A (ASort h2 n3) (ASort (S n1)
+n0))).(let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
+(\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
+\Rightarrow n3])) (ASort h2 n3) (ASort (S n1) n0) H7) in ((let H9 \def
+(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
+[(ASort n4 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h2])) (ASort h2 n3)
+(ASort (S n1) n0) H7) in (eq_ind nat (S n1) (\lambda (n4: nat).((eq nat n3
+n0) \to ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort n4 n3) k)) \to P)))
+(\lambda (H10: (eq nat n3 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A
+(aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n4) k)) \to P)) (\lambda
+(H11: (eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n0) k))).(let
+H12 \def (eq_ind_r A (aplus g (ASort n1 n0) k) (\lambda (a0: A).(eq A a0
+(aplus g (ASort (S n1) n0) k))) H11 (aplus g (ASort (S n1) n0) (S k))
+(aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def (aplus_inj g (S k) k
+(ASort (S n1) n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n4: nat).(le
+n4 k)) (le_n k) (S k) H_y) P)))) n3 (sym_eq nat n3 n0 H10))) h2 (sym_eq nat
+h2 (S n1) H9))) H8))) n2 (sym_eq nat n2 n0 H6))) h1 (sym_eq nat h1 n1 H5)))
+H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
+(AHead a1 a3) (ASort n1 n0))).(\lambda (H4: (eq A (AHead a2 a4) (ASort (S n1)
+n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n1 n0) H3) in (False_ind ((eq A (AHead a2 a4)
+(ASort (S n1) n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H5)) H4 H1
+H2)))]) in (H1 (refl_equal A (ASort n1 n0)) (refl_equal A (ASort (S n1)
+n0)))))]) H))))) (\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to
+(\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1)
+a1) \to (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1))
+(AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (match H1 in leq return
+(\lambda (a2: A).(\lambda (a3: A).(\lambda (_: (leq ? a2 a3)).((eq A a2
+(AHead a0 (asucc g a1))) \to ((eq A a3 (AHead a0 a1)) \to P))))) with
+[(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1)
+(AHead a0 (asucc g a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0
+a1))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead a0 (asucc g a1)) H3) in (False_ind ((eq A
+(ASort h2 n2) (AHead a0 a1)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a2 a3 H2 a4 a5 H3)
+\Rightarrow (\lambda (H4: (eq A (AHead a2 a4) (AHead a0 (asucc g
+a1)))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a0 a1))).((let H6 \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 a2 a4)
+(AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 |
+(AHead a6 _) \Rightarrow a6])) (AHead a2 a4) (AHead a0 (asucc g a1)) H4) in
+(eq_ind A a0 (\lambda (a6: A).((eq A a4 (asucc g a1)) \to ((eq A (AHead a3
+a5) (AHead a0 a1)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda
+(H8: (eq A a4 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a6: A).((eq A
+(AHead a3 a5) (AHead a0 a1)) \to ((leq g a0 a3) \to ((leq g a6 a5) \to P))))
+(\lambda (H9: (eq A (AHead a3 a5) (AHead a0 a1))).(let H10 \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 a0 a1)
+H9) in ((let H11 \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 a0 a1) H9) in (eq_ind A a0 (\lambda
+(a6: A).((eq A a5 a1) \to ((leq g a0 a6) \to ((leq g (asucc g a1) a5) \to
+P)))) (\lambda (H12: (eq A a5 a1)).(eq_ind A a1 (\lambda (a6: A).((leq g a0
+a0) \to ((leq g (asucc g a1) a6) \to P))) (\lambda (_: (leq g a0
+a0)).(\lambda (H14: (leq g (asucc g a1) a1)).(H0 H14 P))) a5 (sym_eq A a5 a1
+H12))) a3 (sym_eq A a3 a0 H11))) H10))) a4 (sym_eq A a4 (asucc g a1) H8))) a2
+(sym_eq A a2 a0 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0
+(asucc g a1))) (refl_equal A (AHead a0 a1)))))))))) a)).
include "leq/defs.ma".
