X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2Fproblems.ma;h=995eeae40ffc48f73ecda4a94707af1d75150d02;hb=b1221a75b73711a6de7880e4b14b1fee52fdd576;hp=de2399b05a957304e078f9d2774ae5af2e9205c0;hpb=f764844fa35ab0bb9c10707151340b924060f069;p=helm.git

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/problems.ma b/matita/contribs/LAMBDA-TYPES/Level-1/problems.ma
index de2399b05..995eeae40 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/problems.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/problems.ma
@@ -25,6 +25,94 @@ include "LambdaDelta/theory.ma".
 
 iso_trans (in iso/props)
 
+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))).
+
+
 drop1_getl_trans
 
 (* Problem 1: does not typecheck a match on an empty type *)