(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props".
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/props".
include "leq/defs.ma".
\lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
-n0) a2) (ASort n n0))).(\lambda (P: Prop).((match n in nat return (\lambda
-(n1: nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) with [O
-\Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O 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 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 n0))).((let H4 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
+n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
+(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
+(ASort O n0) a2) (ASort O 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 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 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 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 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 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 n0)) \to ((leq g (ASort O n0) a3) \to ((leq
+g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O 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 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 n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
+(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
+(ASort (S n1) n0) a2) (ASort (S 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 (S 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
+(S 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 O n0) a2) H2) in (False_ind ((eq A
-(ASort h2 n2) (ASort O 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 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
-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 n0)) \to
-((leq g (ASort O n0) a3) \to ((leq g a a5) \to P)))) (\lambda (H8: (eq A
-(AHead a3 a5) (ASort O 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 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 n0)))))
-| (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort
-(S 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 (S 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 (S 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 (S 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 (S 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 (S 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
-(S 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 (S 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 (S
-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 (S n1) n0)))))]) H)))))) (\lambda (a: A).(\lambda (H:
-((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
+_ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind
+((eq A (ASort h2 n3) (ASort (S 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 (S 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 (S 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 (S 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 (S 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 (S 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 (S n1) n0))))))) n H)))))) (\lambda (a:
+A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H2 \def