+include "leq/props.ma".
+
include "leq/asucc.ma".
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* Problematic objects for disambiguation/typechecking ********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
-
-include "LambdaDelta/theory.ma".
-
-theorem iso_trans:
- \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
-t3) \to (iso t1 t3)))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
-(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
-(TSort n2) t3)).(let H1 \def (match H0 in iso return (\lambda (t: T).(\lambda
-(t0: T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to
-(iso (TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq
-T (TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_:
-nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T
-(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t))
-(iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2)
-\Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T
-(TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in
-(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) |
-(iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1)
-(TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T
-(THead k v1 t1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2)
-t3) \to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2))
-(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3:
-T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 in iso return
-(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef
-i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2)
-\Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T
-(TSort n2) t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef
-_) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in
-(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) |
-(iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef
-i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat
-(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
-(TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef
-i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind
-T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0
-(sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda
-(H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2)
-t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in
-(False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))])
-in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k:
-K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def
-(match H0 in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t
-t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3)
-t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
-(THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T
-(TSort n1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2)
-t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow
-(\lambda (H0: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef
-i2) t5)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0)
-in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1)))
-| (iso_head k0 v0 v3 t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0)
-(THead k v2 t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
-\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H4 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _)
-\Rightarrow k])) (THead k0 v0 t0) (THead k v2 t4) H0) in (eq_ind K k (\lambda
-(k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t4) t5) \to
-(iso (THead k v1 t3) t5))))) (\lambda (H5: (eq T v0 v2)).(eq_ind T v2
-(\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t4) t5) \to (iso (THead
-k v1 t3) t5)))) (\lambda (H6: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq
-T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H7: (eq T
-(THead k v3 t4) t5)).(eq_ind T (THead k v3 t4) (\lambda (t: T).(iso (THead k
-v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7)) t0 (sym_eq T t0 t4 H6))) v0
-(sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H1
-(refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* Problematic objects for disambiguation/typechecking ********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
-
-include "LambdaDelta/theory.ma".
-
-theorem drop1_getl_trans:
- \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
-\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
-i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2
-(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
-C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
-C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
-(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i
-v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2
-c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H
-in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
-(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to
-(ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with
-[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2:
-(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C
-c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))
-(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2:
-C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i
-c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2
-H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3:
-(eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5:
-(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
-\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
-(False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1
-hds c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))
-H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2)
-(refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda
-(hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2
-c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i:
-nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl
-(trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda
-(c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2
-c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
-nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0
-in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
-(_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq
-C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
-[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
-h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
-\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
-\Rightarrow (ctrans hds0 i v)])))))))))))) with [(drop1_nil c) \Rightarrow
-(\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c
-c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
-\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2)
-in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl
-(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
-\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
-(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
-(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3
-H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq
-PList (PCons h0 d0 hds0) (PCons h d hds0))).(\lambda (H5: (eq C c0
-c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda
-(e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
-\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h
-d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
-in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _
-n _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H9
-\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
-(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
-(PCons h0 d0 hds0) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n:
-nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4
-c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2:
-C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i)
-| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match
-(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0
-i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))))
-(\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds0
-hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1
-hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d)
-with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0
-i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
-\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
-\Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds0
-hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1)
-\to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl
-(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
-\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
-(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
-(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda
-(H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d
-c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt
-(trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow
-(plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d)
-with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
-v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4
-c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c)
-\to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true
-\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
-(CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift
-h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans
-hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1
-hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex
-C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) |
-false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
-with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
-v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x:
-bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to
-(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i)
-| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
-with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
-v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H0: (eq bool (blt
-(trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16
-\def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2
-(Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2
-(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
-v)))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind
-b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d
-(le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d)
-(S (trans hds0 i)) H0))) c2 c3 h H14 b x (ctrans hds0 i v) H17) in (let H
-\def H_x0 in (ex2_ind C (\lambda (e1: C).(getl (trans hds0 i) c2 (CHead e1
-(Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda
-(e1: C).(drop h (minus d (S (trans hds0 i))) e1 x)) (ex C (\lambda (e2:
-C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0
-i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H1: (getl (trans hds0
-i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
-v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C
-(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d
-(S (trans hds0 i))) (ctrans hds0 i v))))) x0 H1)))) H))))) H16)))) (\lambda
-(H0: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3 H15 b e1
-v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0
-i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl
-(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))) (\lambda
-(x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0
-i v)))).(let H \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x
-(Bind b) (ctrans hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus
-(trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d
-(trans hds0 i) H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq
-C c0 c2 H12))) hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10)))
-h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
-PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C c1)))))))))))))))
-hds).
