X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fleq%2Fprops.ma;h=ac9ef3ee8aa8d75ea9faf397ace285e323f2e12d;hb=fdda444a05fe4c68c925cd94e4e3a38c93d0c35f;hp=2fda46a6e65e001a88d2be06c4e234bd0d4da5c0;hpb=9376f52b7f5890d924ae7d93bcae2af9e516126d;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
index 2fda46a6e..ac9ef3ee8 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
@@ -165,7 +165,7 @@ 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)))).
 
-theorem leq_ahead_false:
+theorem leq_ahead_false_1:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) 
 \to (\forall (P: Prop).P))))
 \def
@@ -268,3 +268,106 @@ 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)).
 
+theorem leq_ahead_false_2:
+ \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) 
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: 
+A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort 
+n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g 
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead 
+a1 (ASort O n0)) (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 a1 (ASort 
+O n0))) \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 a1 (ASort O 
+n0)))).(\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 a1 (ASort O n0)) 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 a1 (ASort O n0)))).(\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 a1 (ASort O n0)) 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 a1 (ASort O n0)) H3) in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort 
+O n0)) \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 (ASort O n0))).(eq_ind A (ASort O 
+n0) (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a1 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 a1 a3) \to ((leq g (ASort O 
+n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort O n0) H7))) a0 (sym_eq A a0 a1 
+H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead a1 (ASort O n0))) 
+(refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g 
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead 
+a1 (ASort (S n1) n0)) (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 a1 (ASort (S n1) n0))) \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 a1 (ASort (S n1) n0)))).(\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 a1 (ASort (S n1) n0)) 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 a1 (ASort (S n1) 
+n0)))).(\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 a1 (ASort (S n1) n0)) 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 a1 (ASort (S n1) n0)) H3) 
+in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort (S n1) n0)) \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 (ASort (S n1) n0))).(eq_ind A (ASort (S n1) n0) 
+(\lambda (a: A).((eq A (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a1 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 a1 a3) 
+\to ((leq g (ASort (S n1) n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort (S n1) 
+n0) H7))) a0 (sym_eq A a0 a1 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A 
+(AHead a1 (ASort (S n1) n0))) (refl_equal A (ASort (S n1) n0))))))) n H)))))) 
+(\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) \to 
+(\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: 
+A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: 
+A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (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 a1 (AHead a a0))) \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 a1 (AHead a a0)))).(\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 a1 (AHead a a0)) 
+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 a1 
+(AHead a a0)))).(\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 a1 (AHead a a0)) 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 a1 (AHead a a0)) H4) in 
+(eq_ind A a1 (\lambda (a7: A).((eq A a5 (AHead a a0)) \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 (AHead a a0))).(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A 
+(AHead a4 a6) (AHead a a0)) \to ((leq g a1 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 a1 a7) \to ((leq g (AHead a a0) a6) \to P)))) 
+(\lambda (H12: (eq A a6 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g a1 a) \to 
+((leq g (AHead a a0) a7) \to P))) (\lambda (_: (leq g a1 a)).(\lambda (H14: 
+(leq g (AHead a a0) a0)).(H0 a H14 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq 
+A a4 a H11))) H10))) a5 (sym_eq A a5 (AHead a a0) H8))) a3 (sym_eq A a3 a1 
+H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a1 (AHead a a0))) 
+(refl_equal A (AHead a a0))))))))))) a2)).
+