one problem still remains

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props.ma
new file mode 100644 (file)
index 0000000..65f43ac
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props.ma
@@ -0,0 +1,256 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props".
+
+include "leq/defs.ma".
+
+theorem leq_gen_sort:
+ \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
+g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda 
+(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
+h2 n2) k))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
+A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H in leq return 
+(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort 
+h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda 
+(h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) 
+(aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0) 
+\Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2: 
+(eq A (ASort h2 n2) a2)).((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 n0])) (ASort h0 n0) (ASort h1 n1) 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 h0])) (ASort h0 n0) 
+(ASort h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to 
+((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort 
+h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda 
+(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort 
+h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n: 
+nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g 
+(ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda 
+(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort 
+h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2 
+n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: 
+nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
+(k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))) 
+(\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
+(_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda 
+(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort 
+h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq 
+nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0 
+a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1 
+n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((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 h1 
+n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g 
+a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal 
+A a2))))))).
+
+theorem leq_gen_head:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g 
+(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a 
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda 
+(_: A).(\lambda (a4: A).(leq g a2 a4))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
+(H: (leq g (AHead a1 a2) a)).(let H0 \def (match H in leq return (\lambda 
+(a0: A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) 
+\to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a 
+(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda 
+(_: A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k 
+H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda 
+(H2: (eq A (ASort h2 n2) a)).((let H3 \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 a1 a2) H1) in 
+(False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k) 
+(aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: 
+A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) 
+(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head 
+a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1 
+a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \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 a0 a4) (AHead a1 a2) 
+H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A return 
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) 
+\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda 
+(a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to 
+((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a 
+(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda 
+(_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4 
+a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3) 
+\to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a 
+(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda 
+(_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3 
+a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a6: A).((leq g a1 a3) \to ((leq g 
+a2 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a6 (AHead a7 
+a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: 
+A).(\lambda (a8: A).(leq g a2 a8))))))) (\lambda (H8: (leq g a1 a3)).(\lambda 
+(H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A 
+(AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1 
+a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A 
+(AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1 
+H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A 
+a))))))).
+
+theorem leq_refl:
+ \forall (g: G).(\forall (a: A).(leq g a a))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) 
+(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A 
+(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 
+a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 
+H0))))) a)).
+
+theorem leq_eq:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 
+a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 
+a2)).(eq_ind_r A a2 (\lambda (a: A).(leq g a a2)) (leq_refl g a2) a1 H)))).
+
+theorem leq_sym:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
+a2 a1))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: 
+nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: 
+nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g 
+(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: 
+(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: 
+A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 
+H1 a6 a5 H3))))))))) a1 a2 H)))).
+
+axiom leq_trans:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
+(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
+.
+
+theorem leq_ahead_false:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) 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) 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 
+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: 
+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 
+(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 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 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 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 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 
+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 a0)) \to 
+((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9: (eq A 
+(AHead a4 a6) (AHead a 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 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 a0) H9) in (eq_ind A a (\lambda (a7: A).((eq A a6 a0) \to ((leq 
+g (AHead a a0) a7) \to ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 
+a0)).(eq_ind A 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 
+a0)).(H a0 H13 P))) a6 (sym_eq A a6 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 
+a0))))))))))) a1)).
+

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/problems-4.ma b/matita/contribs/LAMBDA-TYPES/Level-1/problems-4.ma
new file mode 100644 (file)
index 0000000..03dfdaf
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/problems-4.ma
@@ -0,0 +1,96 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 leq_trans:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
+(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 
+a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort 
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g 
+(ASort h2 n2) a3)).(let H2 \def (match H1 in leq return (\lambda (a: 
+A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to 
+((eq A a0 a3) \to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3 
+k0 H2) \Rightarrow (\lambda (H3: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda 
+(H4: (eq A (ASort h3 n3) a3)).((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 n0])) (ASort h0 n0) (ASort h2 n2) 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 h0])) (ASort h0 n0) 
+(ASort h2 n2) H3) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to 
+((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort 
+h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H7: (eq nat n0 
+n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A 
+(aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 
+n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) 
+(\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0)) 
+\to (leq g (ASort h1 n1) a))) (\lambda (H9: (eq A (aplus g (ASort h2 n2) k0) 
+(aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3 
+n3)) (\lambda (H10: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) 
+(ASort h2 n2) k k H0 (minus k0 k)) in (let H11 \def (eq_ind_r nat (plus 
+(minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g 
+(ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0 
+H10)))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0) 
+(aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11 H9))))) (\lambda 
+(H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0 
+k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus (minus k k0) k0) 
+(\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g (ASort h3 n3) n))) 
+H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A 
+(aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k) 
+H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0 (sym_eq nat h0 h2 H6))) 
+H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A 
+(AHead a0 a5) (ASort h2 n2))).(\lambda (H5: (eq A (AHead a4 a6) a3)).((let H6 
+\def (eq_ind A (AHead a0 a5) (\lambda (e: A).(match e in A return (\lambda 
+(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
+True])) I (ASort h2 n2) H4) in (False_ind ((eq A (AHead a4 a6) a3) \to ((leq 
+g a0 a4) \to ((leq g a5 a6) \to (leq g (ASort h1 n1) a3)))) H6)) H5 H2 
+H3)))]) in (H2 (refl_equal A (ASort h2 n2)) (refl_equal A a3))))))))))) 
+(\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: 
+((\forall (a5: A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: 
+A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: 
+A).((leq g a6 a7) \to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g 
+(AHead a4 a6) a0)).(let H5 \def (match H4 in leq return (\lambda (a: 
+A).(\lambda (a7: A).(\lambda (_: (leq ? a a7)).((eq A a (AHead a4 a6)) \to 
+((eq A a7 a0) \to (leq g (AHead a3 a5) a0)))))) with [(leq_sort h1 h2 n1 n2 k 
+H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n1) (AHead a4 a6))).(\lambda 
+(H7: (eq A (ASort h2 n2) a0)).((let H8 \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 a4 a6) H6) in 
+(False_ind ((eq A (ASort h2 n2) a0) \to ((eq A (aplus g (ASort h1 n1) k) 
+(aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 a5) a0))) H8)) H7 H5))) | 
+(leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda (H7: (eq A (AHead a7 a9) 
+(AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10) a0)).((let H9 \def 
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
+[(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a])) (AHead a7 a9) 
+(AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e: A).(match e in 
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _) 
+\Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4 (\lambda (a: 
+A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8) \to ((leq g 
+a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9 
+a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a8 a10) a0) \to ((leq g a4 
+a8) \to ((leq g a a10) \to (leq g (AHead a3 a5) a0))))) (\lambda (H12: (eq A 
+(AHead a8 a10) a0)).(eq_ind A (AHead a8 a10) (\lambda (a: A).((leq g a4 a8) 
+\to ((leq g a6 a10) \to (leq g (AHead a3 a5) a)))) (\lambda (H13: (leq g a4 
+a8)).(\lambda (H14: (leq g a6 a10)).(leq_head g a3 a8 (H1 a8 H13) a5 a10 (H3 
+a10 H14)))) a0 H12)) a9 (sym_eq A a9 a6 H11))) a7 (sym_eq A a7 a4 H10))) H9)) 
+H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A 
+a0))))))))))))) a1 a2 H)))).