-
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* Problematic objects for disambiguation/typechecking ********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
-
-include "LambdaDelta/theory.ma".
-
-theorem asucc_inj:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
-g a2)) \to (leq g a1 a2))))
-\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
-(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
-(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
-(asucc g (ASort n1 n2)))).((match n in nat return (\lambda (n3: nat).((leq g
-(asucc g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0)
-(ASort n1 n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O
-n0)) (asucc g (ASort n1 n2)))).((match n1 in nat return (\lambda (n3:
-nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g
-(ASort O n0) (ASort n3 n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc
-g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq return
-(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O
-(next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort O n0)
-(ASort O n2))))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda
-(H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort
-h2 n3) (ASort O (next g n2)))).((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 n1])) (ASort h1 n1) (ASort O (next g n0)) H1) 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 h1]))
-(ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n:
-nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n3) (ASort O (next g n2)))
-\to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g
-(ASort O n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind
-nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2)))
-\to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k)) \to (leq g
-(ASort O n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O
-(next g n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \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 n3)
-(ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3
-(next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n
-n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H9: (eq nat n3
-(next g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g
-(ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to (leq g (ASort O n0)
-(ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus
-g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next
-g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) H10
-(aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H11 \def
-(eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g
-(ASort O n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl
-g n2 k)) in (leq_sort g O O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2)
-H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1
-(sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1)
-\Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g
-n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g n2)))).((let H4
-\def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A return (\lambda
-(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort
-O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0)
-(ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g
-n0))) (refl_equal A (ASort O (next g n2)))))) | (S n3) \Rightarrow (\lambda
-(H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def
-(match H1 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ?
-a a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq
-g (ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n1 n3 k H0)
-\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g
-n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort n3 n2))).((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 n1])) (ASort h1 n1)
-(ASort O (next g n0)) H1) 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 h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in
-(eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2
-n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3)
-k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H5: (eq nat n1
-(next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3)
-(ASort n3 n2)) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k))
-\to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H6: (eq A (ASort h2
-n3) (ASort n3 n2))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n3])) (ASort h2 n3) (ASort n3 n2) H6) in ((let H8 \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 n3) (ASort n3 n2)
-H6) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n2) \to ((eq A (aplus g
-(ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0)
-(ASort (S n3) n2))))) (\lambda (H9: (eq nat n3 n2)).(eq_ind nat n2 (\lambda
-(n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n) k))
-\to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H10: (eq A (aplus g
-(ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H \def (eq_ind_r A
-(aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort n3
-n2) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in
-(let H11 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a: A).(eq A
-(aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort (S n3) n2) (S k))
-(aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H11))))
-n3 (sym_eq nat n3 n2 H9))) h2 (sym_eq nat h2 n3 H8))) H7))) n1 (sym_eq nat n1
-(next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2
-H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g
-n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n3 n2))).((let H4 \def
-(eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort
-n3 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort
-(S n3) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0)))
-(refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0:
-(leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 in
-nat return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g
-(ASort n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O
-\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
-O n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda (a0:
-A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O
-(next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2))))))) with [(leq_sort
-h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n3
-n0))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((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 n1])) (ASort h1 n1)
-(ASort n3 n0) H1) 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 h1])) (ASort h1 n1) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
-(n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to
-((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort
-(S n3) n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0
-(\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A
-(aplus g (ASort n3 n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3)
-n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g
-n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \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 n3)
-(ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3
-(next g n2)) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n n3) k))
-\to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next
-g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g (ASort n3
-n0) k) (aplus g (ASort O n) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))
-(\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2))
-k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A
-a (aplus g (ASort O (next g n2)) k))) H10 (aplus g (ASort (S n3) n0) (S k))
-(aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
-k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in
-(leq_sort g (S n3) O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9)))
-h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1
-n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda
-(H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4)
-(ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e:
-A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
-False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind
-((eq A (AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3
-a4) \to (leq g (ASort (S n3) n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2
-(refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4)
-\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
-(S n4) n2)))).(let H2 \def (match H1 in leq return (\lambda (a: A).(\lambda
-(a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0
-(ASort n4 n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))))) with
-[(leq_sort h1 h2 n3 n4 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n3)
-(ASort n3 n0))).(\lambda (H2: (eq A (ASort h2 n4) (ASort n4 n2))).((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 n3])) (ASort h1 n3)
-(ASort n3 n0) H1) 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 h1])) (ASort h1 n3) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
-(n: nat).((eq nat n3 n0) \to ((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A
-(aplus g (ASort n n3) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3)
-n0) (ASort (S n4) n2)))))) (\lambda (H5: (eq nat n3 n0)).(eq_ind nat n0
-(\lambda (n: nat).((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g
-(ASort n3 n) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0)
-(ASort (S n4) n2))))) (\lambda (H6: (eq A (ASort h2 n4) (ASort n4 n2))).(let
-H7 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_:
-A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n4]))
-(ASort h2 n4) (ASort n4 n2) H6) in ((let H8 \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 n4) (ASort n4 n2) H6) in (eq_ind
-nat n4 (\lambda (n: nat).((eq nat n4 n2) \to ((eq A (aplus g (ASort n3 n0) k)
-(aplus g (ASort n n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))))
-(\lambda (H9: (eq nat n4 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus
-g (ASort n3 n0) k) (aplus g (ASort n4 n) k)) \to (leq g (ASort (S n3) n0)
-(ASort (S n4) n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g
-(ASort n4 n2) k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda
-(a: A).(eq A a (aplus g (ASort n4 n2) k))) H10 (aplus g (ASort (S n3) n0) (S
-k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a))
-H (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 k)) in
-(leq_sort g (S n3) (S n4) n0 n2 (S k) H11)))) n4 (sym_eq nat n4 n2 H9))) h2
-(sym_eq nat h2 n4 H8))) H7))) n3 (sym_eq nat n3 n0 H5))) h1 (sym_eq nat h1 n3
-H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2:
-(eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort
-n4 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
-_) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4)
-(ASort n4 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3)
-n0) (ASort (S n4) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3
-n0)) (refl_equal A (ASort n4 n2)))))]) H0))]) H)))) (\lambda (a: A).(\lambda
-(H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0)
-a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g (ASort n n0)) (asucc g
-a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq g (asucc g (ASort n
-n0)) (asucc g (AHead a a0)))).((match n in nat return (\lambda (n1:
-nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0)
-a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1
-n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to
-(leq g (ASort n1 n0) (AHead a a0)))))) with [O \Rightarrow (\lambda (_:
-(((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0)
-a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g
-(ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
-(AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a1:
-A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort O (next g
-n0))) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a
-a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
-(ASort h1 n1) (ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n2)
-(AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match
-e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _
-_) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H3) in ((let H6 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
-(ASort O (next g n0)) H3) in (eq_ind nat O (\lambda (n: nat).((eq nat n1
-(next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
-(aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
-(AHead a a0)))))) (\lambda (H7: (eq nat n1 (next g n0))).(eq_ind nat (next g
-n0) (\lambda (n: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
-(aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
-(AHead a a0))))) (\lambda (H8: (eq A (ASort h2 n2) (AHead a (asucc g
-a0)))).(let H9 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
-\Rightarrow False])) I (AHead a (asucc g a0)) H8) in (False_ind ((eq A (aplus
-g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
-(AHead a a0))) H9))) n1 (sym_eq nat n1 (next g n0) H7))) h1 (sym_eq nat h1 O
-H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4:
-(eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a2 a4)
-(AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e:
-A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
-False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in
-(False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2)
-\to ((leq g a3 a4) \to (leq g (ASort O n0) (AHead a a0))))) H6)) H5 H2
-H3)))]) in (H5 (refl_equal A (ASort O (next g n0))) (refl_equal A (AHead a
-(asucc g a0)))))))) | (S n1) \Rightarrow (\lambda (_: (((leq g (asucc g
-(ASort (S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda
-(_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S
-n1) n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g
-(AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a1:
-A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort n1 n0)) \to
-((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a
-a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
-(ASort h1 n1) (ASort n1 n0))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a
-(asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
-\Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H3) in ((let H6 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n
-_) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0)
-H3) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2
-n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) (aplus g
-(ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda
-(H7: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n2)
-(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort
-h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H8: (eq A
-(ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2 n2)
-(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
-H8) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))
-\to (leq g (ASort (S n1) n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 n0 H7)))
-h1 (sym_eq nat h1 n1 H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3)
-\Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H5:
-(eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead
-a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
-n0) H4) in (False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq
-g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n1) n0) (AHead a a0)))))
-H6)) H5 H2 H3)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a
-(asucc g a0))))))))]) H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_:
-((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a
-a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0)
-(asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3:
-A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
-a0)) (asucc g (ASort n n0)))).((match n in nat return (\lambda (n1:
-nat).((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g
-(AHead a a0) (ASort n1 n0)))) with [O \Rightarrow (\lambda (H2: (leq g (asucc
-g (AHead a a0)) (asucc g (ASort O n0)))).(let H3 \def (match H2 in leq return
-(\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1
-(AHead a (asucc g a0))) \to ((eq A a2 (ASort O (next g n0))) \to (leq g
-(AHead a a0) (ASort O n0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
-(\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq
-A (ASort h2 n2) (ASort O (next g n0)))).((let H5 \def (eq_ind A (ASort h1 n1)
-(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
-H3) in (False_ind ((eq A (ASort h2 n2) (ASort O (next g n0))) \to ((eq A
-(aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0)
-(ASort O n0)))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow
-(\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc g a0)))).(\lambda (H5: (eq
-A (AHead a2 a4) (ASort O (next g n0)))).((let H6 \def (f_equal A A (\lambda
-(e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
-a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in
-((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_:
-A).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead
-a1 a3) (AHead a (asucc g a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3
-(asucc g a0)) \to ((eq A (AHead a2 a4) (ASort O (next g n0))) \to ((leq g a5
-a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda
-(H8: (eq A a3 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A
-(AHead a2 a4) (ASort O (next g n0))) \to ((leq g a a2) \to ((leq g a5 a4) \to
-(leq g (AHead a a0) (ASort O n0)))))) (\lambda (H9: (eq A (AHead a2 a4)
-(ASort O (next g n0)))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e:
-A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
-False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H9) in
-(False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0)
-(ASort O n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a
-H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a (asucc g a0)))
-(refl_equal A (ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H2:
-(leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H3 \def
-(match H2 in leq return (\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ?
-a1 a2)).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (ASort n1 n0)) \to
-(leq g (AHead a a0) (ASort (S n1) n0))))))) with [(leq_sort h1 h2 n1 n2 k H2)
-\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g
-a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort n1 n0))).((let H5 \def
-(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort h2 n2)
-(ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H5)) H4 H2))) | (leq_head a1
-a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc
-g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort n1 n0))).((let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3)
-(AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
-(AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in
-(eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4)
-(ASort n1 n0)) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0)
-(ASort (S n1) n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A
-(asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g
-a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort (S n1) n0))))))
-(\lambda (H9: (eq A (AHead a2 a4) (ASort n1 n0))).(let H10 \def (eq_ind A
-(AHead a2 a4) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
-n0) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g
-(AHead a a0) (ASort (S n1) n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8)))
-a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a
-(asucc g a0))) (refl_equal A (ASort n1 n0)))))]) H1)))) (\lambda (a3:
-A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g
-(AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc g (AHead a
-a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: (leq g (asucc
-g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H4 \def (match H3 in leq
-return (\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A
-a1 (AHead a (asucc g a0))) \to ((eq A a2 (AHead a3 (asucc g a4))) \to (leq g
-(AHead a a0) (AHead a3 a4))))))) with [(leq_sort h1 h2 n1 n2 k H4)
-\Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a (asucc g
-a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g a4)))).((let H7
-\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
-(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort h2 n2)
-(AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
-h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) | (leq_head
-a3 a4 H4 a5 a6 H5) \Rightarrow (\lambda (H6: (eq A (AHead a3 a5) (AHead a
-(asucc g a0)))).(\lambda (H7: (eq A (AHead a4 a6) (AHead a3 (asucc g
-a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow
-a])) (AHead a3 a5) (AHead a (asucc g a0)) H6) in ((let H9 \def (f_equal A A
-(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g
-a0)) H6) in (eq_ind A a (\lambda (a1: A).((eq A a5 (asucc g a0)) \to ((eq A
-(AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a1 a4) \to ((leq g a5 a6)
-\to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq A a5 (asucc g
-a0))).(eq_ind A (asucc g a0) (\lambda (a1: A).((eq A (AHead a4 a6) (AHead a3
-(asucc g a4))) \to ((leq g a a4) \to ((leq g a1 a6) \to (leq g (AHead a a0)
-(AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a4 a6) (AHead a3 (asucc g
-a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow
-a])) (AHead a4 a6) (AHead a3 (asucc g a4)) H11) in ((let H13 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a4 | (AHead a _) \Rightarrow a])) (AHead a4 a6) (AHead a3 (asucc
-g a4)) H11) in (eq_ind A a3 (\lambda (a1: A).((eq A a6 (asucc g a4)) \to
-((leq g a a1) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (AHead a3
-a4)))))) (\lambda (H14: (eq A a6 (asucc g a4))).(eq_ind A (asucc g a4)
-(\lambda (a1: A).((leq g a a3) \to ((leq g (asucc g a0) a1) \to (leq g (AHead
-a a0) (AHead a3 a4))))) (\lambda (H15: (leq g a a3)).(\lambda (H16: (leq g
-(asucc g a0) (asucc g a4))).(leq_head g a a3 H15 a0 a4 (H0 a4 H16)))) a6
-(sym_eq A a6 (asucc g a4) H14))) a4 (sym_eq A a4 a3 H13))) H12))) a5 (sym_eq
-A a5 (asucc g a0) H10))) a3 (sym_eq A a3 a H9))) H8)) H7 H4 H5)))]) in (H4
-(refl_equal A (AHead a (asucc g a0))) (refl_equal A (AHead a3 (asucc g
-a4)))))))))) a2)))))) a1)).
-
(* Problem 1: disambiguation errors with these objects *)
-(* iso_trans (in problems-1)
- * drop1_getl_trans (in problems-2)
- * asucc_inj (in problems-3)
+(* leq_trans (in problems-4)
*)
(* Problem 2: assertion failure raised by type checker on this object *)
projects / helm.git / commitdiff